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I. Normed algebras

§ 1. Spectra and characters

Lemma 1 Let A be an algebra. For every idempotent j of A, the subspace jAj of A is the set of x∈ A such that xj=jx=x. It is a subalgebra of A having j as unit element. 024R
Definition 1 Let A be a unital algebra over K, and let e be its unit element. For every x∈ A, the spectrum of x relative to A is the set of λ ∈ K such that λ e-x is not invertible. 024S
Remark 1 If A =0, then Sp(0) =∅. 024T
Remark 2 If Anot=0, then Sp(λ e) =λ for every λ ∈ K. 024U
Remark 3 For x∈ A to be invertible, it is necessary and sufficient that 0∉ Sp(x). 024V
Remark 4 Let R∈ K(X) be a rational fraction and let x∈ A be an element which can be substituted in R, that is to say (A, IV, p. 20), that there exist P and Q∈ K[X] such that R = P/Q and Q(x… 024W
Remark 5 Suppose that the algebra A is nonzero. Let x∈ A be a nilpotent element. Let n be an integer such that xⁿ= 0. The spectrum of xⁿ reduces to 0, hence the same is true of the spectrum… 024X
Remark 6 Let A and B be unital algebras over K and φ : A→ B a unital morphism. For every x∈ A, one has Sp_B(φ (x))⊂ Sp_A(x). 024Y
Remark 7 Let A be a unital algebra, R its radical (A, VIII, p. 150, Def. 2), and let φ be the canonical morphism of A onto B = A/R. If x∈ A, one has Sp_B(φ (x)) = Sp_A(x). In fact, it is en… 024Z
Remark 8 Let (Bᵢ)_i∈ I be a family of unital algebras, with Bᵢ= (Aᵢ, eᵢ) for i∈ I. Put A =∏ᵢAᵢ,e= (eᵢ)_i∈ I. Then (A, e) is a unital algebra called the product of the Bᵢ. If x= (xᵢ)_i∈ I∈ A… 0250
Example 1 Let X be a set and let A = K^X be the algebra of functions with values in K defined on X. The spectrum of an element f of A is the set of values of f. 0251
Example 2 Let A be a unital algebra of finite rank over K. For x∈ A to be invertible, it is necessary and sufficient that the linear mapping y↦ xy of A into A should have nonzero determinant… 0252
Definition 2 Let A be a unital algebra over K and x∈ A. For every λ ∈ K Sp(x), one sets 0253
Remark 1 Let (A,1) be a unital algebra. If x∈ A, one has 0254
Remark 2 It follows from Remark 1 that, if B is an algebra over K and if x∈ B, one has: 0255
Remark 3 If x belongs to the radical of A (A, VIII, p. 430, def. 3), one has Sp'_A(x) =0. This follows from Remark 7 of I, p. 3. 0256
Proposition 1 Let A be an algebra and x, y∈ A. One has Sp'(xy) = Sp'(yx). 0257
Definition 3 A full subalgebra of A is called a unital subalgebra B such that every element of B which is invertible in A is invertible in B. 0258
Lemma 2 Let (x_λ)_λ∈Λ be a family of pairwise commuting elements of A. The full subalgebra generated by (x_λ) is the set of elements of the form R((x_λ)), where R∈ K((X_λ)) runs through th… 0259
Definition 4 Let A be a commutative unital algebra over K. A unital character is a unital morphism of A into K. 025A
Definition 5 Let A be a commutative unital algebra over K. For every x∈ A, let G_A(x), or simply G(x), denote the mapping χ ↦ χ (x) of X(A) into K. It is called the Gelfand transform of x. 025B
Definition 6 Let A be a commutative algebra over K. A character of A is, by definition, an algebra morphism of A into K. 025C
Lemma 3 The mapping χ ↦ Ker(χ ) is a bijection of X(A) onto the set of regular ideals of codimension 1 of A. 025D
Example The zero mapping of A into L(E) is a representation, called trivial, of A. It is irreducible if and only if E is of dimension 1. 025E
Lemma 4 Let π be a nontrivial irreducible representation of A in E. For every nonzero element ξ of E, one has π (A)ξ = E. 025F
Definition 7 Let A be an algebra over K. A primitive ideal of A is defined to be the kernel of a nontrivial irreducible representation of A. 025G
Lemma 5 Let π be an irreducible representation of A in a vector space E over K. 025H
Lemma 6 Let I₁ and I₂ be two-sided ideals of A, I a primitive ideal of A. If I contains I₁I₂(in particular, if I contains I₁∩ I₂), then I contains I₁ or I₂. 025I
Lemma 7 Suppose that A admits a unit element. Let I be a maximal two-sided ideal of A. Then I is a primitive ideal. 025J
Proposition 2 Let I₁ and I₂ be distinct points of J(A). Then one of these two points is not adherent to the other. 025K
Proposition 3 Let I∈ J(A). In order that I be closed in J(A), it is necessary and sufficient that I be a maximal primitive ideal. 025L
Proposition 4 If A possesses a unit element, the spaces J(A) and widehatA are quasi-compact. 025M
Proposition 5 The topology induced on X(A) by that of widehatA is coarser than the topology of X(A). 025N

§ 2. Normed algebras

Definition 1 Let A be a normed algebra. For a∈ A, denote by γ ₐ and δ ₐ the mappings x↦ ax, and x↦ xa from A into A. The mapping γ :a↦ γ ₐ is a representation of A in A, called the left regular… 025W
Lemma 1 Let A be a normed algebra. The left regular representation of A and the right regular representation of A are continuous of norm leqslant 1. 025X
Lemma 2 (Fekete Lemma) Let (aₙ)_ngeqslant 1 be a sequence of real numbers. Suppose that 025Y
Proposition 1 Let A be a normed algebra. For every x∈ A, the sequence (|xⁿ|^1/n)_ngeqslant 1 is convergent and its limit varrho (x) is equal to inf_ngeqslant 1|xⁿ|^1/n. Moreover, for every norm… 025Z
Definition 2 For every element x of a normed algebra A, the real number 0260
Definition 3 An element x of A is quasi-nilpotent if varrho (x) = 0. 0261
Remark 1 Let A be a normed algebra. If an element x∈ A satisfies varrho (x) =|x|, one has |xⁿ|=|x|ⁿ for every n∈ N, by (3) and (4). 0262
Remark 2 The function x↦ varrho (x) on A, being the greatest lower bound of the continuous functions x↦ |xⁿ|^1/n for ngeqslant 1, is upper semi-continuous (TG, IV, p. 31, cor.), but in gene… 0263
Proposition 2 Let A be a Banach algebra and x an element of A. The series ∑ₙ₌₁^∞λⁿxⁿ, considered as a power series in λ, has radius of convergence varrho (x)⁻¹. If A is unital and if varrho (x)<… 0264
Corollary 1 If A is a unital Banach algebra, then the group of invertible elements of A contains the open ball with center 1 and radius 1. 0265
Corollary 2 Let A be a Banach algebra and I a regular maximal left ideal (resp. right ideal) of A. Then I is closed. 0266
Corollaire 3 The radical of a Banach algebra is closed. 0267
Proposition 3 Let A be a unital Banach algebra. 0268
Définition 4 Let A be a normed algebra, let x be an element of A. Let us denote by γ ₓ and δ ₓ the mappings y↦ xy and y↦ yx of A into A. We say that x is a topological zero divisor on the left… 0269
Remarque According to TG, IX, p. 36, cor. 2,x is a topological zero divisor on the left (resp. right) if and only if there exists a sequence (zₙ) in A such that |zₙ|= 1 and such that xzₙ te… 026A
Proposition 4 Let A be a unital Banach algebra. Let x be an element of A which is not left invertible. If there exists a sequence (xₙ) of left invertible elements of A which converges to x, then… 026B
Proposition 5 Let A be a unital Banach algebra and let B be a full subalgebra of A. Then B is a full subalgebra of A. 026C
Theorem 1 Let A be a unital Banach algebra and x∈ A. 026D
Remark 1 The spectrum of an element in a unital Banach algebra may be any nonempty compact subset F of C(cf. Example 3 of I, p. 17; for A =C(F;C) and f∈ A the canonical injection of F into… 026E
Remark 2 Let A be a unital Banach algebra and let x∈ A. By Theorem 1 of I, p. 24,C- Sp_A(x) is an open subset of C, hence is locally connected. Therefore the connected components of C- Sp_A… 026F
Corollary 1 Let A be a nonzero unital normed algebra. For every x∈ A, the spectrum Sp_A(x) is nonempty. 026G
Corollary 2 (Gelfand-Mazur Theorem) Let A be a normed algebra over C. If A is a field, then A =C· 1. 026H
Corollary 3 Let A be a unital Banach algebra and x an invertible element of A such that |x|=|x⁻¹|= 1. Then Sp(x)⊂ U. 026I
Corollary 4 Let E be a complex Banach space, L(E) the Banach algebra of continuous endomorphisms of E and A a nonzero subalgebra of L(E) such that E is a simple A-pseudomodule (A, II, p. 176… 026J
Corollary 5 Let A be a Banach algebra and x∈ A. 026K
Lemma 3 Let X₁ and X₂ be compact subsets of C. If X₂ is contained in X₁ and if the boundary of X₁ in C is contained in X₂, then X₁ is the union of X₂ and of certain bounded connected compo… 026L
Proposition 6 Let A be a unital Banach algebra and B a closed unital subalgebra of A. For every x∈ B, one has Sp_B(x)⊃ Sp_A(x), and the boundary of Sp_A(x) in C contains the boundary of Sp_B(x)… 026M
Corollary The set Sp_B(x) is the union of Sp_A(x) and of certain bounded connected components of C- Sp_A(x). 026N

§ 3. Commutative Banach algebras

Theorem 1 Let A be a Banach algebra and let χ : A→ C be an algebra morphism (cf. I, p. 9). Then χ is continuous, of norm at most 1. If A is unital and if χ is a unital morphism, then χ is of… 027P
Remark Let A be a commutative Banach algebra. It follows from this theorem that the topology of simple convergence on X'(A) coincides with the restriction to X'(A) of the weak topology σ… 027Q
Corollary Let A be a commutative Banach algebra. The space X'(A) is compact. The space X(A) is locally compact, and is compact if A admits a unit element. 027R
Theorem 2 Let A be a commutative Banach algebra. The mapping χ ↦ Ker(χ ) is a bijection of X(A) onto the set J(A) of regular maximal ideals of A. 027S
Proposition 1 For every closed subset Φ of X, let I_Φ be the set of f∈ C₀(X) that are zero on Φ. Then Φ ↦ I_Φ is a bijection of the set of closed subsets of X onto the set of closed ideals of C₀… 027T
Corollary 1 For every x∈ X, let Iₓ be the set of f∈ C₀(X) vanishing at x. Then x↦ Iₓ is a bijection of X onto the set of maximal closed ideals of C₀(X). These ideals are regular. 027U
Corollary 2 The mapping x↦ evₓ is a homeomorphism of X' onto X'(C₀(X)), and its restriction to X is a homeomorphism of X onto X(C₀(X)). Moreover, the weak topology and the Jacobson topology co… 027V
Corollary 3 Suppose X compact. Then the mapping x↦ evₓ is a homeomorphism of X onto X(C(X)). The weak topology and the Jacobson topology coincide on X(C(X)). 027W
Definition 1 A proper partial mapping from X into Y is a correspondence f= (Γ ,X,Y) (E, II, p. 10, Def. 2) between X and Y such that 027X
Lemma 1 For every proper partial mapping f of X into Y, with domain of definition U, let us denote by widetildef the mapping of X' into Y' defined by widetildef(x) =f(x) if x∈ U and wideti… 027Y
Proposition 2 Let A and B be commutative complex Banach algebras. For every algebra morphism π : A→ B, the mapping X'(π ) is a proper partial mapping of X(B) into X(A). 027Z
Proposition 3 a) For every proper partial mapping φ of X into Y, the mapping f↦ f∘ φ of C(Y') into C(X') induces an algebra morphism φ^* of C₀(Y) into C₀(X) ; 0280
Proposition 4 Suppose that X and Y are compact. Let us identify the space X (resp. the space Y) with X(C(X)) (resp. X(C(Y))) (corollary 3 of I, p. 33). 0281
Remark In the language of category theory, the preceding results may be interpreted in the following way. Let G be the category whose objects are locally compact topological spaces and wh… 0282
Example 1 Let X be a locally compact topological space and consider the commutative Banach algebra C₀(X) (example 3 of I, p. 17 and No. 2 of I, p. 31). By Corollary 1 of I, p. 32, the charac… 0283
Example 2 Let ngeqslant 0 be an integer. Let Aₙ be the algebra of functions f: [0,1]→ K admitting continuous derivatives on [0,1] up to order n. Endowed with the norm 0284
Example 3 Let Δ be the disk of complex numbers z satisfying |z|leqslant 1 and let A be the complex Banach algebra of continuous functions on Δ analytic in the interior of Δ, endowed with the… 0285
Example 4 Consider the complex Banach algebra A of absolutely convergent Fourier series (example 8 of I, p. 19). For every element u of the unit circle U, the mapping f↦ f(u) is a character… 0286
Example 5 Let Δ be the unit disc of complex numbers z such that |z|leqslant 1. Its boundary in C is U. Let A be the Banach algebra of complex functions f on U such that there exists a contin… 0287
Proposition 5 Let A be a commutative Banach algebra. For every x∈ A, the function G(x) belongs to the commutative Banach algebra C₀(X(A)) of continuous functions on X(A) tending to 0 at infinity… 0288
Proposition 6 Let A be a commutative Banach algebra and let x∈ A. 0289
Example Consider the Banach algebra A of absolutely convergent Fourier series (Example 4). Prop. 6, b), implies that if φ is a function on the unit circle U admitting an absolutely converg… 028A
Proposition 7 Let A be a commutative Banach algebra. 028B
Corollary Let A be a Banach algebra, and let x and y be permutable elements of A. 028C
Proposition 8 Let A be a commutative Banach algebra. The following four sets are equal: 028D
Remark 1 In general, the image of the Gelfand transform is neither closed in C₀(X(A)), nor dense in C₀(X(A)) (Exercise 7 of I, p. 193). 028E
Remark 2 The image of the Gelfand transform separates the points of X(A), since if χ₁not=χ₂ are elements of X(A), there exists x∈ A such that χ₁(x)not=χ₂(x). 028F
Remark 3 If χ ∈ X(A), there exists an element of the image of the Gelfand transform which does not vanish at χ. 028G
Remark 4 If A possesses a unit element, the image of the Gelfand transform is a full subalgebra of the algebra of continuous functions on X(A) (Prop. 6, b)). 028H
Lemma 2 Let A be a commutative Banach algebra. Let M be a subset of X(A). Then M is closed for the Jacobson topology if and only if, for every character χ ∈ X(A) M, there exists an element… 028I
Proposition 9 Let A be a Banach algebra, let B be a commutative Banach algebra without radical, Every morphism of the underlying algebra of A into the underlying algebra of B is continuous. 028J
Corollary On a commutative complex algebra without radical, two norms defining Banach algebra structures are equivalent. 028K
Proposition 10 Let A and B be commutative unital Banach algebras, h: A→ B a unital algebra morphism, not necessarily continuous. If X(h) is surjective, then h(A) is a full subalgebra of B. 028L
Proposition 11 Let A and B be commutative unital Banach algebras, let a be an element of A and let h: A→ B be an injective unital morphism (not necessarily continuous). Suppose that the full clos… 028M
Definition 2 The image of the mapping X(h) is called the simultaneous spectrum of x, and denoted by Sp^Λ_A(x) or Sp^Λ(x). 028N
Example Let K ⊂ C^Λ be a compact subset. Let z= (z_λ)_λ∈Λ be the family in C(K) of the restrictions to K of the coordinate functions of C^Λ. Then the joint spectrum Sp^Λ_C(K)(z) is equal t… 028O
Proposition 12 Let Λ be a set and A a commutative unital Banach algebra. Let x= (x_λ)_λ∈Λ be a family of elements of A. 028P
Definition 3 Let Λ be a set and V a subset of C^Λ. One says that V is polynomially convex if V is the set of points (c_λ)_λ∈Λ of C^Λ such that 028Q
Lemma 3 Let Λ be a set. A subset V of C^Λ is polynomially convex if and only if there exist a family (Pᵢ)_i∈ I of elements of C[(X_λ)_λ∈Λ] and a family (Mᵢ)_i∈ I of elements of [0,+∞ ] suc… 028R
Lemma 4 Let A be a commutative unital Banach algebra. Let Λ be a set and x= (x_λ)_λ∈Λ a family of elements of A. If the unital subalgebra generated by the family x is dense in A, then the… 028S
Definition 4 Let Λ be a set and V a subset of C^Λ. The polynomially convex envelope of V is the smallest polynomially convex subset of C^Λ containing V. 028T
Example Let Λ be a finite set and V⊂ C^Λ a compact convex subset. Then V is polynomially convex. In fact, let W be the polynomially convex envelope of V. Let us prove that W⊂ V, which will… 028U
Lemma 5 Let K be a compact subset of C. We denote by widehatK the union of K and of the connected components of C- K which are relatively compact. Then the set widehatK is compact. 028V
Proposition 13 Let ngeqslant 1 be an integer. Let K⊂ Cⁿ be a closed subset and V its polynomially convex hull. 028W
Proposition 14 Let Λ be a set. Let A be a commutative unital Banach algebra, x= (x_λ)_λ∈Λ a family of elements of A, and A' the unital Banach subalgebra generated by x. Then the joint spectrum Sp… 028X
Proposition 15 Let Λ be a set. Let K be a polynomially convex compact subset of C^Λ. Let A₀ be the set of the restrictions to K of polynomial functions on C^Λ, and let A be the closure of A₀ in t… 028Y
Lemma 6 Let K⊂ C^Λ be a polynomially convex compact subset and U a neighbourhood of K. There exists a finite subset Λ₀ of Λ such that, for every subset Λ ' of Λ containing Λ₀, the set pr_Λ… 028Z
Lemma 7 Let ngeqslant 1 be an integer and K a compact polynomially convex subset of Cⁿ. Then K admits a fundamental system of compact polynomially convex neighbourhoods. 0290

§ 4. Holomorphic functional calculus

Remark If the radical of the algebra A is zero, one can omit the continuity condition in (CF1) (cf. prop. 9 of I, p. 40). 029U
Theorem 1 Let A be a commutative complex unital Banach algebra. There exists a unique holomorphic functional calculus on A. 029V
Definition 1 Let a= (a₁, . . . , aₙ)∈ Aⁿ, let h:Cⁿ→ C be a mapping and let u₁, . . . , uₙ be mappings of Cⁿ into A. We say that the sequence (h, u₁, . . . , uₙ) is adapted to a if 029W
Lemma 1 Let U be an open subset of Cⁿ and K a compact subset of U. There exists an infinitely differentiable mapping h of Cⁿ into C, equal to 1 on K and with compact support contained in U… 029X
Example Suppose that n= 1. Let a∈ A. For every open neighbourhood U of Sp(a), there exists an infinitely differentiable mapping h of C into C with compact support contained in U, equal to… 029Y
Lemma 2 Let a= (a₁, . . . , aₙ)∈ Aⁿ. There exist infinitely differentiable mappings v₁, . . . , vₙ of Cⁿ- Spⁿ(a) into A such that 029Z
Lemma 3 Let a= (a₁, . . . , aₙ)∈ Aⁿ. Let h be a mapping from Cⁿ into C, infinitely differentiable, equal to 1 in a neighbourhood of Spⁿ(a) and with compact support. There exist infinitely… 02A0
Lemma 4 Let a∈ Aⁿ. Let (h, u₁, . . . , uₙ) be a sequence adapted to a and ω the associated differential form. 02A1
Lemma 5 Let a∈ Aⁿ, let (h, u₁, . . . , uₙ) be a sequence adapted to a, and let ω be the associated differential form. 02A2
Lemma 6 Let a∈ Aⁿ. All sequences adapted to a are linked. 02A3
Lemma 7 For every function f∈ O(U; A), the integral ∫_Uf ω is an element of A depending only on a and on the germ of f in a neighbourhood of Spⁿ(a). It satisfies the inequality 02A4
Proposition 1 Let a be an element of A, let U be an open neighbourhood of Sp(a), and let f∈ O(U; A). Let V be a compact neighbourhood of Sp(a) contained in U and such that V is a piece of C. 02A5
Corollary Let a∈ A. One has Θₐ(1) = 1. 02A6
Lemma 8 Let U be an open set in Cⁿ. Let ω₁ be a continuous differential form of degree n in U with compact support and with values in A (resp. ω₂ a continuous differential form of degree 2… 02A7
Lemma 9 Let a= (a₁, . . . , aₙ)∈ Aⁿ. Let p∈ N and a' = (aₙ₊₁, . . . , aₙ₊ₚ)∈ Aᵖ. Then we have Θ_(a,a')∘ π_n,n+p^*= Θ_a. In particular, we have Θ_a(1) = 1. 02A8
Lemma 10 Let a∈ Aⁿ. Let g be a polynomial function on Cⁿ with coefficients in A and let f∈ O(Spⁿ(a); A). Then Θ_a(gf) =g(a)Θ_a(f). In particular, Θ_a(g) =g(a). 02A9
Proposition 2 Let varrho₁, . . . , varrhoₙ be real numbers >0 and let U⊂ Cⁿ be the polydisc product of the open discs with center 0 and radius varrhoᵢ. Let 02AA
Corollary Assume that A is nonzero. Let a∈ Cⁿ⊂ Aⁿ. One has Spⁿ_A(a) =a. For every germ f∈ O(a; A), one has Θ_a(f) =f(a). 02AB
Proposition 3 Let B be a commutative unital Banach algebra and φ a continuous unital morphism of A into B. Let a= (a₁, . . . , aₙ)∈ Aⁿ. Put b= (φ (a₁), . . . , φ (aₙ)), so that Spⁿ_B(b)⊂ Spⁿ_A(a… 02AC
Corollary 1 Let χ ∈ X(A) and a∈ Aⁿ. For every germ f∈ O(Spⁿ(a)), one has χ (Θ_a(f)) =f(χ (a₁), . . . , χ (aₙ)). 02AD
Remark Suppose that the algebra A is without radical. Let a= (a₁, . . . , aₙ)∈ Aⁿ. By Proposition 8 of I, p. 38, the mapping Θᵁ_a is the unique mapping φ from O(U) into A such that χ (φ (… 02AE
Corollary 2 Let p be an integer geqslant For every family (f₁, . . . , fₚ) of elements of O(Spⁿ(a)), one has 02AF
Example Let A be the complex Banach algebra of functions on the unit circle with absolutely convergent Fourier series (I, p. 19, Example 8). Let φ ∈ A. Let f be a germ of a holomorphic fun… 02AG
Proposition 4 Let L be a polynomially convex compact subset of Cⁿ and let U be an open neighbourhood of L. For every function f∈ O(U; A), there exists a sequence of polynomial functions on Cⁿ wi… 02AH
Theorem 2 (Oka–Weil) Let K be a polynomially convex compact subset of Cⁿ and P the set of germs in a neighbourhood of K of polynomial functions on Cⁿ with coefficients in A. Then P is dense in O(K; A)… 02AI
Corollary 1 Let U be an open neighbourhood of K. Let u₁ and u₂ be continuous mappings of O(U; A) into a topological space X factoring through O(K; A). Then u₁=u₂ if and only if u₁ and u₂ coinc… 02AJ
Corollary 2 Let E be a Banach space. Let K be a polynomially convex compact subset of Cⁿ. Let P be the set of germs in a neighbourhood of K of polynomial functions on Cⁿ with values in E. Then… 02AK
Theorem 3 (Runge) Let K be a compact subset of C, and Q the set of germs of rational functions holomorphic in a neighbourhood of K. Then Q is dense in O(K). 02AL
Definition 2 Let ngeqslant 1 be an integer and let a∈ Aⁿ. Let U be an open neighbourhood of Spⁿ(a). One says that a family a' is an envelope of (a,U) if a'∈ Cⁿ⁺ᵖ extends a and if U× Cᵖ contains… 02AM
Lemma 11 Let ngeqslant 1 be an integer. Let a∈ Aⁿ. For every open neighbourhood U of Spⁿ(a), there exists an envelope of (a,U). 02AN
Proposition 5 The data of the mappings Θ_a, for ngeqslant 1 and a∈ Aⁿ, constitute a holomorphic functional calculus on A, that is to say that conditions (CF1), (CF2) and (CF3) of I, p. 51 are sa… 02AO
Proposition 6 Let a∈ Aⁿ. Suppose that Spⁿ(a) is polynomially convex. Let z₁, . . . , zₙ be the germs in a neighbourhood of Spⁿ(a) of the coordinate functions on Cⁿ. Then the mapping Θ_a is the u… 02AP
Theorem 4 Let A be a commutative complex unital Banach algebra, let ngeqslant 1 be an integer, and let a= (a₁, . . . , aₙ)∈ Aⁿ. Let f= (f₁, . . . , fₚ), where f₁, . . . , fₚ are elements of… 02AQ
Lemma 12 Let K be the polynomially convex envelope of Spᵖ(f(a)). One has g(f(a)) = (g∘ f)(a) for every germ g∈ O(K; A). 02AR
Theorem 5 Let A be a unital Banach algebra, not necessarily commutative. Let a be an element of A and z the germ of the identity function of C in a neighbourhood of Sp_A(a). There exists a u… 02AS
Proposition 7 Let A and B be unital Banach algebras and φ a continuous unital morphism from A into B. Let a∈ A. Then Sp_B(φ (a))⊂ Sp_A(a) and one has φ (f(a)) = f(φ (a)) for every f∈ O(Sp_A(a))… 02AT
Proposition 8 Let A be a unital Banach algebra and a∈ A. Let f∈ O(Sp(a)). One has f(Sp_A(a)) = Sp_A(f(a)). Moreover, for every g∈ O(Sp_A(f(a))), one has g∘ f∈ O(Sp_A(a)) and g(f(a)) = (g∘ f)(a). 02AU
Proposition 9 Let A be a unital Banach algebra and a∈ A. Let U be an open neighbourhood of Sp_A(a) and f∈ O(U). Let moreover V be a compact neighbourhood of Sp_A(a) contained in U such that V is… 02AV
Proposition 10 Let A be a unital Banach algebra and U an open subset of C. 02AW
Proposition 11 Let A be a unital Banach algebra, a∈ A and U an open neighbourhood of Sp_A(a). Let δ denote the distance from Sp_A(a) to C- U. Let f∈ O(U). 02AX
Proposition 12 Let A be a commutative unital Banach algebra. The image of the exponential mapping is the identity component of the group G of invertible elements of A. 02AY
Proposition 13 Let A be a commutative unital Banach algebra. Let U₁ and U₂ be open sets of X(A) forming a partition of X(A). Then there exists a unique idempotent j of A such that the Gelfand tra… 02AZ
Corollary Let A be a commutative unital Banach algebra. The following assertions are equivalent: 02B0
Proposition 14 Let A be a commutative Banach algebra without radical. In order that A admit a unit element, it is necessary and sufficient that X(A) be compact. 02B1
Proposition 15 Let A be a commutative Banach algebra, let I₁ be an ideal of A and F₁ the set of χ ∈ X(A) which are zero on I₁. Let F₂ be a subset of X(A) disjoint from F₁, closed for the Jacobson… 02B2
Corollary Let A be a commutative Banach algebra. Let F₁ and F₂ be two disjoint subsets of X(A), closed for the Jacobson topology. Assume that F₂ is weakly compact. Then there exists u∈ A suc… 02B3
Proposition 16 We retain the preceding notation. Let (Hᵢ)_1leqslant ileqslant n be a partition of Sp_A(x) into elements of Π. 02B4
Proposition 17 Let μ be an isolated point of Sp_A(x). Then 02B5
Corollary 1 Let μ be an isolated point of Sp_A(x) and p a strictly positive integer. In order that μ be a pole of order p of the resolvent of x(cf. VAR, R1, p. 30, 3.3.9), it is necessary and… 02B6
Corollary 2 Let μ be an isolated point of Sp_A(x). Let Γ be the oriented boundary of an open disk Δ with center μ such that 02B7
Proposition 18 The complex topological vector space E_(C) is normable (resp. complete) if and only if E is normable (resp. complete). 02B8
Remark Let E and F be normable topological vector spaces over K. The vector space L(E; F) of continuous linear mappings from E into F, endowed with the topology of bounded convergence, is… 02B9
Proposition 19 Let z denote the germ in O(S) of the identity mapping of C. Then O_R(S) is the smallest closed full sub-R-algebra of O(S) containing z. 02BA
Lemma For every f∈ O(Sp_A_(C)(x)), one has f^*(1⊗ x) =overlinef(1 ⊗ x). 02BB
Proposition 20 For every f∈ O_R(Sp_A_(C)(x)), there exists a unique element f(x) of A such that f(1⊗ x) = 1⊗ f(x) in A_(C). The mapping f↦ f(x) from O_R(Sp_A_(C)(x)) into A is the unique continuo… 02BC

§ 5. Regular commutative Banach algebras

Proposition 1 Let A be a commutative Banach algebra. The following conditions are equivalent: 02BV
Definition 1 Let A be a commutative Banach algebra. It is said to be regular if it satisfies the equivalent conditions of Proposition 1. 02BW
Remark Let widetildeA be the Banach algebra deduced from A by adjunction of a unit element e. Condition (ii) of prop. 1 shows that if widetildeA is regular, then A is regular. Suppose tha… 02BX
Example Let us take up again the examples of nᵒ2 of I, p. 17. 02BY
Proposition 2 Let A be a unital commutative regular Banach algebra. Let ngeqslant 1 be an integer and (U₁, . . . ,Uₙ) an open covering of X(A). There exist elements x₁, . . . , xₙ of A of sum 1… 02BZ
Corollary 1 Let A be a commutative regular unital Banach algebra, let I be an ideal of A and let f:X(A)→ C be a continuous function. Suppose that, for every χ ∈ X(A), there exists an element y… 02C0
Corollaire 2 Let A be a commutative regular Banach algebra, I an ideal of A and f:X'(A)→ C a continuous function. Suppose that, for every χ ∈ X'(A), there exists an element y_χ∈ I such that f=G… 02C1
Proposition 3 Let A be a commutative regular Banach algebra, I an ideal of A and K a compact subset of X(A) disjoint from V(I). There exists an element x∈ I such that G(x) = 1 for every x in K. 02C2
Proposition 4 Let A be a commutative regular Banach algebra without radical. Let F be a closed subset of X(A). The set of ideals I of A such that V(I) = F, ordered by inclusion, has a greatest e… 02C3
Corollary 1 Let A be a commutative regular Banach algebra without radical. Let J be the set of x∈ A such that G(x) has compact support. Suppose that J = A. Then every closed ideal of A distinc… 02C4
Corollary 2 Let A be a commutative regular Banach algebra without radical. Let x, y∈ A. If the support of G(x) is compact and contained in the set of characters χ such that G(y)(χ )not= 0, the… 02C5
Definition 2 Let A be a commutative Banach algebra. 02C6
Remark Let A be a commutative Banach algebra. 02C7
Lemma 1 Let A be a commutative regular Banach algebra without radical, satisfying Ditkin's condition. Let I be a closed ideal of A and x an element of Υ(V(I)). Let K be the set of χ ∈ X'(A… 02C8
Lemma 2 Let X be a topological space. Let F and D be disjoint subspaces of X such that F is closed and D discrete. If F contains no nonempty perfect subspace, the same is true of F∪ D. 02C9
Proposition 5 Let A be a regular commutative Banach algebra without radical, satisfying Ditkin's condition. Let I be a closed ideal of A such that the boundary F of V(I) contains no nonempty per… 02CA

§ 6. Stellar algebras

Lemma 1 Let E be a complex vector space and u a semilinear involution on E. Let x∈ E; put 02CI
Proposition 1 Let E₁ and E₂ be complex vector spaces, and let u₁ and u₂ be semilinear involutions on E₁ and E₂ respectively. Let f be a linear mapping of E₁ into E₂. Then f∘ u₁=u₂∘ f if and only… 02CJ
Definition 1 Let A be an algebra over C. An involution in A is called a mapping x↦ x^* of A into A such that: 02CK
Lemma 2 Let A be an involutive algebra and x∈ A. The elements 02CL
Remark Let A be an involutive algebra and B a maximal commutative involutive subalgebra of A. Then B is a maximal commutative subalgebra. In particular, if A is unital, then the algebra B… 02CM
Lemma 3 Let A be a commutative involutive algebra. The Gelfand transform of A into C₀(X(A)) is a morphism of involutive algebras if and only if every character of A is hermitian. 02CN
Example 1 Let A be the algebra of complex-valued functions on a set X. The mapping f↦ overlinef is an involution in A. The subalgebra of bounded functions on X is an involutive subalgebra of… 02CO
Example 2 Let X be a locally compact topological space and μ a positive measure on X. The mapping f↦ overlinef is an involution on the algebra L^∞(X, μ); by passing to the quotient it induce… 02CP
Example 3 Let E be a complex Hilbert space. On the Banach algebra L(E), the mapping x↦ x^* (EVT, V, p. 37, prop. 1) is an involution. 02CQ
Example 4 Let G be a locally compact group. Let M¹(G) be the Banach algebra of bounded complex measures on G (example 6 of I, p. 19). 02CR
Example 5 Let U be an open subset of C stable under complex conjugation. Consider the algebra O(U) of complex-valued holomorphic functions on U. For every function f∈ O(U), the mapping f^*:z… 02CS
Definition 2 An involutive normed algebra is a normed algebra A endowed with an involution x↦ x^* such that |x^*|=|x| for every x. If A is a Banach algebra, one says that A is an involutive Ban… 02CT
Example 1 Let X be a locally compact topological space. The Banach algebra C_b(X) of complex-valued continuous bounded functions on X, endowed with the norm |f| = sup_x∈ X|f(x)| and the invo… 02CU
Example 2 Let X be a locally compact topological space and μ a positive measure on X. The involutive algebra L^∞(X, μ) (example 2 of I, p. 99) is an involutive Banach algebra, since |f|=|ove… 02CV
Example 3 The involutive algebra L(E) of continuous endomorphisms of a complex Hilbert space E (I, p. 99, example 3), endowed with the norm 02CW
Example 4 The involutive algebra M¹(G) of bounded measures on a locally compact group (I, p. 99, example 4), endowed with the usual norm (example 6 of I, p. 19), is an involutive Banach alge… 02CX
Example 5 Let (Aᵢ) be a family of involutive normed algebras. Let A be the product normed algebra of the Aᵢ (No.ᵒ1 of I, p. 15). The algebra A, endowed with the involution (xᵢ)^*= (x^*ᵢ), is… 02CY
Example 6 Let A be an involutive normed algebra and let widetildeA be the normed algebra deduced from A by adjunction of a unit element. Endowed with the involution defined in No.ᵒ2, the alg… 02CZ
Lemma 4 Let A be an involutive normed algebra. For every continuous linear form f on A, one has |f^*|=|f|. If moreover f is hermitian, then |f|=|f|Aₕ|. 02D0
Lemma 5 Let A be an involutive Banach algebra. 02D1
Definition 3 A Banach involutive algebra A such that |x|²=|x^*x| for every x∈ A is called a stellar algebra. 02D2
Example 1 The Banach involutive algebra of continuous endomorphisms of a complex Hilbert space (example 3 of I, p. 100) is a stellar algebra (EVT, V, p. 39, prop. 2). 02D3
Example 2 Let X be a locally compact topological space. The Banach involutive algebra C_b(X) of continuous bounded complex-valued functions on X (example 1 of I, p. 100) is a stellar algebra… 02D4
Example 3 Let X be a compact topological space and let x₀∈ X be a fixed element of X. The subalgebra C'(X) of C(X) consisting of continuous functions f: X→ C such that f(x₀) = 0 is a stellar… 02D5
Example 4 Let X be a Hausdorff topological space and μ a positive measure on X. The Banach involutive algebra L^∞(X, μ) is a commutative unital stellar algebra. 02D6
Example 5 Let (Aᵢ) be a family of stellar algebras. The Banach involutive algebra A product of the Aᵢ (example 5 of I, p. 101) is a stellar algebra, called the product stellar algebra of the… 02D7
Example 6 Let A be a stellar algebra. If B is a closed involutive subalgebra of A, then B is a stellar algebra. We shall see (V, to appear) that every stellar algebra is isomorphic to a clos… 02D8
Example 7 Let A be a stellar algebra. If M⊂ A is any subset, then the closed involutive subalgebra of A generated by M is a stellar algebra, called the stellar subalgebra of A generated by M… 02D9
Example 8 In general, the Banach involutive algebra M¹(G) (example 4 of I, p. 100) is not a stellar algebra (I, p. 181, exerc. 8). 02DA
Lemma 6 Let A be a Banach algebra endowed with an involution satisfying 02DB
Lemma 7 Let A be a stellar algebra. 02DC
Remark 1 Let A be a unital stellar algebra. One has 02DD
Remark 2 Let A be a normed involutive algebra. If |x|²=|x^*x| for all x∈ A, the completion widehatA of A is a stellar algebra. 02DE
Proposition 2 Let A be an involutive Banach algebra, let B be a stellar algebra, and let π be a morphism of involutive algebras from A into B. One has |π (x)|leqslant |x| for all x∈ A, and in pa… 02DF
Proposition 3 Let A be a stellar algebra and let widetildeA be the involutive algebra deduced from A by adjunction of a unit element. There exists a unique norm on widetildeA extending that of A… 02DG
Definition 4 One says that widetildeA, endowed with the norm of Prop. 3, is the star algebra deduced from A by adjunction of a unit element. 02DH
Proposition 4 Let A be a star algebra. 02DI
Proposition 5 Let A be a unital star algebra and B a star subalgebra of A containing the unit element of A. Then B is a full subalgebra of A. In particular, one has Sp_B(x) = Sp_A(x) for every x… 02DJ
Corollary Let A be a star algebra and let B be a star subalgebra of A. Then one has Sp'_B(x) = Sp'_A(x) for every x in B. 02DK
Corollary Let A be a star algebra and a a normal element of A. The commutant (resp. the bicommutant) of a, a^* coincides with the commutant (resp. the bicommutant) of a. 02DL
Lemma 8 Let X and Y be metric spaces, the space X being complete. Let f be a mapping from X into Y such that 02DM
Lemma 9 Let A be a commutative star algebra. Every character of A is hermitian and the Gelfand transform is a morphism of star algebras of A into C₀(X(A)). 02DN
Theorem 1 Let A be a commutative star algebra. The Gelfand transform is an isometric isomorphism of the star algebra A onto the star algebra C₀(X(A)) of continuous functions on X(A) tending… 02DO
Corollary 1 Let A be a star algebra and x a normal element of A. Then |x|=varrho (x). 02DP
Corollary 2 Let A be a commutative star algebra. 02DQ
Corollary 3 Let A be a commutative unital star algebra. 02DR
Remark Let G be the category whose objects are the locally compact spaces and whose morphisms are the proper partial mappings (def. 1 of I, p. 33), and let S be the category of commutativ… 02DS
Lemma 10 The mapping evₓ:χ ↦ χ (x) induces a homeomorphism of X(B) onto Sp_A(x). 02DT
Definition 5 The involutive morphism f↦ (G_B⁻¹∘ φₓ)(f) of C(Sp_A(x)) into A is called the mapping of continuous functional calculus of x in A. It is denoted by f↦ f(x). 02DU
Remark The mapping f↦ f(x) is isometric; its image is the unital star-subalgebra B generated by x, which is contained in the bicommutant of x. 02DV
Example 1 Suppose that there exists λ ∈ C such that Sp_A(x) is reduced to λ. Then one has x=λ · 1. In fact, the identity function of Sp_A(x) is equal to λ, therefore its image under the func… 02DW
Example 2 For x to be hermitian, it is necessary and sufficient that Sp_A(x) be contained in R. In fact, let f be the continuous mapping on Sp_A(x) given by f(z) =z-overlinez. Then x is herm… 02DX
Example 3 For x to be unitary, it is necessary and sufficient that its spectrum be contained in the unit circle of C. In fact, let f∈ C (Sp_A(x)) be the function defined by f(z) =zz-1; the e… 02DY
Proposition 7 The mapping f↦ f(x) is the unique unital morphism of involutive algebras from C (Sp_A(x)) into A such that the identity mapping z of Sp_A(x) has image x. 02DZ
Corollary 1 Let f∈ O(Sp_A(x)) be a germ of a holomorphic function in a neighbourhood of Sp_A(x) and let widetildef∈ C(Sp_A(x)) be the continuous function on Sp_A(x) associated with f. Then wid… 02E0
Corollary 2 Let f∈ C (Sp_A(x)). 02E1
Example 4 Let X be a locally compact space and let A =C_b(X) be the commutative unital star algebra of continuous bounded functions on X (Example 2 of I, p. 102). Let g∈ A; its spectrum S is… 02E2
Proposition 8 Let f∈ C(Sp_A(x)). Denoting again by f the restriction of f to Sp_A'(π (x)), one has the equality π (f(x)) =f(π (x)). In particular, for every χ ∈ X(A), one has χ (f(x)) =f(χ (x)). 02E3
Corollary Suppose that A is commutative. For every f∈ C(Sp_A(x)), one has G_A(f(x)) =f∘ G_A(x). 02E4
Proposition 9 Every injective morphism of stellar algebras is isometric and, in particular, has closed image. 02E5
Lemma 11 Let X be a completely regular topological space, that is to say uniformizable and separated (TG, IX, p. 8, Definition 4), containing at least two points. There exist non-zero conti… 02E6
Proposition 10 Let A be a unital stellar algebra. Suppose that for every pair (x, y) of permutable elements of A, the condition xy= 0 implies that x= 0 or y= 0. Then A =C· 1. 02E7
Proposition 11 Let A be a unital involutive algebra, and let a,x and y be elements of A. Suppose that x and y are normal. If xa= ay, then one has f(x)a=af(y) for every function f continuous in th… 02E8
Proposition 12 The mapping f↦ f(x) is the unique morphism of involutive algebras from the stellar algebra C'(Sp'_A(x)) into A such that the identity mapping z of Sp'_A(x) has image x. 02E9
Proposition 13 We have the following properties: 02EA
Remark Let A be a unital involutive algebra and let widetildeA be the unital involutive algebra deduced from A by adjunction of a unit element e. For every x∈ A, one has Sp'_A(x) = Sp_A(x… 02EB
Definition 6 Let A be an involutive algebra. An element x of A is said to be positive if it is hermitian and if Sp'_A(x)⊂ R₊. The set of positive elements of A is denoted by A₊. It is a subset… 02EC
Example 1 Let X be a locally compact space. In the involutive algebra C₀(X) of continuous functions on X tending to 0 at infinity, resp. in the involutive algebra C_b(X) of bounded continuou… 02ED
Example 2 Let A be a commutative involutive algebra. Let a be in A. Since Sp'_A(x) is the union of 0 and of the image of the Gelfand transform G(a) (prop. 6 of I, p. 37), the element a is po… 02EE
Example 3 Let E be a complex Hilbert space. An element x of the involutive algebra L(E) (example 1 of I, p. 102) is positive if and only if it is a positive endomorphism of E in the sense of… 02EF
Lemma 12 Let A be a unital involutive algebra and let x∈ A be a hermitian element. 02EG
Proposition 14 Let A be an involutive algebra. The set A₊ is a salient closed pointed convex cone in the real Banach space Aₕ (EVT, II, p. 11). 02EH
Proposition 15 Let A be a star-algebra. Let x be a normal element of A. 02EI
Proposition 16 Let x be a positive element of A. Let α ∈ R^*₊. Then x^1/α is the unique positive element y of A such that y^α=x. 02EJ
Lemma 13 Let A be a unital stellar algebra. Every element of A is a sum of unitary elements. 02EK
Theorem 2 Let A be a stellar algebra. An element x of A is positive if and only if there exists y∈ A such that x=y^*y. 02EL
Remark Let A be a star-algebra and let x∈ A. The element x^*x of A is positive; one then sets |x|= (x^*x)^1/2. We have |x|²=|x^*x|= ||x|²|=||x||², hence |x|=||x||. 02EM
Lemma 14 Let A be a star-algebra, and let x and y be hermitian elements of A. 02EN
Remark In general, if x and y are positive elements of a -algebra A, the condition 0leqslant xleqslant y does not imply x²leqslant y²(cf. exercise 15 of I, p. 182). 02EO
Definition 7 Let A be a normed algebra. An approximate unit of A is a filter basis 𝔉 on the unit ball of A such that, for every x in A, the filter bases x𝔉 and 𝔉x on A converge to x, in other w… 02EP
Proposition 17 Let A be a -algebra, and let 𝔉 be a filter basis on A^leqslant 1₊. For 𝔉 to be an increasing approximate unit of A, it is necessary and sufficient that one have 02EQ
Proposition 18 Let A be an involutive algebra. The ordered set A^<1₊ is directed to the right (E, III, p. 12, def. 7), and the filter of its sections (TG, I, p. 38, example 2) is an increasing ap… 02ER
Lemma 15 Let A be a stellar algebra and let I be a closed two-sided ideal of A. Then I is self-adjoint. 02ES
Proposition 19 Let A be a stellar algebra and let I be a closed two-sided ideal of A. Then the quotient involutive Banach algebra A/I is a stellar algebra. 02ET
Lemma 16 Let A be an involutive algebra and let p be a seminorm on A. The following conditions are equivalent : 02EU
Definition 8 The stellar algebra Stell(A) is called the enveloping stellar algebra of the normed involutive algebra A. 02EV
Proposition 20 Let A be a normed involutive algebra and let j be the canonical morphism of A into Stell(A). For every stellar algebra B and every morphism φ of involutive algebras from A into B… 02EW
Corollary Let A be a commutative involutive Banach algebra and j the canonical morphism of A into Stell(A). The mapping X(j) is a homeomorphism of X(Stell(A)) onto the subspace X(A)ₕ of X(A)… 02EX
Proposition 21 Let A be an involutive Banach algebra and j the canonical morphism of A into Stell(A). The radical of A is contained in the kernel of j. 02EY
Definition 9 Let G be a locally compact group and let A be the involutive Banach algebra of bounded measures on G admitting a density with respect to a Haar measure on G (example 4 of I, p. 99)… 02EZ
Remark Let ν be a left Haar measure on G and let Δ be its module. The mapping f↦ f· ν is an isometric isomorphism of the algebra L¹(G, ν ) onto A (loc. cit.). Thus one may also define Ste… 02F0
Remark In general, the left regular representation γ ' of Stell(G) in L²(G, ν ) is not faithful. One can prove that it is so if and only if there exists on L^∞_R(G) a positive linear form… 02F1
Proposition 22 The canonical mapping j of L¹(G, ν ) into Stell(G) is injective and has dense image. 02F2
Corollary The algebra L¹(G, ν ) is semisimple. 02F3

§ 7. Spectrum of endomorphisms of Banach spaces

Definition 1 Let E be a topological vector space and let u be an endomorphism of E. The spectrum of u, denoted by Sp(u), is the spectrum of u relative to the unital algebra L(E). 02G0
Lemma 1 Let E be a Banach space and let u be an endomorphism of E. Let (Eᵢ)_i∈ I be a finite family of closed subspaces of E, stable under u, such that E =⨁_i∈ IEᵢ. For each i∈ I, let uᵢ d… 02G1
Proposition 1 Let E be a complex Banach space and u an endomorphism of E. Let λ ∈ C and f∈ O(Sp(u)). We have 02G2
Proposition 2 Let E be a Banach space and u an endomorphism of E. Let E₁ and E₂ be closed subspaces of E invariant under u such that E = E₁⊕ E₂. Suppose that the endomorphisms u₁ and u₂ of E₁ an… 02G3
Proposition 3 Let E be a Banach space and let E' be its dual. Let u be an endomorphism of E. 02G4
Remark Let H be a subset open and closed in Sp(u), whose associated spectral decomposition is E = E_H(u)⊕ widetildeE_H(u). It follows from assertion c) that if one identifies E' with E_H(… 02G5
Lemma 2 Let E be a complex Hilbert space and let u be a normal endomorphism of E. Let (Eᵢ)_i∈ I be a finite family of closed subspaces of E, stable under u and pairwise orthogonal, such th… 02G6
Proposition 4 Let E be a complex Hilbert space and let u be a normal endomorphism of E. For every function f∈ C (Sp(u)) and every λ ∈ C, one has 02G7
Lemma 3 Let E be a Hilbert space and p∈ L(E) a projector. The following assertions are equivalent: 02G8
Proposition 5 Let E be a complex Hilbert space and u a normal endomorphism of E. 02G9
Lemma 4 Let E be a Hilbert space and let u be a normal endomorphism of E. Let F be a closed subspace of E containing a total set of proper vectors of u. Then F^∘ is stable under u and the… 02GA
Definition 2 Let E be a complex Hilbert space and let u be an endomorphism of E. The numerical image of u is the set of complex numbers of the form ⟨ x|u(x)⟩, where x ranges over the unit spher… 02GB
Proposition 6 Let E be a complex Hilbert space and let u be an endomorphism of E. 02GC
Lemma 5 Let E be a complex Hilbert space of dimension 2. Let us make the real vector space L(E)ₕ of Hermitian endomorphisms of E into a prehilbertian normed space by means of the norm u↦ T… 02GD
Lemma 6 Let E be a normed real vector space, let F be a real vector space, and let u: E→ F be a non-injective affine mapping. Let B be a ball in E and S the corresponding sphere. Then u(S)… 02GE
Proposition 8 Let E be a complex Hilbert space. Let u be an endomorphism of E. The following conditions are equivalent: 02GF
Proposition 9 Let E be a complex Hilbert space and let u be a hermitian element of L(E). 02GG
Definition 3 One says that (u^*u)^1/2 is the absolute value of u, and it is denoted by |u|. 02GH
Proposition 10 Let E₁ and E₂ be complex Hilbert spaces and let u∈ L(E₁; E₂). 02GI
Definition 4 Let E₁ and E₂ be complex Hilbert spaces and u∈ L(E₁; E₂). The pair (j,|u|), where j is the unique partially isometric mapping of E₁ into E₂ such that u=j|u| and Ker(j) = Ker(u), is… 02GJ
Proposition 11 Let E₁ and E₂ be complex Hilbert spaces and u∈ L(E₁; E₂). Let (j,|u|) be the polar decomposition of u. 02GK
Corollary Let E₁ and E₂ be complex Hilbert spaces and let u∈ L(E₁; E₂). Then Im(u) = Im(|u^*|). 02GL
Proposition 12 Let E₁ and E₂ be complex Hilbert spaces and u∈ L(E₁; E₂). Let (j,|u|) be the polar decomposition of u. For u to be bijective, it is necessary and sufficient that |u| be invertible… 02GM
Proposition 13 Let E be a complex Hilbert space and u an endomorphism of E. The following conditions are equivalent: 02GN

§ 8. Algebras of continuous functions on a compact space

Proposition 1 Let f↦ |f|_B be a norm making B into a Banach algebra. 02H4
Proposition 2 Let f↦ |f|_B be a norm such that (B,|· |_B) is a Banach algebra. 02H5
Example Let X = [0,1] and let ngeqslant 0 be an integer. Let B be the algebra of functions f: X→ C admitting continuous derivatives on [0,1] up to order n, endowed with the norm considered… 02H6
Proposition 3 Let X be a compact space. Let B be a unital Banach subalgebra of C(X), endowed with the induced norm, separating the points of X. We identify X with a closed subset of X(B) (prop… 02H7
Proposition 4 Let X be a compact space. Let B be a unital Banach subalgebra of C(X), endowed with the induced norm and separating the points of X. We identify X with a closed subset of X(B) (Pro… 02H8
Corollary 1 Let Λ be a set and X a compact subset of C^Λ. We identify X with a subset of X(P(X)) (prop. 1 of I, p. 142, d)). Let (z_λ)_λ∈Λ be the family of coordinate functions on C^Λ. 02H9
Corollary 2 Let Λ be a set and let X⊂ C^Λ be a compact set. If X is connected, then its polynomially convex envelope is connected. 02HA
Lemma 1 Let X be a compact subset of the plane and let O be a bounded connected component of C- X. The boundary of O is contained in X. 02HB
Proposition 5 With the above notation: 02HC
Corollary 1 The following conditions are equivalent: 02HD
Corollary 2 (Runge's Theorem) For each i∈ I, let λᵢ be a point of Oᵢ. Let f be a holomorphic complex function in an open neighbourhood of X. Then f|X is the uniform limit of restrictions to X of rational fracti… 02HE

II. Commutative locally compact groups

§ 1. Fourier transform

Definition 1 A continuous homomorphism of G into the multiplicative group U of complex numbers of modulus 1 is called a unitary character of G. 02HQ
Definition 2 The topological group widehatG is called the dual group of G. 02HR
Lemma 1 For every χ ∈ widehatG, the mapping μ↦ χ (μ) is a hermitian character of the involutive Banach algebra M¹(G). 02HS
Proposition 1 The canonical mapping of widehatG into X(L¹(G)) is a homeomorphism. 02HT
Remark 1 The bijectivity of the mapping from widehatG into X(L¹(G)) of Proposition 1 is a particular case of the correspondence between continuous representations of a locally compact group… 02HU
Remark 2 The canonical mapping from widehatG into X(M¹(G)) is not surjective in general (II, p. 308, exercise 14). 02HV
Corollary 1 Every character of L¹(G) is hermitian. The canonical mapping from X(Stell(G)) (Definition 9 of I, p. 125) into X(L¹(G)) is a homeomorphism. 02HW
Corollary 2 The topological group widehatG is locally compact. 02HX
Lemma 2 Let G and H be locally compact commutative groups and let f: H→ G be a morphism of topological groups. The kernel of widehatf is the orthogonal of the image of f. 02HY
Proposition 2 Let ngeqslant 0 be an integer and let G₁, . . . ,Gₙ be locally compact commutative groups. Let G be the product group of the groups Gⱼ for 1leqslant jleqslant n. For 1leqslant jleq… 02HZ
Remark The calculation of the dual group of an infinite product of commutative compact groups is the object of Corollary 4 of II, p. 234 below. The case of a commutative locally compact g… 02I0
Definition 3 Let μ∈ M¹(G) be a bounded complex measure on G. The Fourier transform of μ is called the function F_G(μ) on widehatG defined by 02I1
Proposition 3 For every measure μ∈ M¹(G), the functions F_G(μ) and overlineF_G(μ) are continuous and bounded. The mappings F_G:μ↦ F_G(μ) and overlineF_G:μ↦ overlineF_G(μ) are continuous morphism… 02I2
Proposition 4 The Fourier transform and the Fourier cotransform are injective morphisms of involutive algebras of L¹(G) into the algebra C₀(widehatG) of continuous functions vanishing at infinit… 02I3
Remark The Fourier transform on the space L¹(G) depends on the choice of the Haar measure dx, unlike the Fourier transform on M¹(G). If one replaces dx by the measure a· dx (with a >0), t… 02I4
Proposition 5 By continuity, the Fourier transform and the cotransform of Fourier extend uniquely to isomorphisms of stellar algebras of Stell(G) onto C₀(widehatG). 02I5
Corollary The image of L¹(G) under the Fourier transform of G is dense in C₀(widehatG). 02I6
Proposition 6 Suppose that G is compact. The normalized Haar measure dx belongs to M¹(G), and its Fourier transform is φₑ, the characteristic function of e. 02I7
Proposition 7 The space A(G) is a self-adjoint ideal of L¹(G). It is contained in L¹(G)∩ L²(G), and in the image of C(G) in L¹(G). 02I8
Proposition 8 There exists a filter basis 𝔅 on A(G)∩ K₊(G) such that the following conditions are satisfied: 02I9
Corollary 1 The space A(G) is dense in L¹(G) and in L²(G). It is also dense in Stell(G), and its image widehatA(G) under the Fourier transform is dense in C₀(widehatG). 02IA
Corollary 2 For f∈ A(G), let Ω_f be the set of χ ∈ widehatG such that F(f)(χ )not= 0. The sets Ω_f form an open covering of widehatG. 02IB
Lemma 3 For every f∈ A(G), there exists a unique bounded measure μ_f on widehatG such that 02IC
Lemma 4 There exists a unique measure ν on widehatG such that 02ID
Proposition 9 The measure ν characterized by lemma 4 is a Haar measure on widehatG. 02IE
Definition 4 The Haar measure ν on widehatG of proposition 9 is called the dual measure of the given Haar measure dx on G. 02IF
Remark Let a be a real number >0. If one replaces dx by the measure a· dx, the convolution product of the functions f and g∈ L¹(G) is replaced by a(f*g). We have seen (II, p. 209, remark)… 02IG
Lemma 5 The space A(widehatG) is dense in L²(widehatG). 02IH
Theorem 1 (Plancherel) The restriction of the Fourier transform to the subspace A(G) of L²(G) extends in a unique way to an isometry Φ of L²(G) onto L²(widehatG). 02II
Corollary Suppose that G is compact and that dx is the normalized Haar measure on G. Then the family of unitary characters of G is an orthonormal basis of L²(G). 02IJ
Remark 1 Certain formulas concerning the Fourier transform on L¹(G) extend to the Fourier transform on L²(G). In particular, for f∈ L²(G) and χ ∈ widehatG, we have 02IK
Remark 2 The formulas 02IL
Proposition 10 Let ngeqslant 0 be an integer and let G₁,⋯ ,Gₙ be locally compact commutative groups. Let μⱼ, for 1leqslant jleqslant n, be Haar measures on Gⱼ. Let G be the product group of the g… 02IM
Proposition 11 Let f∈ A(G). Then F(f)∈ L¹(widehatG) and, for every x∈ G, one has 02IN
Lemma 6 Let φ ∈ L¹(widehatG)∩ L²(widehatG). Then f=overlineF_widehatG(φ )∘ η belongs to L²(G) and F_G(f) =φ in L²(widehatG). 02IO
Corollary 1 For every closed subset P of widehatG and every χ ∈ widehatG not belonging to P, there exists a function f∈ L¹(G) such that F(f) is zero on P and non-zero at χ. 02IP
Corollary 2 The Banach algebra L¹(G) is regular (I, p. 89, def. 1). 02IQ
Theorem 2 (Pontryagin) The canonical mapping η of G into widehatwidehatG is an isomorphism of topological groups. It transforms the Haar measure dx into the bidual Haar measure dwidehatwidehatx. 02IR
Corollary The Fourier cotransform of L²(widehatG) onto L²(G) and the Fourier transform of L²(G) onto L²(widehatG) are reciprocal isometries of one another. 02IS
Remark Let f∈ L²(G) and g∈ L²(widehatG). Applying the Plancherel formula (24) to f and overlineF_widehatG(g), one obtains the formula 02IT
Proposition 13 Let α ∈ M¹(G) and β ∈ M¹(widehatG). Then one has (30) F_G(overlineF_widehatG(β )· α ) =β *F_G(α ) 02IU
Corollary The Fourier transform F_G is injective on M¹(G). 02IV
Theorem 3 The restriction of the Fourier transform to B(G) induces an isomorphism of vector spaces of B(G) onto B(widehatG), whose inverse is induced by the restriction of the co-Fourier tra… 02IW
Corollary 1 Let f∈ L¹(G). Then f∈ B(G) if and only if f belongs to the image of the Fourier transform F_widehatG on L¹(widehatG). In particular, one has A(G)⊂ B(G). 02IX
Corollary 2 The vector space B(G) is an algebra both for multiplication and for convolution. The Fourier transform exchanges convolution and multiplication in B(G) and B(widehatG). 02IY
Proposition 14 Let f and g be in L²(G). Then F_G(f g) = F_G(f)*F_G(g). 02IZ
Remark See nᵒ9 and exercises 22 of II, p. 270 and 31 of II, p. 275 for other examples of function spaces on which the Fourier transform is an isomorphism, in the case of particular groups… 02J0
Lemma 7 Let (Gᵢ)_i∈ I and (Hᵢ)_i∈ I be finite families of locally compact topological groups. For i∈ I, let θᵢ: Gᵢ× Hᵢ→ U be a mapping which puts the groups Gᵢ and Hᵢ in duality. The mappi… 02J1
Definition 5 Let G, H and K be topological groups. Let f: H→ G and g: G→ K be morphisms of topological groups. The pair (f, g) is said to be an exact sequence of topological groups if it is an… 02J2
Example 1 Let f: H→ G be a strict injective morphism whose image is a distinguished subgroup. The sequence 02J3
Example 2 Let g: G→ K be a strict surjective morphism. The sequence 02J4
Theorem 4 A sequence 02J5
Lemma 8 Let g: G→ K be a morphism of commutative locally compact topological groups. If the morphism g is surjective and strict, then widehatg is injective and strict. 02J6
Lemma 9 Let f: H→ G be a morphism of locally compact topological groups. If the morphism f is injective and strict, then widehatf is surjective and strict. 02J7
Lemma 10 Let 02J8
Corollary 1 Let 02J9
Corollary 2 Let f: G→ H be a morphism of locally compact commutative topological groups. The morphism f is strict if and only if widehatf is strict. 02JA
Corollary 3 Let H be a subgroup of G. Then (H^bot)^bot= H. 02JB
Corollary 4 Let I be a set and let (Hᵢ)_i∈ I be a family of closed subgroups of G. The orthogonal of the closed subgroup generated by the Hᵢ is ⋂_i∈ IH^botᵢ. The orthogonal of ⋂ᵢHᵢ is the clos… 02JC
Corollary 5 Let φ : G→ H be a morphism of locally compact commutative groups. Then the subgroup Im(φ ) of H and the subgroup Ker(widehatφ) of widehatH are orthogonal to one another. In particu… 02JD
Corollary 6 Let k∈ Z. Then the kernel of the homomorphism x↦ xᵏ of G into G and the closure of the image of the morphism χ ↦ χᵏ of widehatG into widehatG are orthogonal to one another. 02JE
Corollary 7 Let G be a locally compact commutative group. 02JF
Remark There exist commutative locally compact groups G which are not divisible and such that widehatG is torsion-free (Exercise 63 of II, p. 299). 02JG
Proposition 15 Let f∈ L¹(G) be such that the restriction to H^bot of the continuous function F_G(f) is integrable with respect to widehatγ. Then, for almost every x∈ G, the function y↦ f(xy) on H… 02JH
Corollary (Poisson Formula) Let f∈ L¹(G). Suppose that the following conditions are satisfied: 02JI
Proposition 16 The measure widehatα on widehatH =widehatG/H^bot is equal to widehatβ /widehatγ. 02JJ
Proposition 17 Let ngeqslant 1 be an integer. Let us denote by μₙ the group of n-th roots of unity in C. The groups Z/nZ and μₙ are in duality relative to the mapping induced by passing to the qu… 02JK
Corollary 1 Let G be a finite commutative group. The dual group widehatG is isomorphic to G. 02JL
Corollary 2 Let G be a locally compact commutative group. The group G is finite if and only if widehatG is finite. A closed subgroup H of G is of finite index if and only if its orthogonal is… 02JM
Proposition 18 In order that widehatG be compact, it is necessary and sufficient that G be discrete. If G is discrete, the dual measure of the counting measure on G is the normalised Haar measure… 02JN
Corollary 1 (Orthogonality relations) Suppose G discrete and endowed with the counting measure. For x and y in G, one has 02JO
Corollary 2 Let H be a closed subgroup of G. 02JP
Corollary 3 Let (Hᵢ)_i∈ I be a decreasing filtered family of compact subgroups of G. In order that G be identified with the projective limit of the groups G/Hᵢ, it is necessary and sufficient… 02JQ
Corollary 4 Let I be a set and let (Hᵢ)_i∈ I be a family of compact groups. The dual of the product group of the Hᵢ is the discrete direct sum group of the groups widehatHᵢ. 02JR
Proposition 19 Let K be a locally compact non-discrete field, not necessarily commutative, and let G be the additive group of K, whose group law is denoted additively. Let χ be a unitary characte… 02JS
Corollary 1 Let K be a locally compact non-discrete field not necessarily commutative and χ a nontrivial unitary character of the additive group of K. Let E be a finite-dimensional topological… 02JT
Corollary 2 a) The group R is in duality with itself relative to the map (x, y)↦ exp(2iπ xy), and the dual measure of Lebesgue measure is Lebesgue measure ; 02JU
Remark In particular one recovers the determination of X(L¹(Z)) given in Example 4 of I, p. 36. 02JV
Corollary 3 Let ngeqslant 1 be an integer. The group Rⁿ is in duality with itself with respect to the mapping (x, y)↦ exp(2iπ x· y), and the dual measure of Lebesgue measure on Rⁿ is Lebesgue… 02JW
Remark Given a subgroup H of Rⁿ, there corresponds to it its orthogonal H^bot, a subgroup of widehatRⁿ=Rⁿ, which is none other than the subgroup associated with H defined in TG, VII, p. 6… 02JX
Corollary 4 The group R^* is in duality with the group -1,1 × R by the mapping (x,(σ , t))↦ σ (x/|x|)|x|ⁱᵗ. The group R^*₊ is in duality with R by the mapping (x, t)↦ xⁱᵗ. 02JY
Proposition 20 The mapping x↦ exp(2iπ λ (x)) is a unitary character of Qₚ whose kernel is Zₚ. 02JZ
Corollary a) The group Qₚ is in duality with itself with respect to the mapping (x, y)↦ exp(2iπ λ (xy)). The normalised Haar measure on Qₚ is then its own dual ; 02K0
Proposition 21 The restriction of the Fourier transform to S(Rⁿ) is an automorphism of topological vector spaces whose inverse is the restriction of the Fourier cotransform. 02K1
Remark 1 More generally, by the corollary to proposition 15 of II, p. 230, this formula is valid for every integrable complex function on Rⁿ such that 02K2
Remark 2 There exist functions f∈ B(R) such that the series ∑_n∈Zf(n) diverges (exercise 4 of II, p. 263). 02K3
Example Let Q be a positive definite quadratic form on Rⁿ. The function defined by φ (x) = exp(-π Q(x)) belong to S(Rⁿ). There exists σ ∈ GL(n,R) such that Q(x) =|σ (x)|² for every x∈ Rⁿ… 02K4
Definition 6 The space of tempered distributions on Rⁿ is, by definition, the dual space of S(Rⁿ) endowed with the topology of bounded convergence. It is denoted by S'(Rⁿ). 02K5
Example A measure ν on Rⁿ is said to be tempered if there exists a positive integer r such that the continuous mapping x↦ (1+|x|)⁻ʳ is ν-integrable on Rⁿ. The restriction of ν to S(Rⁿ) is… 02K6
Definition 7 The transpose of the Fourier transform on S(Rⁿ) (resp. of the inverse Fourier transform) is called the Fourier transform on S'(Rⁿ), and is denoted by F (resp. is called the inverse… 02K7
Proposition 22 Let f be a tempered distribution belonging to M¹(Rⁿ) (resp. to L²(Rⁿ)). The Fourier transform of f in S'(Rⁿ) is the tempered distribution associated with the Fourier transform of f… 02K8
Remark The elementary formulas concerning the Fourier transform of measures remain valid for the Fourier transform of tempered distributions. 02K9
Proposition 23 (Fejér's theorem) Let ngeqslant 1 be an integer. For every h= (hᵢ)∈ Zⁿ, put |h|= supᵢ|hᵢ|. Let f∈ C(Tⁿ). For every integer Ngeqslant 1, let us denote by f_N the function on Tⁿ such that 02KA
Lemma 11 For every Ngeqslant 1, let μ_N be the measure on Tⁿ with density the continuous mapping 02KB
Remark There exist functions f∈ L¹(T) whose Fourier series diverges at every point x∈ T (Kolmogorov's theorem, cf. exercise 51 of II, p. 289). 02KC

§ 2. Classification

Lemma 1 Let H be a locally compact group, and let R be one of the groups R or Z. Let φ be a continuous morphism from R into H. If φ is not a topological isomorphism of R onto a subgroup of… 02M9
Lemma 2 If G is generated by a compact subset V, there exists an integer ngeqslant 0 and a discrete subgroup D of G isomorphic to Zⁿ such that G/D is compact. 02MA
Lemma 3 Let A and B be commutative groups such that A is divisible. Let C be a subgroup of B and φ a morphism of C into A. There exists a morphism of B into A which extends φ. 02MB
Remark In the language of categories, the lemma says that divisible groups are injective objects in the category of commutative groups; cf. A, VII, p. 53, exerc. 3. 02MC
Proposition 1 The following conditions are equivalent: 02MD
Corollary 1 Suppose that G is generated by a compact neighbourhood of e. 02ME
Remark By duality, widehatG is isomorphic to Rᵖ× T^q× D (prop. 3 (iv)) where the subgroups Rᵖ× T^q and T^q, and the integers p and q, are determined in a unique manner. 02MF
Corollary 2 The following conditions are equivalent: 02MG
Remark With the notations of cor. 2, if one identifies G with Rᵖ× T^q× Zʳ× A, the subgroup Rᵖ× T^q is the neutral component of G, the subgroup T^q× A is its greatest compact subgroup and… 02MH
Proposition 2 Suppose G compact. There exists a decreasing filtrant family (Hᵢ)_i∈ I of closed subgroups of G such that 02MI
Corollary If the group G is generated by a compact subset, then it is a projective limit of groups isomorphic to groups of the form Rᵖ× T^q× Zʳ× A, where A is a finite group and p,q,r are po… 02MJ
Proposition 3 a) There exists an integer ngeqslant 0 and a subgroup L such that G is direct product of L and a subgroup isomorphic to Rⁿ, and moreover L admits a compact open subgroup K such tha… 02MK
Proposition 4 Let B_G be the set of elements of G which generate a relatively compact subgroup of G. Then B_G is a closed subgroup of G and B^bot_G is the neutral component of widehatG. 02ML
Corollary 1 Suppose G compact. Then the following conditions are equivalent: 02MM
Corollary 2 Suppose G compact. Then G is totally discontinuous if and only if widehatG is a torsion group. 02MN
Corollary 3 If G is connected, then G is divisible. 02MO

§ 3. Invariant subspaces

Proposition 1 a) Let M be a measurable subset of widehatG. For every x∈ G, the space E_M is stable under the mapping f↦ εₓ*f; 02N3
Lemma 1 The space of functions of L¹(G) whose Fourier transform has compact support is dense in L¹(G). 02N4
Proposition 2 Let I be a closed ideal of L¹(G), and let f∈ L¹(G). If overlineF_G(f) vanishes on a neighbourhood of V(I), then f belongs to I. 02N5
Theorem 1 Let I be a closed ideal of L¹(G) distinct from L¹(G). There exists a character widehatx∈ widehatG such that F_G(f)(widehatx) = 0 for every f∈ I. 02N6
Definition 1 Let g be a complex function on G and let Φ be a filter on G. We say that g is slowly oscillating following Φ if, for every ε >0, there exists a set M∈ Φ and a neighbourhood V of e… 02N7
Corollary 2 Let Φ be a filter on G invariant under translation. Let f∈ L¹(G) such that the Fourier transform of f does not vanish and such that ∫_Gf(x)dx= 1. Let g∈ L^∞(G). Suppose that f*g ha… 02N8
Lemma 2 Let K be a compact subset of G. For every η >0, there exists a function j∈ L¹(G) such that: 02N9
Proposition 3 The algebra L¹(G) satisfies the Ditkin condition (I, p. 92, def. 2). 02NA
Theorem 2 Let I be a closed ideal of L¹(G) such that the boundary of V(I) contains no nonempty perfect set. Then I is the set of functions f∈ L¹(G) such that F(f) vanishes on V(I). 02NB
Corollary If a closed ideal I of L¹(G) is contained in a single regular maximal ideal, then I itself is regular maximal. 02NC
Proposition 4 Let W be a weakly closed vector subspace of L^∞(G), invariant under translation and nonzero. Then W contains at least one character of G. 02ND
Proposition 5 Let W be a weakly closed vector subspace of L^∞(G) invariant under translation. 02NE

III. Compact linear mappings and perturbations

§ 1. Compact linear mappings

Definition 1 Let E be a topological vector space and F a separated topological vector space. A linear mapping u from E into F is said to be compact if there exists a neighbourhood V of 0 in E s… 02NY
Remark 1 A neighbourhood of 0 in E absorbs every bounded subset of E; the image of a bounded subset of E by a compact linear mapping from E into F is therefore a relatively compact subset o… 02NZ
Remark 2 Let E be a seminormed space and let B be the unit ball of E. In order that a linear mapping u from E into a separated topological vector space F be compact, it is necessary and suf… 02O0
Remark 3 Let F be a separated topological vector space. In order that a vector subspace E of F be of finite dimension, it is necessary and sufficient that the canonical injection of E into… 02O1
Remark 4 For every topological vector space E over the field R, we denote by E_(C) the complexified topological vector space of E (EVT, II, p. 65). We identify E with a real topological vec… 02O2
Remark 5 Let E be a topological vector space, F a separated topological vector space. Let F₁ be a closed vector subspace of F and let j be the canonical injection of F₁ into F. In order tha… 02O3
Remark 6 Let (Gᵢ)_i∈ I be a family of separated topological vector spaces, and for every i∈ I, let uᵢ be a compact linear mapping from a topological vector space E into Gᵢ. If the set I is… 02O4
Proposition 1 Let E be a topological vector space and F a separated topological vector space. 02O5
Proposition 2 Let E be a semi-normed space and F a separated topological vector space. Suppose that every closed bounded subset of F is complete. Then the vector subspace Lᶜ(E; F) of L(E; F) is… 02O6
Corollary Let E and F be Banach spaces. The set Lᶜ(E; F) is a closed vector subspace of the Banach space L(E; F). It contains the closure of the space Lᶠ(E; F) of continuous linear mappings… 02O7
Proposition 3 Let E₁, E, F, F₁ be topological vector spaces, F and F₁ being supposed separated. Let v: E₁→ E,u: E→ F and w: F→ F₁ be linear mappings. If v and w are continuous and u is compact… 02O8
Corollary Let E be a Hilbert space. The space Lᶜ(E) is a closed self-adjoint two-sided ideal of L(E). In particular, it is a star-algebra. 02O9
Proposition 4 Let E be a topological vector space, widehatE its separated completion and j the canonical mapping of E into widehatE. Let u be a compact linear mapping of E into a separated topol… 02OA
Proposition 5 Let E be a locally convex space, F a separated topological vector space and u a compact linear mapping of E into F. Then there exist a Banach space G, a continuous linear mapping v… 02OB
Proposition 6 Let E be a locally convex space, F a separated locally convex space, u: E→ F a compact linear mapping and B a bounded subset of E. The restriction of u to B is a continuous mapping… 02OC
Corollary Let (xₙ)_n∈N be a sequence of points of E, which converges to a point x of E for the weakened topology. The sequence (u(xₙ))_n∈N converges in F to u(x). 02OD
Proposition 7 Let E be a semi-normed space, E' its dual, B the unit ball of E' and u a linear mapping of E' into a locally convex separated space F. If the restriction of u to B endowed with the… 02OE
Corollary 1 Let E be a semi-normed space of countable type, let E' be its dual and let B be the unit ball of E'. Let u be a linear mapping of E' into a locally convex separated space F. Suppos… 02OF
Corollary 2 Let E be a semi-normed space and 𝔖 a set of precompact subsets of E whose union is dense in E. Let us denote by E'_b the strong dual of E and by E'_𝔖 the dual of E endowed with the… 02OG
Example Let E be a semi-normed space. The identical mapping of E'_b into the space E', endowed with the weak topology, the topology of compact convergence or the topology of precompact con… 02OH
Proposition 8 Let E be a reflexive Banach space, B its unit ball, F a locally convex separated space and u: E→ F a linear mapping. The following conditions are equivalent : 02OI
Proposition 9 Let E be a locally convex space, F a separated locally convex space and u a continuous linear mapping of E into F. 02OJ
Corollary 1 (Schauder) Let E be a seminormed space, F a Banach space, E' and F' their respective strong duals, and u a continuous linear mapping of E into F. The following properties are equivalent: 02OK
Corollary 2 Let E be a seminormed space and F a Banach space of countable type. Let u be a continuous linear mapping of E into F. Suppose that for every sequence (yₙ) of elements of F' converg… 02OL
Remark Let E and F be separated locally convex spaces. The transpose of a compact linear mapping of E into F is not always a compact linear mapping of F'_b into E'_b(cf. III, p. 108, exer… 02OM
Theorem 1 (Brouwer) Let B be a nonempty compact convex subset of a finite-dimensional normed vector space. Every continuous mapping of B into B has a fixed point. 02ON
Theorem 2 (Leray-Schauder) Let X be a nonempty compact topological space, homeomorphic to a convex subset of a locally convex space. Every continuous mapping of X into X has a fixed point. 02OO
Theorem 3 Let E be a separated locally convex space over C and A a subset of L(E). Make the following assumptions: 02OP
Lemma 1 There exists an element a of A such that the kernel of ha-1_E is not reduced to 0. 02OQ
Corollary 1 Let us retain the assumptions of Theorem 3 and suppose moreover that the elements of A are pairwise permutable. Then E is of dimension 1 over C. 02OR
Corollary 2 (Lomonosov) Let E be a separated locally convex space of dimension at least 2 over the field C and u an endomorphism of E. Suppose that there exists a compact endomorphism hnot = 0 of E permut… 02OS
Corollary 3 Let E be a separated locally convex space over the field C and u a compact endomorphism of E. If E is of dimension at least 2, there exists a closed vector subspace F of E, distinc… 02OT
Definition 2 A locally convex space E is said to have the approximation property, or again to be an approximation space, if the identity mapping 1_E of E is adherent in the space L_pc(E) to the… 02OU
Remark 1 In order that the topological direct sum of locally convex spaces E and F be an approximation space, it is necessary and sufficient that E and F be approximation spaces. 02OV
Remark 2 Let E be a locally convex space. Let N be the closure of 0 in E and P an algebraic supplementary subspace of N in E. Then P is a topological supplementary subspace of N in E, and i… 02OW
Remark 3 Let E be a locally convex space, A an equicontinuous subset of Lᶠ(E) and T a total subset of E. If 1_E is adherent to A for the topology of simple convergence in T, then 1_E is adh… 02OX
Remark 4 Let E be a locally convex space over C. Let us denote by E₀ the locally convex space over R underlying E. In order that E be an approximation space, it is necessary and sufficient… 02OY
Remark 5 Let E be a locally convex space over R. In order that the complexified locally convex space E_(C) of E be an approximation space, it is necessary and sufficient that E be one. This… 02OZ
Proposition 10 Let E be a Hilbert space. Then E is an approximation space. 02P0
Lemma 2 Let E be a locally convex space. Then E is an approximation space if and only if, for every precompact subset C of E and every neighbourhood U of 0 in E, there exist an integer nge… 02P1
Remark 6 There exist Banach spaces which do not possess the approximation property, as was proved by P. Enflo (A counterexample to the approximation problem in Banach spaces, Acta Math. 130… 02P2
Proposition 11 Let E and F be locally convex spaces. Suppose that E is an approximation space. 02P3
Corollary Let E be a separated approximation space and F a locally convex space. Every compact linear mapping of F into E is adherent to Lᶠ(F; E) in the space L(F; E) endowed with the topolo… 02P4
Proposition 12 Let E be a locally convex space, I a set, and for each i∈ I, let Fᵢ be a locally convex space and vᵢ: E→ Fᵢ a continuous linear mapping with dense image. Suppose that for every nei… 02P5
Corollary 1 A dense vector subspace of an approximation space is an approximation space. 02P6
Corollary 2 If the separated completion of a locally convex space E is an approximation space, then E is an approximation space. 02P7
Corollary 3 The product of a family of approximation spaces is an approximation space. 02P8
Corollary 4 Let E be a locally convex space. If every continuous seminorm on E is majorized by a continuous prehilbertian seminorm, then E is an approximation space. 02P9
Lemma 3 Let E be a metrisable locally convex space and (xₙ)_n∈N a sequence of elements of E converging to 0. There exist a sequence (yₙ)_n∈N of elements of E converging to 0 and a sequence… 02PA
Lemma 4 Let E be a Fréchet space and A a compact subset of E. There exists a compact convex balanced subset L of E containing A such that the topologies induced on A by those of E and of E… 02PB
Theorem 4 Let E be a Fréchet space. 02PC
Lemma 5 Let X be a compact topological space, F a finite subset of C(X) and ε >0 a real number. 02PD
Proposition 13 Let X be a locally compact topological space. The Banach space C₀(X) is an approximation space. 02PE
Corollary Every commutative star algebra is an approximation space. 02PF
Remark If E is an infinite-dimensional Hilbert space, the star algebra L(E) does not have the approximation property (A. Szankowski, B(H) does not have the approximation property, Acta Ma… 02PG
Lemma 6 Let X be a locally compact topological space. Let μ be a positive measure on X and p∈ [1,+∞ [. Let F be a finite subset of Lᵖ_K(X, μ) and ε >0 a real number. 02PH
Proposition 14 Let X be a locally compact topological space, μ a positive measure on X, and p∈ [1,+∞ ]. The space Lᵖ_K(X, μ) is an approximation space. 02PI

§ 2. Examples of compact linear mappings

Proposition 1 Let λ = (λᵢ)_i∈ I be a bounded family of elements of C. Let p be an element of [1,+∞ ]. Let E be the Banach space ℓᵖ_C(I). 02QK
Remark 1 The assumption of assertion b) is always valid if I is finite. When the set I is infinite, it means that the family (λᵢ)_i∈ I tends to 0 according to the filter of complements of f… 02QL
Remark 2 It will be seen later (cf. IV, p. 149, theorem 1) that when p= 2, every normal compact endomorphism of the Hilbert space ℓ²_C(I) is of the form u=w∘ v∘ w⁻¹, where v is a diagonal c… 02QM
Proposition 2 The mapping widetildek of L_C^∞(X, μ) into C(Y) is compact. 02QN
Corollary Let X be a compact topological space, Y a locally compact topological space, μ a complex measure on X, and k a continuous mapping of X× Y into C. Then formula (1) defines a compact… 02QO
Remark Let f∈ Lᵖ(X) and g∈ L^q(Y). Since p <+∞ and q <+∞, the functions f and g are moderate, and the function defined by (x, y)↦ f(x)g(y) is (μ⊗ ν)-moderate (cf. INT, V, p. 92, § 8, nᵒ3… 02QP
Lemma 1 Let k∈ N^p,q(X× Y). The function k is locally (μ⊗ ν )-integrable. 02QQ
Proposition 3 a) Let k∈ N^p,q(X× Y). For f∈ Lᵖ(X) and g∈ L^q(Y), the function 02QR
Definition 1 Let k∈ N^p,q(X× Y). One says that the mapping uₖ is the integral operator with kernel k from Lᵖ(X) into L^q'(Y). 02QS
Proposition 4 Let k∈ N^p,q(X× Y) and f∈ Lᵖ(X). The mapping defined on X by x↦ k(x, y)f(x) is μ-integrable for every y outside a locally ν-negligible set, and uₖ(f) coincides with the class of th… 02QT
Proposition 5 The space L^p'(X× Y) is contained in N^p,p(X× Y). The linear mapping from L^p'(X× Y) into L(Lᵖ(X); L^p'(Y)) defined by k↦ uₖ induces, after passing to the quotient, a continuous in… 02QU
Lemma 2 For every k∈ L^p'(X× Y), one has N_p', p'(k) =|k|_p'. 02QV
Proposition 6 a) The space L^p'(X× Y) is contained in L^p',p'(X,Y), and for k∈ L^p'(X× Y), one has N_p', p'(k) =|k|_p'; 02QW
Proposition 7 Let k∈ L^p',q'(X,Y). The linear mapping uₖ of Lᵖ(X) into L^q'(Y) is compact. 02QX
Corollary 1 (Hilbert–Schmidt) Let k∈ L²(X× Y). The linear mapping uₖ is compact from L²(X) into L²(Y). 02QY
Remark In the case p=q= 2, it is generally more convenient to express the characterization of the integral operator uₖ by means of the scalar product: it is the unique mapping of L²(X) in… 02QZ
Corollary 2 Let G be a compact topological group endowed with a Haar measure μ. Let p be a real number such that 1< p <+∞ and q the conjugate exponent of p. Let f∈ L^q(G). Put k(x, y) =f(x⁻¹y)… 02R0
Lemma 3 Let A be the set of multi-indices α ∈ Nⁿ such that |α |leqslant r and let u be the linear mapping f↦ (∂^αf)_α∈ A of Cʳ(U; F) into C(U; F)ᴬ. 02R1
Proposition 8 Suppose that F is finite-dimensional. Let s be an integer such that 0leqslant s < r and V a relatively compact open subset of U. The linear mapping f↦ f|V of Cʳ(U; F) into Cˢ(V; F)… 02R2
Lemma 4 Let U be an open covering of X. The mapping u:f↦ (f|U)_U∈U of Sʳ(X; E) into ∏_U∈USʳ(U; E) is linear, injective, continuous and strict. Its image is closed. 02R3
Proposition 9 The space Sʳ(X; E) is complete. 02R4
Remark Let U be an open covering of X by chart domains c_U= (U, φ_U,R^n_U) such that for every U∈ U, the restriction of E to U is trivializable of type F_U, where F_U is a Banach space. F… 02R5
Proposition 10 Suppose that the vector bundle E is locally of finite rank. Let s be an integer such that 0leqslant s < r and Y a relatively compact open subset of X. The linear mapping f↦ f|Y of… 02R6
Lemma 5 Let X₀ be the underlying real analytic manifold of X and E₀ the complex vector bundle on X₀ underlying E. 02R7
Proposition 11 Suppose that the vector bundle E is locally of finite rank. Let Y be an open relatively compact subset of X. The restriction mapping f↦ f|Y of S^ω(X; E) into S^ω(Y; E) is compact. 02R8
Corollary Let X be a compact complex analytic manifold and E an analytic vector bundle over X, locally of finite rank. The vector space S^ω(X; E) is finite-dimensional. 02R9

§ 3. Fredholm endomorphisms and riesz endomorphisms

Proposition 1 Let E and F be locally convex spaces, E₁ a closed subspace of finite codimension of E and u a continuous linear mapping of E into F. In order that u be a strict morphism with close… 02RE
Corollary 1 Suppose that F is separated. Let v∈ Lᶠ(E; F). If u is a strict morphism with closed image from E into F, then the same is true of u+v. 02RF
Corollary 2 Let T be a separated vector subspace of finite dimension of F. Let π : F→ F/T be the canonical surjection. In order that u be a strict morphism with closed image, it is necessary a… 02RG
Definition 1 Let E and F be locally convex spaces. An element u of L(E; F) is said to be a Fredholm mapping[^1] if it possesses a quasi-inverse. A Fredholm mapping from E into E is called a Fre… 02RH
Remark Let E, F and G be locally convex spaces, u: E→ F and v: F→ G continuous linear mappings. 02RI
Lemma 1 Let E and F be locally convex spaces and let u be a mapping of Fredholm from E into F. The kernel and the cokernel of u are of finite dimension. 02RJ
Proposition 2 Let E and F be separated locally convex spaces and let u be an element of L(E; F). The following properties are equivalent: 02RK
Remark 5 Let E, F be separated locally convex spaces and u: E→ F a Fredholm mapping. If u is bijective, then u is an isomorphism (in fact, u is a strict morphism by Prop. 2, (ii)). 02RL
Proposition 3 Let E and F be locally convex spaces and u∈ L(E; F). Let E₁(resp. F₁) be a closed subspace of finite codimension of E (resp. F). We suppose that u maps E₁ into F₁ and denote by u₁∈… 02RM
Proposition 4 Let E and F be separated locally convex spaces, u: E→ F a Fredholm mapping, and widehatu:widehatE→ widehatF the extension of u to the completions. Then widehatu is a Fredholm mappi… 02RN
Lemma 2 Let E be a vector space and let u be an endomorphism of E which possesses an index (A, V, p. 126). 02RO
Lemma 3 Let E be a vector space and let u be an endomorphism of E which possesses an index. Suppose that the index of u is zero. Let N be the nilespace of u and I its conilespace. 02RP
Remark Suppose there exists an integer ngeqslant 0 such that Ker(uⁿ) = Ker(uⁿ⁺¹) (resp. Im(uⁿ) = Im(uⁿ⁺¹)); one then has Ker(uᵐ) = Ker(uⁿ) (resp. Im(uᵐ) = Im(uⁿ)) for every integer mgeqsl… 02RQ
Definition 2 Let E be a separated locally convex space. A Riesz endomorphism of E means any Fredholm endomorphism of E whose nilspace is finite-dimensional and whose conilspace is of finite cod… 02RR
Proposition 5 Let E be a separated locally convex space. Every Riesz endomorphism of E is of index zero. 02RS
Example 1 Every automorphism of E is a Riesz endomorphism of E. If E is finite-dimensional, every element of L(E) is a Riesz endomorphism of E. 02RT
Example 2 If (Eᵢ)_i∈ I is a finite family of separated locally convex spaces and uᵢ an element of L(Eᵢ) for every i in I, the endomorphism u=⨁_i∈ Iuᵢ of the locally convex space ⨁_i∈ IEᵢ is… 02RU
Example 3 Let E be a separated locally convex space over R, and let E_(C) be its complexification. Let u be an endomorphism of E. In order that u be a Riesz endomorphism, it is necessary and… 02RV
Proposition 6 Let E be a separated locally convex space and let u be a Riesz endomorphism of E. Let N denote the nilspace of u and I its conilspace. 02RW
Proposition 7 Let E be a separated locally convex space and let u be an endomorphism of E. 02RX
Proposition 8 Let E be a separated locally convex space and u an endomorphism of E. The following conditions are equivalent: 02RY
Proposition 9 Let E be a separated locally convex space and let u,v be permutable elements of L(E). The following conditions are equivalent: 02RZ
Proposition 10 Let E and F be locally convex spaces, p∈ L(E; F) and q∈ L(F; E). Put u= 1_E-q∘ p and v= 1_F-p∘ q. 02S0
Lemma 4 Let n∈ N. Put 02S1
Lemma 5 Let u' be an element of L(E). Put v'= 1_F+p∘ u'∘ q. If u' is an inverse of u, then v' is an inverse of v. If u' is a quasi-inverse of u, then v' is a quasi-inverse of v. If u' comm… 02S2
Proposition 11 Let E and F be Fréchet spaces. In order that a continuous linear mapping u of E into F be a Fredholm mapping, it is necessary and sufficient that its kernel and its cokernel be fin… 02S3
Lemma 6 Let E and F be Fréchet spaces and u∈ L(E; F). Suppose that the image of u is of finite codimension in F. Then the image of u is closed in F and u is a strict morphism. 02S4
Proposition 12 Let E and F be Fréchet spaces. Endow their duals E' and F' with the weak topology, the topology of compact convergence, or the topology of bounded convergence. Let u∈ L(E; F). In o… 02S5
Proposition 13 Let E be a Fréchet space and u∈ L(E). 02S6
Proposition 14 Let E be a complex Banach space. In order that an element u of L(E) be a Riesz endomorphism of E, it is necessary and sufficient that one of the following two mutually exclusive co… 02S7
Remark Let E be a real Banach space and let u be an endomorphism of E. Suppose that 0 is an isolated point of the complex spectrum of u, that is to say, of the spectrum of the endomorphis… 02S8

§ 4. Perturbations in Banach spaces

Definition 1 Let E be a topological vector space. A vector subspace of E is said to be direct if it admits a topological supplement. 02SH
Proposition 1 Let E be a locally convex space. Every finite-dimensional vector subspace of E is direct. 02SI
Definition 2 Let E and F be locally convex spaces, and let u be a continuous linear mapping of E into F. One says that u is a direct morphism if u is a strict morphism whose kernel is a direct… 02SJ
Proposition 2 Let E and F be locally convex spaces, let u∈ L(E; F) and v∈ L(F; E). Suppose that u=u∘ v∘ u. Then u is a direct morphism. Moreover, v∘ u is a continuous projector in E with kernel… 02SK
Proposition 3 Let E and F be Banach spaces. The following sets are open subsets of the Banach space L(E; F): 02SL
Theorem 1 Let E and F be Banach spaces. The set F(E; F) of Fredholm mappings of E into F is open in the Banach space L(E; F), and the mapping u↦ ind(u) of F(E; F) into Z is locally constant. 02SM
Proposition 4 Let E and F be Banach spaces, let u₀: E→ F be a Fredholm mapping and v₀ a quasi-inverse of u₀. There exist an open neighbourhood U of u₀ in L(E; F) and an analytic mapping φ : U→ L… 02SN
Lemma 1 Let E be a Banach space. Let p and q be continuous projectors of E such that |q-p|<1. Then p induces an isomorphism of Im(q) onto Im(p), and 1_E-p induces an isomorphism of Ker(q)… 02SO
Theorem 2 Let E be a Banach space. The set R(E) of Riesz endomorphisms of E is open in L(E). The mapping which associates with an element of R(E) the dimension of its nilspace is upper semi-… 02SP
Proposition 5 Let E be a Banach space and u₀ a Riesz endomorphism of E. Let N (resp. I) be the nilspace (resp. the conilspace) of u₀. Let us denote by d the dimension of N. There exist an open n… 02SQ
Remark By definition, in order that one have ((u))>0, it is necessary and sufficient that the bijective linear mapping widetildeu be bicontinuous, that is to say, that u be a strict morph… 02SR
Lemma 2 Let c be a real number. If c <((u)), then for every element z∈ Im(u), there exists an element x of E such that u(x) =z and c|x|leqslant |z|. Conversely, if for every z∈ Im(u) there… 02SS
Proposition 6 Let E and F be normed spaces and u∈ L(E; F). Let B denote the set of elements of E of norm <1. Put 02ST
Proposition 7 Let E be a Banach space and F a normed space. The mapping u↦ ((u)) from L(E; F) into overlineR is upper semi-continuous. 02SU
Proposition 8 Let E be a Banach space, F a normed space. For every u∈ L(E; F), one has ((u)) = ((ᵗu)). 02SV
Theorem 3 Let n be a positive integer, V a real normed vector space of dimension n+ 1 and W a vector subspace of V of dimension n. Let S be the unit sphere of V. There does not exist a conti… 02SW
Theorem 4 (Krein, Krasnoselskii, Milman) Let E be a normed space, and let F and G be vector subspaces of E. Suppose that the dimension of G is finite and strictly less than that of F. There exists an element of F of norm… 02SX
Proposition 9 Let E and F be normed spaces, n∈ N and u a continuous linear mapping from E into F whose kernel is of dimension n. The conorm of u is equal to the distance d from u to the set of m… 02SY
Lemma 3 Let c be a real number such that c >((u)). There exists a continuous linear mapping h: E→ F, of rank 1 and norm < c such that the kernel of u+h contains that of u and is of dimensi… 02SZ
Corollary 1 Let E and F be normed spaces, and let u,v be non-zero continuous linear mappings of E into F whose kernels have the same finite dimension. One then has 02T0
Corollary 2 Let E and F be Banach spaces, and let u and v be non-zero continuous linear mappings of E into F whose images have the same finite codimension in F. Then 02T1
Proposition 10 Let E and F be normed spaces. The set M(E; F) is open in L(E; F). It is the interior of the set of mappings in L(E; F) which are injective. 02T2
Proposition 11 Let E and F be Banach spaces. The set QM(E; F) is open in L(E; F). It is the interior of the set of mappings in L(E; F) whose kernel is of finite dimension. 02T3
Proposition 12 Let E and F be Banach spaces and u an element of QM(E; F). Every v∈ L(E; F) such that |v-u|<((u)) belong to QM(E; F) and satisfy the relations 02T4
Corollary 1 The functions defined by u↦ dim Ker(u) and by u↦ dim Coker(u) on QM(E; F) are upper semi-continuous. The function u↦ ind(u) is locally constant on QM(E; F). 02T5
Corollary 2 The function u↦ dim Coker(u) is locally constant on the set M(E; F) of strict injective morphisms of E into F. 02T6
Lemma 4 Let E and F be Banach spaces. In order that an element u of L(E; F) be a strict morphism of E into F, it is necessary and sufficient that ᵗu be a strict morphism of F' into E'. In… 02T7
Proposition 13 Let E and F be Banach spaces. The sets E(E; F) and QE(E; F) are open in L(E; F). More precisely, if u is an element of QE(E; F) and v an element of L(E; F) such that |v-u|<((u)), t… 02T8
Corollary 1 The functions defined by u↦ dim Ker(u) and by u↦ dim Coker(u) on QE(E; F) are upper semi-continuous. The function u↦ ind(u) is locally constant on QE(E; F). 02T9
Corollary 2 The function defined by u↦ dim Ker(u) is locally constant on E(E; F). 02TA

§ 5. Perturbation by a compact linear mapping

Lemma 1 Let E and F be topological vector spaces, u a continuous linear mapping of E into F and U a neighbourhood of 0 in E. Suppose that u induces a homeomorphism of U onto a closed subse… 02TK
Proposition 1 Let E be a separated locally convex space, F a locally convex space and u a continuous linear mapping of E into F. The following conditions are equivalent: 02TL
Theorem 1 Let E and F be separated locally convex spaces and u a strict morphism of E into F, whose kernel is of finite dimension and whose image is closed. Let h be a compact linear mapping… 02TM
Lemma 2 Let E and F be Fréchet spaces and u∈ L(E; F). The following conditions are equivalent: 02TN
Theorem 2 Let E and F be Fréchet spaces, u: E→ F a continuous linear mapping whose cokernel is of finite dimension, and h: E→ F a compact linear mapping. The linear mapping u+h is a strict m… 02TO
Theorem 3 Let E be a locally convex space, F a separated locally convex space, u a Fredholm mapping from E into F and h a compact linear mapping from E into F. Then u+h is a Fredholm mapping… 02TP
Corollary 1 Let E and F be separated locally convex spaces and u∈ L(E; F). In order that u be a Fredholm mapping, it is necessary and sufficient that there exist a mapping v∈ L(F; E) such that… 02TQ
Corollary 2 Let E and F be separated locally convex spaces and u∈ L(E; F). The following conditions are equivalent: 02TR
Proposition 2 Let u∈ L(E) be such that |1_E-π (u)|<1 in the algebra Calk(E). Then u is a Riesz endomorphism of E. 02TS
Corollary 1 Let E be a separated locally convex space and let u∈ L(E). In order that u be a Riesz endomorphism, it is necessary and sufficient that there exist an element v of L(E) which commu… 02TT
Corollary 2 Let E be a separated locally convex space, u a Riesz endomorphism of E and h∈ L(E) a compact linear mapping. If u and h commute, then u+h is a Riesz endomorphism of E. 02TU
Proposition 3 Let E be a Banach space. Let u be an endomorphism of E, and let A be its commutant in L(E). The closure B of π (A) in the algebra Calk(E) is a Banach algebra. In order that u be a… 02TV
Lemma 3 Let X be a topological space whose topology possesses a countable basis. Every discrete subset of X is countable. 02TW
Theorem 4 Let E be a separated locally convex space and let h be a compact endomorphism of E. The set Σ of the λ ∈ K such that 1_E-λ h is not an automorphism of E is a countable, closed and… 02TX
Proposition 4 (Fredholm Alternative) Let E be a Banach space and let h be a compact endomorphism of E. Let λ be an element of K -0. Let F be the kernel of 1_E-λ h and G the kernel of 1_E'-λᵗh. 02TY
Example Let X be a compact space, μ a measure on X, real or complex according as K is equal to R or C, and N : X× X→ K a continuous function. 02TZ
Lemma 4 The linear mapping from G_λ into E' defined by g↦ g· μ is injective, and its image is the kernel of 1_E'-λᵗh. 02U0
Theorem 5 Let λ ∈ K. 02U1
Corollary The set of the λ ∈ K for which the space F_λ is not zero is a countable, closed and discrete subset of K. 02U2

§ 6. Spectral properties of endomorphisms of Banach spaces

Definition 1 One says that λ is of finite spectral multiplicity for u if the spectral projector e_λ(u) is of finite rank; in this case, the integer m_λ(u) = dim(Im(e_λ(u))) is called the spectr… 02UE
Definition 2 The isolated points of Sp(u) of finite spectral multiplicity are called the sensitive points of the spectrum of u. The set of sensitive points of Sp(u) is called the sensitive spec… 02UF
Proposition 1 The sensitive points of the spectrum of u are eigenvalues of finite spectral multiplicity. 02UG
Proposition 2 Let E be a complete normable space and u an endomorphism of E. Let H be a finite subset of Spₛ(u). The set H is open and closed in Sp(u). The spectral projector e_H(u) is of finite… 02UH
Corollary 1 Let V be a neighbourhood of Spₑ(u). 02UI
Corollary 2 Let f∈ O(Sp(u)) and let μ be a complex number. Let H be the set of λ ∈ Sp(u) such that f(λ ) =μ. Then μ is a point of the discrete spectrum of f(u) if and only if H is finite, none… 02UJ
Example 1 Let E be a finite-dimensional vector space and u an endomorphism of E. The spectrum of u is finite and coincides with Spₛ(u). Its elements are the roots of the characteristic polyn… 02UK
Example 2 Let X be a compact subset of C, and C(X) the Banach space of continuous complex-valued functions on X. Let u denote the endomorphism of C(X) which to f∈ C(X) associates the functio… 02UL
Definition 3 Let E be a complete normable space and u an endomorphism of E. For every n∈ overlineZ, denote by Spₙ(u) the set of complex numbers λ ∈ Spₑ(u) such that u-λ 1_E is a strict morphism… 02UM
Remark Let π denote the canonical homomorphism of L(E) onto the Calkin algebra Calk(E) of E (cf. No. 4 of III, p. 75). The spectrum of π (u) relative to the algebra Calk(E) is sometimes c… 02UN
Theorem 1 Let E be a complete normable space and u an endomorphism of E. 02UO
Corollary a) Let Ω be the unbounded connected component of C- Sp_ω(u). We have Ω ∩ Sp(u)⊂ Spₛ(u); 02UP
Proposition 3 Let E and F be complete normed spaces, u: E→ F and v: F→ E continuous linear mappings. 02UQ
Proposition 4 Let E be a complete normed space, E' the dual space of E and u an endomorphism of E. 02UR
Theorem 2 Let E be a complete normable space, u an endomorphism of E and h a compact endomorphism of E. One has Sp_ω(u+h) = Sp_ω(u) and Spₙ(u+h) = Spₙ(u) for every n∈ overlineZ-0. 02US
Corollary Suppose that the unbounded connected component of the complement of Sp_ω(u) contains 0. Then u+h is a Riesz endomorphism of E. 02UT
Lemma 1 Let E be a separated topological vector space of dimension geqslant 2 over R, and let X be a denumerable subset of E. The complementary set E - X is connected. 02UU
Lemma 2 Let S⊂ C be an infinite, discrete, bounded and closed set in C-0. Then S is the set of values of a sequence (λₙ)_n∈N of non-zero complex numbers, pairwise distinct, such that the s… 02UV
Proposition 5 Let E be a complete normable space and let u be a compact endomorphism of E. Every element of Spₛ(u) is an eigenvalue of finite spectral multiplicity. Moreover: 02UW
Proposition 6 Let E be a complete normed space over C and u a compact endomorphism of E. 02UX
Corollary Let E and F be Hilbert spaces and let u be a continuous linear mapping from E into F. The linear mapping u is compact if and only if the endomorphism |u| of E is compact. 02UY
Proposition 7 Let u∈ L(E). 02UZ
Corollary Let u∈ L(E). If u is normal, then Sp₀(u) is empty, and if u is hermitian, then Spₑ(u) coincides with the spectrum of π (u) relative to the algebra Calk(E). 02V0
Theorem 3 (Weyl) Let u∈ L(E) be a normal endomorphism of u. The essential spectrum of u is the intersection of the sets Sp(u+h), where h ranges over Lᶜ(E). 02V1
Lemma 3 Let (aₙ)_ngeqslant 0 be a sequence of positive real numbers such that the power series 02V2
Lemma 4 Let E be a real Banach space and u∈ L(E_(C)) a nonzero endomorphism of E_(C). Let C be a total convex cone in E. There exists x∈ C such that u(x)not = 0. 02V3
Theorem 4 (Krein–Rutman) Let E be a complete normed space over R. Let C be a closed total salient convex cone in E and let u∈ L(E) be a compact linear mapping such that u(C)⊂ C. If the spectral radius varr… 02V4
Corollary Let E be a complete normable space over R. Let C be a total salient closed convex cone in E and u∈ L(E) a compact linear mapping such that u(C)⊂ C. If the spectral radius varrho (u… 02V5
Remark The assumption varrho (u)>0 cannot in general be omitted in the Krein-Rutman theorem. For example, let V be the endomorphism of the Banach space C([0,1]) defined by 02V6
Proposition 8 Let E be a nonzero complete normable space over R. Let C be a closed salient convex cone with nonempty interior mathringC in E, and let u∈ L(E) be a compact linear mapping such tha… 02V7
Lemma 5 Let E be a real Banach space, C a convex cone in E, and u∈ L(E) such that u(C-0)⊂ mathringC. Let ℓ ∈ C^∘ be an eigenvector of ᵗu. Then the kernel of ℓ is stable under u and does no… 02V8
Lemma 6 Let E be a finite-dimensional real vector space and B a compact convex neighbourhood of 0 in E. Let u be an endomorphism of E such that u(B)⊂ mathringB. Then the complex spectrum o… 02V9
Corollary Let E be a nonzero complete normable space over R. Let C be a closed salient convex cone with nonempty interior mathringC in E, and let u be a compact endomorphism of E such that u… 02VA
Lemma 7 We have u_A(Rⁿ₊)⊂ R₊ⁿ if and only if a_i,jgeqslant 0 for all i and j, and u_A(Rⁿ₊-0)⊂ (R₊^*)ⁿ if and only if a_i,j>0 for all i and j. 02VB
Theorem 5 (Perron–Frobenius) a) If a_i,jgeqslant 0 for all i and j in 1, . . . , n, then the real number varrho is an eigenvalue of A; 02VC
Remark Assume that there exists an integer kgeqslant 1 such that all the coefficients of Aᵏ are >0 (such a matrix is sometimes called primitive). Then, by the cor. to Prop. 8, the spectra… 02VD

IV. Hilbertian spectral theory

§ 1. Compact operators on a Hilbert space

Definition 1 Let B = (eᵢ)_i∈ I be an orthonormal basis of E. An endomorphism u is said to be diagonal in the basis B or diagonal relative to B if there exists a family λ = (λᵢ)_i∈ I of elements… 02W8
Lemma 1 Let u∈ D_B(E) and λ = (λᵢ)_i∈ I the family of its eigenvalues. The family λ is bounded and one has supᵢ|λᵢ|=|u|. 02W9
Proposition 1 a) The mapping α from D_B(E) into B(I) which associates with a diagonal endomorphism u the family of the eigenvalues of u is an isometric isomorphism of Banach algebras over K. If… 02WA
Remark The Banach algebra D_B(E) is a maximal commutative closed subalgebra of L(E). Indeed, let u be an endomorphism of E which commutes with D_B(E). Let i∈ I. Since the orthogonal proje… 02WB
Proposition 2 Let u∈ D_B(E) and λ = (λᵢ)_i∈ I be the family of its eigenvalues. 02WC
Remark When I is infinite, the condition λ ∈ C₀(I; K) may also be stated as follows: “the family λ tends to 0 according to the filter of the complements of the finite subsets of I.” 02WD
Theorem 1 Suppose K =C. Let u be a compact and normal endomorphism of E. There exists an orthonormal basis B of E such that u is diagonal in the basis B. 02WE
Corollary 1 Let u be a compact hermitian endomorphism of E. There exists an orthonormal basis B of E such that u is diagonal in the basis B and its eigenvalues are real. 02WF
Corollary 2 Let F be a Hilbert space and let u be a compact continuous linear mapping from E into F. There exists a countable set I, an orthonormal basis (eᵢ)_i∈ I of the initial space Ker(u)^… 02WG
Definition 2 With the notations of the corollary, the family (αᵢ)_i∈ I is the family of singular values of u, relative to the orthonormal basis (eᵢ)_i∈ I of the initial space of u. 02WH
Remark 1 This corollary generalizes Theorem 2 of EVT, V, p. 54, which corresponds to Hilbert–Schmidt mappings. 02WI
Remark 2 With the notations of the corollary, one has the formula 02WJ
Remark 3 Let u be a positive compact endomorphism of E. Let B = (fᵢ)_i∈ I be an orthonormal basis of E such that u is diagonal in the basis B (Theorem 1), and let (λᵢ)_i∈ I be the family of… 02WK
Proposition 3 Let u be a compact positive endomorphism of E. There exists an orthonormal family (eₙ)_n∈ I in E such that, for every x∈ E, one has 02WL
Proposition 4 Let u be a positive compact endomorphism of E. Let f∈ C(R₊) be a continuous increasing mapping such that f(0) = 0. 02WM
Proposition 5 Let n∈ I. 02WN
Proposition 6 For every n∈ I, one has 02WO
Proposition 7 Let u and v be positive compact endomorphisms of E. 02WP
Proposition 8 Let u be a positive compact endomorphism of E. Let H be a closed subspace of E and i_H: H→ E the canonical injection. Denote by u_H the endomorphism i^*_Hui_H of H. It is compact a… 02WQ
Definition 3 Let F be a Hilbert space and let u be a compact linear mapping of E into F. 02WR
Proposition 9 Let F be a Hilbert space and let u be a compact linear mapping of E into F. 02WS
Corollary Let F be a Hilbert space and u a compact linear mapping of E into F. Let w∈ L(E) and v∈ L(F). For every n∈ I_E, one has αₙ(w∘ u∘ v)leqslant |v| |w|αₙ(u). 02WT
Remark 1 Since the sequence (αₙ(u))_n∈ I_E is decreasing, the set J is either equal to I_E, or equal to a segment 0, . . . , m in I_E where m∈ N. This latter case holds if and only if u is… 02WU
Remark 2 As |u(x)|=||u|(x)| for every x∈ E (I, p. 139, prop. 10), one has αₙ(u) =αₙ(|u|) for every n∈ I_E. 02WV
Remark 3 If u is positive, then αₙ(u) =λₙ(u) for every n∈ I_E (indeed, one then has αₙ(u)²=λₙ(u^*u) =λₙ(u²) =λₙ(u)² by Prop. 4 of IV, p. 153). 02WW
Remark 4 It is possible that αₙ(v∘ u∘ w) = 0 even if αₙ(u) is nonzero; in this case, αₙ(v∘ u∘ w) is not a singular value of v∘ u∘ w. 02WX
Lemma 2 Let n∈ N. The mapping widehat∧ⁿi_H is an isometric linear mapping of widehat∧ⁿH into widehat∧ⁿE. 02WY
Lemma 3 Let F be a Hilbert space and u∈ L(E; F). Let n∈ N. 02WZ
Proposition 10 One has the equality 02X0
Corollary Let G be a Hilbert space and v∈ Lᶜ(F; G). Then 02X1
Lemma 4 Let A be a ring. Let n∈ N and let (aᵢ)_0leqslant ileqslant n and (bᵢ)_0leqslant ileqslant n be families of elements of A. For 0leqslant jleqslant n, put 02X2
Lemma 5 Let I⊂ overlineR be an interval not containing +∞, and let f be an increasing convex function from I into [-∞ ,+∞ [. 02X3
Proposition 11 (Weyl Inequalities) Let G be a Hilbert space and v∈ Lᶜ(F; G). Let g:R₊→ [-∞ ,+∞ [ be an increasing function such that the function g∘ exp is convex. We have 02X4
Lemma 6 Let g and h be convex functions defined on intervals I and J of R, respectively. If g is increasing and defined on the image of h, then the function g∘ h is convex on J. 02X5
Corollary Let G be a Hilbert space and v∈ Lᶜ(F; G). Let n∈ I_E∩ I_F. 02X6
Proposition 12 Let B = (eᵢ)_i∈ I be an orthonormal basis of E. Let u∈ D_B(E) and let us denote by λ = (λᵢ)_i∈ I the family of its eigenvalues. The endomorphism u is of finite trace if and only if… 02X7
Corollary 1 Let u be an endomorphism of finite trace of E. 02X8
Corollary 2 Let F be a Hilbert space. Let u∈ L(E; F) be a Hilbert-Schmidt mapping. Then u is a compact linear mapping. 02X9
Corollary 3 Suppose that K =C. Let u be a positive compact endomorphism of E. The endomorphism u is of finite trace if and only if the decreasing family of its eigenvalues (λₙ(u))_n∈ I_E is su… 02XA
Corollary 4 Let F be a Hilbert space. Let u∈ L(E; F) be a compact linear mapping. Let B = (eᵢ)_i∈ I be an orthonormal basis of the initial space Ker(u)^∘ of u, C = (fᵢ)_i∈ I an orthonormal fam… 02XB
Lemma 7 Let λ = (λᵢ)_i∈ I be a family of complex numbers. For each t∈ C^*, let nₜ be the cardinal of the set of i∈ I such that λᵢ=t. The family (λᵢ)_i∈ I is summable if and only if nₜ is f… 02XC
Proposition 13 Suppose that K =C. Let u be a compact normal endomorphism of E. For t∈ Spₛ(u), let nₜgeqslant 1 be the spectral multiplicity of the eigenvalue t of u. In order that u be of finite… 02XD
Proposition 14 Let u∈ L₂(E; F). Then 02XE
Definition 4 One says that the vector space L₁(E; F) endowed with the norm u↦ |u|₁ is the space of nuclear mappings of E into F. If u∈ L₁(E; F), one says that u is nuclear. 02XF
Remark 1 Suppose that K =R. Let u∈ L(E; F). One has u∈ L₁(E; F) if and only if u_(C)∈ L₁(E_(C); F_(C)); in this case, one has |u|₁=|u_(C)|₁. 02XG
Remark 2 Let u be a nuclear mapping of E into F. Since the mapping u is of Hilbert–Schmidt type, it is compact (cor. 2 of IV, p. 165). Moreover, Proposition 14 implies that for every v∈ L₂(… 02XH
Remark 3 Suppose that K =C. Let u∈ L₁(E; E) be such that u is positive. The norm |u|₁ of u is the sum of the sequence (λₙ(u))_n∈ I_E (cor. 3 of IV, p. 165). 02XI
Remark 4 The canonical inclusion of L₁(E; F) in L(E; F) is of norm leqslant 1 (inequality (9), p. 167). 02XJ
Proposition 15 Let G be a Hilbert space over K. The mapping (u, v)↦ v∘ u of L(E; F)× L(F; G) into L(E; G) defines by passing to the subspaces a continuous bilinear mapping of norm leqslant 1 of L… 02XK
Lemma 8 The mapping u↦ u^* of L(E; F) into L(F; E) defines by passing to the subspaces an isometric linear mapping of L₁(E; F) into L₁(F; E). 02XL
Proposition 16 Let E₁ and F₁ be Hilbert spaces. Let u be in L₁(E; F),v be in L(E₁; E) and v₂ be in L(F; F₁). Then v₂uv₁∈ L₁(E₁; F₁) and |v₂uv₁|₁leqslant |v₂| |v₁| |u|₁. 02XM
Proposition 17 The space L₁(E; E) coincides with the space L₁(E) of endomorphisms of finite trace of E, and one has |Tr(u)|leqslant |u|₁ for every endomorphism u of E of finite trace. 02XN
Lemma 9 Let u∈ L₁(E; F). We have 02XO
Proposition 18 The mapping θ is an isometric isomorphism. 02XP
Corollary The normed space L₁(E; F) is a Banach space. 02XQ
Lemma 10 The mapping θ defines by passing to quotients a linear mapping widetildeθ from L²(X)⊗ L²(Y) into L²(X× Y), and there exists a unique isometric isomorphism from L²(X)widehat⊗₂L²(Y)… 02XR
Proposition 19 The mapping N↦ u_N is an isometric isomorphism of L²(X× Y) onto the space L₂(L²(X); L²(Y)) of mappings 02XS
Corollary For every N∈ L²(X× Y), the linear mapping u_N is a Hilbert–Schmidt mapping and one has Tr(u^*_Nu_N) =|N|². 02XT
Remark Let N∈ L²(X× Y). By corollary 2 of IV, p. 150, there exist a countable set I, orthonormal families (fᵢ)_i∈ I in L²(X) and (gᵢ)_i∈ I in L²(Y), as well as a family (αᵢ)_i∈ I in R^*₊… 02XU
Proposition 20 Let (fₙ)_n∈N be a sequence of measurable mappings of X into a metrisable space F. Suppose that the limit of fₙ(x) exists in the complement of a μ-negligible subset of X. There exis… 02XV
Proposition 21 Suppose that u_N is of finite trace. Then there exist a set widetildeX⊂ X whose complement X -widetildeX is μ-negligible and a function H∈ L²(X× X) satisfying the following conditi… 02XW
Theorem 2 Suppose that u_N is of finite trace. Then the function x↦ N(x, x) belongs to L¹(X) and we have 02XX
Lemma 11 If the endomorphism u_N of L²(X) is positive, then N(x, x)geqslant 0 for every x in the support of μ. 02XY
Remark The converse assertion of the lemma is not valid (exercise 14 of IV, p. 316). 02XZ
Proposition 22 Suppose that u_N is a positive endomorphism of L²(X). Then the endomorphism u_N is of finite trace if and only if the function x↦ N(x, x) is μ-integrable. In that case, one has 02Y0
Remark Even when X is compact and N continuous, the endomorphism u_N of L²(X) is not always of finite trace (cf. exercise 8 of IV, p. 314). 02Y1

§ 2. Normal endomorphisms

Proposition 1 Let X be a locally compact topological space. Let μ be a positive measure on X and let p∈ [1,+∞ [. Let (Xᵢ)_i∈ I be a locally countable family (INT, IV, p. 190, § 5, nᵒ9, def. 7) o… 02YO
Proposition 2 Let X be a locally compact space countable at infinity. Let μ be a positive measure on X. For all p∈ [1,+∞ [, the space Lᵖ(X, μ) is of countable type. 02YP
Definition 1 Let Y be a topological space. For every measurable function g from X into Y, the μ-essential image of g is the subset of the y∈ Y such that, for every open neighbourhood U of y, th… 02YQ
Lemma 1 Let g be a μ-measurable function from X into a topological space Y, and S its μ-essential image. The set of elements x∈ X such that g(x) does not belong to S is locally μ-negligibl… 02YR
Lemma 2 Let g be a continuous function from X into C. The μ-essential image of g is the closure of the image under g of the support of μ. 02YS
Lemma 3 Let g be a continuous function from X into C and S its μ-essential image. Suppose that S is not empty and that 0∉ S; denote by δ the distance from 0 to S. Then δ >0. 02YT
Lemma 4 Let Y and Z be locally compact topological spaces, f: X→ Y and g: Y→ Z universally measurable mappings. The mapping g∘ f: X→ Z is universally measurable. 02YU
Proposition 3 a) The set Lᵤ(X) is an involutive subalgebra of the involutive algebra of functions from X into C; 02YV
Lemma 5 Let μ be a positive measure on X. Let g be a μ-measurable function from X into C, and let S be its μ-essential image. For every function f∈ Lᵤ(S), the mapping h from X into C such… 02YW
Remark Under the hypotheses of the lemma, let us denote by abuse of language the function h defined in the lemma by f∘ g. 02YX
Lemma 6 Let g∈ L^∞(X, μ). The spectrum of the class of g in L^∞(X, μ) is equal to the μ-essential image of g. 02YY
Proposition 4 Let g∈ L^∞(X, μ). Denote by widetildeg the class of g in L^∞(X, μ) and by S the spectrum of widetildeg. The mapping f↦ f∘ g (remark, p. 184) is the functional calculus mapping of C… 02YZ
Lemma 7 Let g∈ L^∞(X, μ). The endomorphism widetildem_g of Lᵖ(X, μ) is injective if and only if the set of x∈ X such that g(x) = 0 is locally μ-negligible. 02Z0
Proposition 5 Let p∈ [1,+∞ [. The map m:g↦ widetildem_g of L^∞(X, μ) into L(Lᵖ(X, μ)) is an isometric morphism of unital Banach algebras. 02Z1
Lemma 8 Let (gₙ) be a bounded sequence in L^∞(X, μ) converging pointwise μ-almost everywhere. Let widetildem∈ L^∞(X, μ) be the class of its limit. Then widetildem_gₙ converges to widetilde… 02Z2
Proposition 6 The mapping m:g↦ widetildem_g is an isometric unital morphism of the involutive algebra L^∞(X, μ) into the involutive algebra L(L²(X, μ)). 02Z3
Corollary Let g∈ L^∞(X, μ). 02Z4
Proposition 7 The image of the morphism widetildem from L^∞(X, μ) into L(L²(X, μ)) is a maximal commutative subalgebra of the algebra L(L²(X, μ)). 02Z5
Corollary Let u be an endomorphism of the Hilbert space L²(X, μ) commuting with widetildem_g for every function g∈ K(X). Then there exists a unique element f∈ L^∞(X, μ) such that u=widetilde… 02Z6
Remark In the sequel, we shall often denote simply by m_g the endomorphism of multiplication by g on Lᵖ(X, μ). 02Z7
Lemma 9 Let A be a commutative unital stellar subalgebra of L(E). For every x and y in E, the mapping μ of X(A) into C defined by μ(f) =⟨ x|H_A(f)y⟩ for every f∈ C(X(A)) is a bounded measu… 02Z8
Definition 2 Let A be a commutative unital stellar subalgebra of L(E). For every x and y in E, the linear form f↦ ⟨ x|H_A(f)y⟩ on C(X(A)) is called the spectral measure of (x, y) relative to A… 02Z9
Remark Let A be a commutative unital stellar subalgebra of L(E). For every x and y in E, let μ_x,y denote the spectral measure of (x, y) relative to A. The mapping defined by (x, y)↦ μ_x… 02ZA
Definition 3 Let A be an algebra over C and E a complex topological vector space. Let π be a representation of A in E. An element x of E is said to be a cyclic vector for π if the set of elemen… 02ZB
Proposition 8 Let E be a complex Hilbert space. Let A be a commutative unital stellar subalgebra of L(E). Let x be an element of E. Put Eₓ= A· x and let μₓ denote the spectral measure of x relat… 02ZC
Corollary Let E be a complex Hilbert space. Let A be a commutative unital stellar subalgebra of L(E) admitting a cyclic vector x. Let μₓ be the spectral measure of x relative to A. There exi… 02ZD
Proposition 9 There exists a subset C of E such that E is the Hilbert sum of the spaces Eₓ for x∈ C. If E is of countable type, the set C is countable. 02ZE
Theorem 1 Let A be a commutative unital stellar subalgebra of L(E). There exist a locally compact topological space X, a positive measure μ on X, an isometric isomorphism θ of L²(X, μ) onto… 02ZF
Corollary (Spectral Theorem) Let E be a complex Hilbert space. Let u∈ L(E) be a normal endomorphism of E. There exist a locally compact topological space X, a positive measure μ on X, an isometric isomorphism… 02ZG
Proposition 10 Let A be a unital involutive algebra. Let U be a relatively compact open subset of C. Let Ωₙ denote the set of normal elements of A such that Sp_A(a) is contained in U. The mapping… 02ZH
Corollary 1 Let A be a unital star algebra and let Aₙ be the set of normal elements of A. Endow the space C(C) with the topology of compact convergence. The mapping (f, a)↦ f(a) of C(C)× Aₙ in… 02ZI
Corollary 2 Let E be a complex Hilbert space and let L(E)ₙ be the set of normal endomorphisms of E. Endow the space C(C) with the topology of compact convergence. The mapping of C(C)× L(E)ₙ in… 02ZJ

§ 3. Distributions and tempered distributions

Lemma 1 Let U be an open subset of Rⁿ. 030C
Proposition 1 Let U be an open subset of Rⁿ and k∈ N. Let E, F and G be topological vector spaces and let b: E× F→ G be a continuous bilinear mapping. Let f and g be functions of class Cᵏ from U… 030D
Proposition 2 Let I be an interval of R and f a mapping from X× I into E such that 030E
Corollary 1 Let U be an open set of Rⁿ. Let k∈ N and f be a mapping of X× U into E satisfying the following conditions: 030F
Corollary 2 Let k be a natural number. Let f∈ L¹(Rⁿ, μ) and g∈ Cᵏ(Rⁿ). Suppose that for every α ∈ Nⁿ such that |α |leqslant k, the function ∂^αg is bounded. Then the convolution product f*g be… 030G
Proposition 3 Let N be a norm on Rⁿ,r a real number and p∈ [1,+∞ [. 030H
Remark One defines analogously the space D_R(U) of real-valued test functions in U. The linear mapping such that z⊗ φ ↦ zφ for every z∈ C and every φ ∈ D_R(U) is an isomorphism of C⊗ D_R(… 030I
Lemma 2 Let U be an open covering of U. There exists a locally finite open covering of U finer than U and formed of relatively compact sets. 030J
Lemma 3 Let f∈ K(Rⁿ) and let V be an open neighbourhood of the support K of f. There exists an integer m₀geqslant 1 such that, for every x∈ K, the set V contains the ball of radius m⁻¹₀ wi… 030K
Proposition 4 a) The canonical injection of D(U) into K(U) is continuous and D(U) is dense in K(U); 030L
Definition 1 The dual space of D(U), endowed with the topology of bounded convergence, is called the space of distributions on U. It is denoted by D'(U). 030M
Lemma 4 Let f be a linear mapping of D(U) into C. Then f is a distribution if and only if, for every compact subset K of U and every family (M_α)_α∈Nⁿ in R₊, the linear form f is bounded o… 030N
Lemma 5 Let 𝔉 be a filter on D'(U) having a countable basis or containing a simply bounded set. Then 𝔉 converges to a distribution if and only if ⟨ f, φ ⟩ converges according to 𝔉 for ever… 030O
Proposition 5 The presheaf D'(U) is a sheaf. 030P
Proposition 6 Let ν be a measure on U. The restriction of ν to D(U) is a distribution, which is zero if and only if the measure ν is zero. 030Q
Proposition 7 Let p∈ [1,+∞ ]. The injection of Lᵖ(U) into D'(U) is continuous. 030R
Proposition 8 The space D(U) is dense in D'(U). 030S
Definition 2 If f∈ D'(U) is a distribution, ∂^αf is called the iterated partial derivative of order α of f. 030T
Lemma 6 Let k be a natural number. Let f∈ Cᵏ(U) and let λ be the distribution associated with f. For every β ∈ Nⁿ such that |β |leqslant k, the distribution ∂^βλ is the distribution associ… 030U
Proposition 9 (Leibniz Formula) Let f be a distribution on U and g an indefinitely differentiable function on U. Let α ∈ Nⁿ. One has the relation 030V
Remark Let φ ∈ S(Rⁿ). For every k∈ N, one has 030W
Example The function γₙ defined on Rⁿ by γₙ(x) = exp(-|x|²) belongs to S(Rⁿ). In fact, one proves by induction on k that, for every integer k∈ N, there exists a polynomial Pₖ∈ R[X] such th… 030X
Lemma 7 Let k∈ N and α ∈ Nⁿ. For every function φ ∈ S(Rⁿ) and every real number T>0, one has 030Y
Proposition 10 Let B be a bounded subset of S(Rⁿ). The topology induced on B by S(Rⁿ) coincides with the topology induced by C^∞(Rⁿ). 030Z
Corollary Let (φₘ)_m∈N be a bounded sequence in S(Rⁿ). For every function φ ∈ S(Rⁿ), the following assertions are equivalent: 0310
Remark A sequence (φₘ) in C^∞(Rⁿ) converges if and only if, for every α ∈ Nⁿ, the sequence (∂^αφₘ) converges to a function φ^(α) in C(Rⁿ) endowed with the topology of compact convergence… 0311
Proposition 11 The space S(Rⁿ) is a Fréchet space and a Montel space. 0312
Proposition 12 The space D(Rⁿ) is contained in S(Rⁿ), and the inclusion of D(Rⁿ) in S(Rⁿ) is continuous with dense image. 0313
Lemma 8 Let B be a bounded subset of D(Rⁿ). The topology induced on B by the topology of S(Rⁿ) coincides with the topology induced by D(Rⁿ). 0314
Proposition 13 Let p∈ [1,+∞ ]. The space S(Rⁿ) is contained in Lᵖ(Rⁿ) and the canonical injection of S(Rⁿ) into Lᵖ(Rⁿ) is continuous. The image of S(Rⁿ) in Lᵖ(Rⁿ) is dense if pnot = +∞. 0315
Definition 3 A function f:Rⁿ→ C has polynomial growth if there exists an integer kgeqslant 1 such that the mapping defined by x↦ (1 +|x|)⁻ᵏf(x) is bounded on Rⁿ. 0316
Proposition 14 Let f∈ C^∞(Rⁿ). Suppose that for every α in Nⁿ, the function ∂^αf has polynomial growth. The linear mapping of the space S(Rⁿ) into itself defined by φ ↦ f φ is continuous. 0317
Definition 4 The dual space of S(R) endowed with the topology of bounded convergence is called the space of tempered distributions on Rⁿ. It is denoted by S'(Rⁿ). 0318
Lemma 9 A linear mapping f of S(Rⁿ) into C is a tempered distribution if and only if for every family (M_α ,k)_(α ,k)∈Nⁿ×N in R₊, the linear form f is bounded on the set of functions φ ∈ S… 0319
Lemma 10 Let 𝔉 be a filter on S'(Rⁿ) having a countable base or containing a simply bounded set. Then 𝔉 converges to a tempered distribution if and only if ⟨ f, φ ⟩ converges along 𝔉 for ev… 031A
Definition 5 A measure ν on Rⁿ is said to be tempered if there exists an integer r∈ N such that the continuous mapping x↦ (1 +|x|)⁻ʳ is ν-integrable on Rⁿ. 031B
Proposition 15 Let ν be a tempered measure on Rⁿ. The restriction of ν to S(Rⁿ) is a tempered distribution. It is zero if and only if the measure ν is zero. 031C
Proposition 16 Let p∈ [1,+∞ ] and f∈ Lᵖ(Rⁿ). Then the measure f· μ with density f with respect to Lebesgue measure is tempered. The mapping f↦ f· μ of Lᵖ(Rⁿ) into S'(Rⁿ) resulting is continuous. 031D
Proposition 17 The spaces D(Rⁿ) and S(Rⁿ) are dense in S'(Rⁿ). 031E
Lemma 11 Let φ ∈ S(Rⁿ). The function F(φ ) is indefinitely differentiable on Rⁿ and one has 031F
Proposition 18 The restriction to S(Rⁿ) of the Fourier transform is an automorphism of topological vector spaces whose inverse is the restriction of the cotransform of Fourier. 031G
Definition 6 The transpose of the Fourier transform on S(Rⁿ) (resp. of the Fourier cotransform) is called the Fourier transform (resp. Fourier cotransform) on S'(Rⁿ). 031H
Proposition 19 Let f be a tempered distribution associated with a bounded measure ν ∈ M¹(Rⁿ) (resp. with g∈ L²(Rⁿ)). The Fourier transform of f in S'(Rⁿ) is the tempered distribution associated w… 031I
Remark The elementary formulas concerning the Fourier transform of measures remain valid for the Fourier transform of tempered distributions. For example, for f∈ S'(Rⁿ) and α ∈ Nⁿ, one ha… 031J
Proposition 20 The Sobolev spaces W^k,p(U) and W^k,p₀ (U) are Banach spaces of countable type. In particular, the spaces Hᵏ(U) and Hᵏ₀(U) are Hilbert spaces of countable type. 031K
Proposition 21 Let N be the euclidean norm on Rⁿ. Let k be an integer geqslant 0. The Sobolev space Hᵏ(Rⁿ) is the space of f∈ S'(Rⁿ) such that (1 + Nᵏ)F(f) belongs to L²(Rⁿ). 031L

§ 4. Partial operators

Definition 1 Let E and F be vector spaces over K. A partial operator u from E into F is a correspondence (Γ ,E,F) between E and F satisfying the following conditions: 032J
Definition 2 Let E and F be topological vector spaces over K (EVT, I, p. 1, def. 1). Let u∈ P(E; F) be a partial operator from E into F. 032K
Example Let E and F be topological vector spaces over K. 032L
Proposition 1 Let E and F be topological vector spaces over K. A partial operator u of E into F is closable if and only if the closure of the graph Γᵤ of u in E× F is a functional graph. There t… 032M
Definition 3 Let E and F be topological vector spaces over K. Let u be a closable operator of E into F. The closed operator whose graph is Γᵤ is called the closure of u. It is denoted by overli… 032N
Remark Let E and F be topological vector spaces over K. Let u be a closable operator of E into F. The domain of the closure of u is contained in the closure of the domain of u in E. In ge… 032O
Proposition 2 Let K =R and let E and F be topological vector spaces over R. Let u be a partial operator of E into F. Then u has dense domain (resp. is closed, is closable) if and only if u_(C) h… 032P
Lemma 1 Let E, F and G be topological vector spaces over K. Let u be a closed operator from E into F. 032Q
Proposition 3 Let E and F be topological vector spaces over K, F being separated. Let u be a closed partial operator from E into F. The kernel of u is a closed subspace of E. 032R
Definition 4 Let E and F be normed spaces over K and let u be a partial operator from E into F. For x in dom(u), one writes 032S
Remark Let E and F be normed spaces over K and let u be a partial operator from E into F. 032T
Proposition 4 Let E and F be Banach spaces (resp. Hilbert spaces), and let u be a partial operator from E into F. Then u is closed if and only if the normed space Eᵤ is a Banach space (resp. a H… 032U
Lemma 2 Let E, F and G be normed spaces over K, F being a Banach space. Let u be a closed partial operator from E into F and let v∈ L(F; G). If there exists C∈ R₊ such that 032V
Example 1 Let X be a locally compact topological space and let μ be a positive measure on X. Let p₁ and p₂ be fixed elements of [1,+∞ [. 032W
Proposition 5 The multiplication operator widetildem_g from L^p₁_K(X, μ) into L^p₂_K(X, μ) is a closed operator with dense domain. 032X
Proposition 6 a) Let g' be a μ-measurable function on X such that |g|leqslant |g'|. One has dom(widetildem_g')⊂ dom(widetildem_g) and dom(widetildem_g') is a core of the partial operator widetil… 032Y
Example 2 Let E be a Hilbert space over K and B = (eᵢ)_i∈ I an orthonormal basis of E. Let (λᵢ)_i∈ I be a family of elements of K. Let D be the vector subspace of E consisting of the element… 032Z
Example 3 The set N_R=N∪ ∞ , ω is endowed with the total ordering described in VAR, R2, p. 10, such that n <∞ < ω for every n∈ N. Let r∈ N_R. Let n∈ N and let U be an open subset of Rⁿ. Let… 0330
Definition 5 Let E be a vector subspace of D'(U) containing D(U). A differential operator associated with D on E is any partial operator on E which is an extension of the partial operator with… 0331
Definition 6 The partial operator from F into E with domain D^* defined by y↦ u^*(y) is called the adjoint of u. It is denoted by u^*. 0332
Remark Thus y∈ D^* if and only if there exists c∈ R₊ such that |⟨ y|u(x)⟩ |leqslant c|x| for every x∈ D. The element u^*(y) is then characterised by the relation 0333
Proposition 7 Let u be a partial operator with dense domain from E into F. 0334
Proposition 8 Let u be an operator with dense domain from E into F. Then u^* has dense domain if and only if u is closable. When this is so, the closure overlineu of u is equal to u^**, and the… 0335
Corollary If u is a closed partial operator with dense domain from E into F, then u^* has dense domain and one has u^**=u. 0336
Definition 7 Let u be a partial operator on E. One says that u is symmetric if u has dense domain and if u^* is an extension of u. One says that u is self-adjoint if u has dense domain and u^*=… 0337
Remark 1 For an operator u with dense domain on E to be symmetric, it is necessary and sufficient that one have 0338
Remark 2 A self-adjoint partial operator u is essentially self-adjoint (cf. prop. 8). 0339
Remark 3 Let u be a symmetric partial operator on E. The operator u is closable (prop. 7, b)). It satisfies dom(u)⊂ dom(u^*), and u is self-adjoint if and only if dom(u) = dom(u^*). Moreove… 033A
Remark 4 Suppose K =C. Let u∈ P(E; E) be a partial operator with dense domain. The condition ⟨ x|u(x)⟩ ∈ R for every x∈ dom(u) implies that u is symmetric (EVT, V, p. 2, remark); in particu… 033B
Remark 5 Let u and v be symmetric partial operators on E. If u is self-adjoint and if u⊂ v, then v⊂ v^*⊂ u^*=u, hence u=v. 033C
Remark 6 An essentially self-adjoint partial operator u is symmetric, since u⊂ overlineu implies overlineu=overlineu^*⊂ u^*, hence u⊂ u^*. 033D
Remark 7 Let u and v be symmetric partial operators on E. If u+v has dense domain, for example if u or v belong to L (E), then u+v is symmetric. In general, the partial operator u+v is not… 033E
Remark 8 Let u be a symmetric partial operator on E. A real number c is a lower bound of u if and only if the operator u-c· 1_E is positive. 033F
Lemma 3 Suppose that K =R. Let u be a partial operator with dense domain from E into F. 033G
Proposition 9 Let v∈ L(F; E) be an injective continuous linear mapping of F into E whose image is dense in E. The adjoint of v is an injective continuous linear mapping of E into F and one has (… 033H
Proposition 10 (Hellinger–Toeplitz) Let u be a symmetric partial operator on the Hilbert space E. If the domain of u is equal to E, then u∈ L(E) and u is hermitian. 033I
Corollary Let u be a symmetric partial operator on E. If the partial operator u induces a bijective linear mapping of dom(u) onto E, then u is self-adjoint. 033J
Proposition 11 Let u be a symmetric partial operator on E, and let λ ∈ C. If u+λ 1_E and u+λ 1_E are surjective, then u is self-adjoint. 033K
Proposition 12 Let u be a closed operator with dense domain from E into F. The partial operator u^*∘ u on E is self-adjoint and positive. Its domain is a core for u. 033L
Proposition 13 Let u be a partial operator on L²(U). If D₋⊂ u⊂ D₊, then u is closable and (ᵗD)₋⊂ u^*⊂ (ᵗD)₊. 033M
Remark The proposition means that the adjoint of a partial operator u such that D₋⊂ u⊂ D₊ can always be calculated in the sense of distributions: the elements f of the domain of u^* are e… 033N
Lemma 4 Let E be a complex Banach space and let u be a closed injective partial operator on E such that u⁻¹∈ L(E). Let v∈ L(E) be such that |v|<|u⁻¹|⁻¹. Then the partial operator u+v is in… 033O
Definition 8 Let u be a closed operator with dense domain on a complex Banach space E. The resolvent set of u is the set of complex numbers λ such that the partial operator λ 1_E-u is injective… 033P
Remark Let E be a complex Banach space and let u be a closed operator with dense domain on E. 033Q
Proposition 14 Let E be a complex Banach space. Let u be a closed operator with dense domain on E and U its resolvent set. 033R
Proposition 15 Let u be a closed operator with dense domain on a complex Banach space E and λ a complex number belonging to the resolvent set of u. 033S
Proposition 16 Let u be a closed operator with dense domain on a complex Hilbert space E. The spectrum of u^* is the image of the spectrum of u under complex conjugation and, for every element λ… 033T
Corollary Let u be a self-adjoint partial operator on a complex Hilbert space E. If E is not zero, then the spectrum of u is not empty. 033U
Lemma 5 Let E be a complex Hilbert space and let u be a closed partial operator with dense domain on E. Let λ ∈ C. Suppose that there exists a real number c >0 such that 033V
Proposition 17 Let E be a complex Hilbert space and let u be a self-adjoint partial operator on E. 033W
Corollary Let u be a self-adjoint operator on E. 033X
Remark If u is a closed symmetric non-self-adjoint operator, its spectrum is not contained in R(cf. cor. 10 of IV, p. 257 below), and it is possible that the eigenspaces of u relative to… 033Y
Definition 9 Let E be a complex Banach space and let u be a closed partial operator on E. Let ε be a strictly positive real number. The ε-pseudospectrum of u is by definition the union of the s… 033Z
Proposition 18 Let E be a complex Banach space and u a closed partial operator on E. Let ε ∈ R^*₊. The set PSp_ε(u) is an open subset of C. It is nonempty if E is nonzero. 0340
Proposition 19 Let E be a complex Banach space and u a closed partial operator on E. One has PSp_ε(u)⊂ PSp_δ(u) if 0< ε < δ and 0341
Proposition 20 Let E be a complex Banach space and u a closed partial operator on E. Let ε ∈ R^*₊. For every λ ∈ C, the following conditions are equivalent: 0342
Corollary Let E be a complex Banach space and u a closed partial operator on E. Let ε >0. 0343
Proposition 21 Let E be a complex Banach space and u a closed partial operator on E. Let ε ∈ R^*₊. Every bounded connected component of PSp_ε(u) meets the spectrum of u. 0344
Proposition 22 Let g be a μ-measurable function on X. 0345
Proposition 23 Let g be a μ-measurable function on X. The adjoint of the multiplication operator m_g is m_overlineg. 0346
Corollary Let g be a μ-measurable function on X. The multiplication operator m_g on L²(X, μ) is self-adjoint (resp. positive ) if and only if the function g is locally μ-almost everywhere re… 0347
Lemma 6 Let g₁ and g₂ be μ-measurable functions on X. 0348
Proposition 24 Let g be a μ-measurable function on X. 0349
Definition 10 Let u be a closed symmetric operator on a complex Hilbert space E. Let E₊= Ker(u^*-i) and E₋= Ker(u^*+i). The pair (E₊,E₋) of subspaces of dom(u^*) is called the deficiency pair of… 034A
Proposition 25 Let u be a closed symmetric operator on a complex Hilbert space E. We have 034B
Lemma 7 Let u be a closed symmetric operator on a complex Hilbert space E and v a closed symmetric extension of u. The domain of u is a closed subspace of the Hilbert space Eᵥ. 034C
Proposition 26 Let u be a closed symmetric operator on a complex Hilbert space E. Let (E₊,E₋) be the deficiency pair of u. The spaces E₊,E₋ and dom(u) are closed mutually orthogonal subspaces of… 034D
Corollary Let u be a closed symmetric operator on E. Then u is self-adjoint if and only if the deficiency index of u is equal to (0,0). 034E
Definition 11 Let u be a closed symmetric operator on a complex Hilbert space E. A boundary condition for u is a partially isometric linear mapping (TVS, V, p. 41, Def. 3) of Ker(u^*-i) into Ker… 034F
Theorem 1 (von Neumann) Let u be a closed symmetric operator on a complex Hilbert space E and (E₊,E₋) its deficiency pair. 034G
Corollary 1 Let u be a closed symmetric operator on a complex Hilbert space E. The mapping b↦ u_b induces a bijection of the set of isometries of Ker(u^*-i) onto Ker(u^*+i) onto the set of sel… 034H
Corollary 2 Let E be a real Hilbert space. Let u be a closed symmetric operator on E. The partial operator u_(C) on E_(C) is symmetric and closed, and admits at least one self-adjoint extensio… 034I
Corollary 3 Let u be a symmetric operator on E. The following conditions are equivalent: 034J
Example Let U be an open set of Rⁿ endowed with the Lebesgue measure μ. Let Δ be the scalar differential operator 034K

§ 5. Normal partial operators and the spectral theorem

Definition 1 The endomorphism p₂∘ j of E is called the bornification of u; we denote this endomorphism by b(u). 0359
Proposition 1 We have the formulas 035A
Corollary We have b(u)^*=b(u^*). 035B
Lemma 1 The subset Ω(E) is self-adjoint in L(E). 035C
Proposition 2 The mapping u↦ b(u) defines a bijection of the set of closed partial operators with dense domain on E onto the set Ω(E). The inverse bijection is given by v↦ B(v). 035D
Corollary The endomorphism b(u) is hermitian if and only if u is self-adjoint. 035E
Definition 2 Let u be a partial operator on E. One says that u is normal if u is closed with dense domain and if its bornification b(u) is a normal endomorphism of E. 035F
Lemma 2 Let X be a locally compact topological space and let μ be a positive measure on X. 035G
Theorem 1 (“Spectral Theorem”) Let u be a normal partial operator on the Hilbert space E. There exist a locally compact topological space X, a positive measure μ on X, an isometric isomorphism θ of L²(X, μ) onto… 035H
Lemma 3 Let u be a normal partial operator on E. One has 035I
Definition 3 For every function f∈ K (Sp(u)), one defines the endomorphism f(u) of E by f(u) =f_β(b(u)). 035J
Remark Suppose that u is a normal endomorphism of E, so that Sp(u) is a compact subset of C. Since the bounded transform b(u) coincides with β (u) (formula (5)), and Sp(b(u)) =β(Sp(u)) (C… 035K
Definition 4 Let u be a normal partial operator on a complex Hilbert space E. Let x and y be in E. The bounded measure on Sp(u) defined by f↦ ⟨ x|f(u)y⟩ for f∈ K (Sp(u)) is called the spectral… 035L
Remark 1 When u is continuous, this definition coincides with that of Definition 2 of IV, p. 190. 035M
Remark 2 Let j be the canonical inclusion of Sp(u) in C. Since Sp(u) is closed, the mapping j is μ-proper for every bounded measure μ on Sp(u) (INT, V, p. 68, § 6, nᵒ1, Proposition 1). One… 035N
Lemma 4 Let X be a locally compact topological space, let μ be a positive measure on X, and let g be a μ-measurable function on X. Let m_g be the operator of multiplication by g on L²(X, μ… 035O
Example Let n∈ N. We endow Rⁿ with Lebesgue measure, and we identify Rⁿ and its dual group as in corollary 3 of II, p. 236. Let N denote the euclidean norm on Rⁿ and F the Fourier transfor… 035P
Proposition 3 Let f be a universally measurable function defined on the spectrum of u. The set D_f is a dense vector subspace of E. The mapping y↦ ev_y(f) is a normal partial operator on E whose… 035Q
Proposition 4 Let f be a universally measurable function on the spectrum of u. 035R
Definition 5 The mapping f↦ f(u) of Lᵤ(Sp(u)) into the set of normal partial operators on E defined by prop. 3 of IV, p. 271 is called the universally measurable functional calculus mapping ass… 035S
Corollary 1 a) For every f∈ Lᵤ(Sp(u)), the partial operator f(u) is normal and f(u)^*=overlinef(u). Moreover, f(u) is positive if fgeqslant 0, and self-adjoint if f is real-valued; 035T
Remark For f= Id_Sp(u), one has f(u) =u (assertion b) for k= 1). The domain of u therefore coincides with the set of x∈ E such that the identity function of Sp(u) belongs to L²(Sp(u), μₓ)… 035U
Corollary 2 Let u be a normal partial operator on the Hilbert space E. It is supposed that E is nonzero. 035V
Corollary 3 Let u be a partial normal operator on a Hilbert space E. Let f∈ Lᵤ(Sp(u)). For all x and y in E, the spectral measure of (x, y) relative to f(u) is the image measure f(μ), where μ… 035W
Corollary 4 Let g∈ Lᵤ(Sp(f(u))). We have g(f(u)) = (g∘ f)(u). 035X
Proposition 5 Let u be a normal partial operator on a Hilbert space E. 035Y
Corollary Let f and g be in Lᵤ(Sp(u)) and let (x, y)∈ D_f× D_g. The spectral measure of (f(u)x, g(u)y) relative to u is the measure f g· μ_x,y, where μ_x,y is the spectral measure of (x, y)… 035Z
Proposition 6 Let u be a normal partial operator on a Hilbert space E. Let (fₙ)_n∈N be a sequence in Lᵤ(Sp(u)) which converges simply to f∈ Lᵤ(Sp(u)) and such that there exists g∈ Lᵤ(Sp(u)) sati… 0360
Proposition 7 Let X be a locally compact topological space and ν a measure on X. Let g:C× X→ C be a continuous function with compact support. For z∈ C, put 0361
Proposition 8 Let (Aᵢ)_i∈ I be a denumerable family of pairwise disjoint universally measurable subsets of Sp(u), and let pᵢ be the spectral projector of u defined by Aᵢ. The union A of the sets… 0362
Proposition 9 Let A be a closed subset of Sp(u). Let φ_A be the characteristic function of A and p_A=φ_A(u) the spectral projector of u defined by A. Let us denote by E_A the image of p_A. It is… 0363
Corollary Let λ ∈ Sp(u). The image of p_λ is the proper subspace of u relative to λ. 0364
Proposition 10 Let λ ∈ C. One has λ ∈ Sp(u) if and only if, for every open neighbourhood V of λ in C, the spectral projector p_V of u relative to V is nonzero. 0365
Corollary Let A be an open set in C and n∈ N. If A contains n elements of Sp(u), then the dimension of the image of the spectral projector p_A of u relative to A is at least equal to n. 0366
Lemma 5 Let f∈ D(R). There exists a function widetildef in D(C) which coincides with f on R and satisfies 0367
Lemma 6 Let ε >0. The function σ_ε defined on R by 0368
Lemma 7 Let f∈ D(R) and let widetildef∈ D(C) be an almost analytic extension of f. For ε >0, define f_ε on R by 0369
Theorem 2 (Helffer–Sjöstrand) Let u be a self-adjoint partial operator on E. Let f∈ D(R) and widetildef∈ D(C) be an almost-analytic extension of f. Let h be the mapping from C into L(E) defined by 036A
Definition 6 Let T be a locally convex topology on L(E). The T -resolvent topology on A(E) is the least fine topology such that the mappings u↦ R(u, λ ) of A(E) into L(E) endowed with the topol… 036B
Proposition 11 Let T be a locally convex topology on L(E) which is less fine than the Banach-space topology of L(E). 036C
Example 1 Let 𝔖 be a set of bounded parts of E. The 𝔖-topology on L(E) (EVT, III, p. 13) is a locally convex topology less fine than the Banach-space topology of L(E). 036D
Example 2 Let T_b be the Banach-space topology of L(E). For the T_b-resolvent topology, every u∈ A(E) admits a countable fundamental system of neighbourhoods. 036E
Corollary Let Tₛ be the topology of simple convergence on L(E). Let f∈ C_b(R). For every sequence (uₙ)_n∈N in A(E) which converges to u for the Tₛ-resolvent topology, the sequence (f(uₙ))_n∈… 036F
Lemma 8 Let E be a separated topological vector space. Let (E₁,|· |₁) and (E₂,|· |₂) be dense subspaces of E endowed with Hilbert space structures such that the canonical injections of E₁… 036G
Proposition 12 Let u be a closed operator with dense domain from E into a complex Hilbert space F. 036H
Definition 7 Let u be a closed operator with dense domain from E into a complex Hilbert space F. The pair (j,|u|) determined by Prop. 12 is called the polar decomposition of u. 036I
Definition 8 Let D be a vector subspace of E. A partial Hermitian form on E with domain D is a correspondence q= (Γ ,E× E,C) whose domain of definition is D× D, whose graph Γ is functional, and… 036J
Definition 9 Let q be a positive partial Hermitian form on E. We denote by E_q the separated prehilbertian space dom(q) endowed with the scalar product 036K
Proposition 13 Let u be a positive self-adjoint partial operator on E. Let qᵤ be the associated positive partial form. 036L
Remark Let q be a closed positive partial form and q' a continuous positive Hermitian form on E. The positive Hermitian form defined on dom(q) by (x, y)↦ q(x, y) +q'(x, y) is a closed pos… 036M
Proposition 14 Let q be a closed positive partial form with dense domain on E and u the partial operator representing q. 036N
Theorem 3 The mapping which associates with a positive self-adjoint operator u on E the positive partial form qᵤ is a bijection between the set of positive self-adjoint partial operators on… 036O
Corollary Let q be a closed positive partial form on E and u the positive self-adjoint operator representing q. The domain of q is equal to the domain of (1_E+u)^1/2, and we have 036P
Example Let u be a positive partial operator on E which is not necessarily closed. One defines a positive partial form q with domain dom(u) by 036Q
Remark Let c∈ R₊. Let q be a closed partial hermitian form with dense domain such that q(x, x)geqslant -c|x|² for every x belonging to the domain of q. This means that the closed partial… 036R
Proposition 15 Let u be a self-adjoint partial operator on E. Suppose that u is bounded below by c∈ R (Def. 7 of IV, p. 237). Then 036S
Definition 10 Let u be a normal partial operator on E. A complex number λ ∈ Sp(u) belongs to the discrete spectrum of u if λ is isolated in Sp(u) and if λ is an eigenvalue of finite multiplicity… 036T
Lemma 9 Let E be a complex Hilbert space and u a normal partial operator on E. Let λ ∈ Sp(u). Then λ ∈ Spₛ(u) if and only if there exists an open neighbourhood V of λ in C such that the sp… 036U
Lemma 10 We have 036V
Lemma 11 Suppose that E_b is finite-dimensional. Then, for every real number ε > 0, the image of the spectral projector of u defined by [varrhoₑ, varrhoₑ+ε ] is infinite-dimensional. 036W
Proposition 16 The set Sp_b(u)⊂ [c, varrhoₑ[ is countable, discrete and closed in [0, varrhoₑ[. It is the set of values of a strictly increasing sequence (νₖ)_0leqslant k<Card(Sp_b(u)) of positiv… 036X
Lemma 12 Let F∈ Fₙᵘ be a subspace adapted to u. Every eigenvector of u belonging to the space F^∘∩ E_b has an eigenvalue geqslant λₙ(u). 036Y
Proposition 17 a) For every integer n∈ N, one has 036Z
Lemma 13 Let I be an orthonormal family in E. The family I converges weakly to 0 with respect to the filter of complements of finite subsets of I. 0370
Proposition 18 Let u be a self-adjoint partial operator on E and λ a real number. Then λ ∈ Spₑ(u) if and only if there exists an orthonormal sequence (xₙ)_n∈N in E such that u(xₙ)-λ xₙ tends to 0… 0371
Theorem 4 Let u be a self-adjoint partial operator on E. The essential spectrum of u is the intersection of the sets Sp(u+v), where v runs through the set of compact hermitian endomorphisms… 0372
Corollary Let u be a self-adjoint partial operator on E and v a compact hermitian endomorphism of E. Then Spₑ(u+v) = Spₑ(u). 0373
Definition 11 Let u be a partial operator on E. A partial operator v on E is said to be bounded relatively to u if dom(u)⊂ dom(v) and if v defines, by passage to the subspace, a continuous linea… 0374
Remark Let u be a partial operator on E. Every endomorphism v∈ L(E) is bounded relatively to u and its relative norm is zero, since |v(x)|leqslant |v| |x| for every x∈ dom(u), which makes… 0375
Theorem 5 (Kato-Rellich) Let u be a self-adjoint partial operator on E and v a symmetric partial operator bounded relatively to u. If the relative norm |v|ᵤ is <1, then the partial operator u+v with domain… 0376
Proposition 19 Let u be a self-adjoint partial operator on E. The following conditions are equivalent : 0377
Definition 12 Let u be a self-adjoint partial operator on E. One says that u has compact resolvent if the equivalent conditions of prop. 19 are satisfied. 0378
Proposition 20 Let u be a self-adjoint partial operator on E. Then u has compact resolvent if and only if the canonical injection j of the Hilbert space Eᵤ into E is compact. 0379
Corollary Let q be a closed positive partial form on E and u the positive self-adjoint operator representing q. The following conditions are equivalent: 037A
Example Let n∈ N. Let U be an open set in Rⁿ. Endow U with Lebesgue measure, denoted by μ. Let Δ be the scalar differential operator -∑ⁿᵢ₌₁∂ᵢ² on U. The partial operator Δ₋ with domain D(U… 037B

V. Unitary representations

§ 1. Unitary representations

Remark If K =R and if varrho is a continuous linear representation of G in an R-vector space E, then the mapping varrho_(C):g↦ 1⊗ varrho (g) is a continuous linear representation of G in… 038J
Definition 1 Let varrho be a continuous linear representation of a topological group G in a topological vector space E. A continuous linear representation π of G in F is a subrepresentation of… 038K
Lemma 1 Let G and H be topological groups and E a Banach space. Let varrho be a bounded representation of G in E and π a bounded representation of H in E. Suppose that varrho (g) commutes… 038L
Definition 2 A representation varrho of G in a topological K-vector space E is said to be irreducible if 0 is not dense in E and if the only subrepresentations of varrho are varrho and the repr… 038M
Lemma 2 Let π and varrho be continuous linear representations of G in separated topological K-vector spaces. Suppose that π is irreducible. Every nonzero G-morphism from π into varrho is i… 038N
Lemma 3 Let varrho be a continuous linear representation of G in a separated topological K-vector space E of finite dimension. If E is nonzero, then there exists an irreducible subrepresen… 038O
Remark A nonzero representation E of G does not always contain an irreducible subrepresentation (cf. V, p. 426, remark). We shall see, however, that this is the case if G is compact and i… 038P
Definition 3 Let G be a topological group and E a Hilbert space over K. A unitary representation of G in E is a continuous linear representation varrho of G in E such that, for every g in G, th… 038Q
Definition 4 Let G be a topological group, and let varrho and π be unitary representations of G in Hilbert spaces E and F respectively. A G-invariant sesquilinear form on E× F is a continuous s… 038R
Example Let G be a topological group, and let varrho be a unitary representation of G in E. The scalar product q on E is a G-invariant sesquilinear form on E× E. 038S
Lemma 4 Let G be a topological group, and let varrho be a homomorphism of G into the unitary group U(E) of a Hilbert space E. Then varrho is a unitary representation of G if and only if va… 038T
Lemma 5 Let G be a topological group, and let varrho be a unitary representation of G in a complex Hilbert space E. The space Hom_G(varrho , varrho ) is a unital involutive subalgebra of L… 038U
Lemma 6 Let π and varrho be unitary representations of a topological group G in Hilbert spaces E and F respectively. Let D be a dense subspace of E which is stable under π. Let u be a clos… 038V
Proposition 1 Let varrho₁ and varrho₂ be unitary representations of a topological group G in Hilbert spaces E₁ and E₂. The mapping from Hom_G(varrho₁, varrho₂) into Sesq_G(varrho₂, varrho₁) whic… 038W
Proposition 2 Let varrho be a unitary representation of a topological group G in a Hilbert space E. Let u denote the isometric isomorphism of overlineE onto E' which associates to x the linear f… 038X
Corollary Let varrho be a unitary representation of a topological group G in a Hilbert space E. The following conditions are equivalent: 038Y
Proposition 3 Let varrho be a unitary representation of a topological group G in a Hilbert space E. Let π be a subrepresentation of E and F the space of π. The orthogonal complement F^∘ of F in… 038Z
Proposition 4 Let varrho be a unitary representation of a topological group G in a Hilbert space E. A closed subspace F of E is a subrepresentation of varrho if and only if the orthogonal projec… 0390
Lemma 7 The mapping g↦ varrho (g) is a unitary representation of G in E. 0391
Lemma 8 a) The mapping varrho : (g, h)↦ varrho (g, h) is a unitary representation of G× H in E ; 0392
Definition 5 Let G be a topological group and let varrho be a unitary representation of G in a Hilbert space E. Let x and y be elements of E. The function from G into K given by g↦ ⟨ x|varrho (… 0393
Proposition 5 Let G be a topological group. The sets Θ(G) and Υ(G) are unital involutive subalgebras of C_b(G). 0394
Proposition 6 (Schur’s Lemma) Let varrho be a unitary representation of a topological group G in a nonzero Hilbert space E. Then varrho is irreducible if and only if Hom_G(varrho , varrho ) is equal to C· 1_E. 0395
Corollary 1 Let π be an irreducible unitary representation of a topological group G in a Hilbert space E. Let u be a closed partial operator on E. Suppose that u has dense domain, that dom(u)… 0396
Corollary 2 Let varrho and π be irreducible unitary representations of a topological group G in Hilbert spaces E and F respectively. The space Hom_G(varrho , π ) is of dimension 1 if varrho is… 0397
Corollary 3 Let varrho be a unitary representation of a topological group G in a Hilbert space E. The representation varrho is irreducible if and only if the space Sesq_G(varrho , varrho ) is… 0398
Corollary 4 Let varrho₁ and varrho₂ be non-isomorphic irreducible unitary representations of a topological group G in Hilbert spaces E₁ and E₂. The space Sesq_G(varrho₁, varrho₂) is zero. 0399
Corollary 5 Let varrho be a unitary representation of a topological group G in a Hilbert space E. Let π be an irreducible unitary representation of G in a Hilbert space F. 039A
Corollary 6 Let G₁ and G₂ be topological groups. Let varrho₁ be an irreducible unitary representation of G₁ in a Hilbert space E₁ and varrho₂ an irreducible unitary representation of G₂ in a H… 039B
Corollary 7 Let π be an irreducible unitary representation of a topological group G in a Hilbert space E. There exists a continuous homomorphism χ of the center C of G into U such that π (z) =… 039C
Definition 6 Let π be an irreducible unitary representation of a topological group G in a Hilbert space E. The homomorphism χ of the center C of G into U such that π (z) =χ (z)· 1_E for all z i… 039D
Remark Let π (resp. varrho ) be an irreducible unitary representation of a topological group G (resp. H) in a Hilbert space E (resp. F). Let χ (resp. η ) denote the central character of π… 039E
Definition 7 Let varrho be a unitary representation of a topological group G in a Hilbert space E. One says that varrho is semisimple if there exists a family (Fᵢ)_i∈ I of irreducible subrepres… 039F
Proposition 7 Let G be a topological group and let varrho be a unitary representation of G in a complex Hilbert space E. The representation varrho is semisimple if and only if every non-zero sub… 039G
Corollary 1 Let G be a topological group and let varrho be a semisimple unitary representation of G in a complex Hilbert space E. Every subrepresentation of varrho (resp. every quotient repres… 039H
Corollary 2 Every finite-dimensional unitary representation varrho of a topological group G is semisimple. 039I
Corollary 3 Let varrho₁ and varrho₂ be finite-dimensional unitary representations of a topological group G. The representations varrho₁ and varrho₂ are isomorphic if and only if χ_varrho₁=χ_va… 039J
Lemma 9 Let E be a Hilbert space over K. Let F⊂ E be a vector subspace dense in E. Then the Hilbert dimension of E is less than or equal to the dimension of F. 039K
Definition 8 We denote by widehatG the set of classes of irreducible unitary representations of G in a complex Hilbert space. One says that widehatG is the unitary dual of G. 039L
Remark 1 Suppose that G is a commutative group. By corollary 7 of V, p. 390, every irreducible unitary representation of G in a complex Hilbert space is of dimension 1. If G is locally comp… 039M
Remark 2 If G is finite, the set widehatG is in bijection with the set of classes of simple C[G]-modules (A, VIII, p. 47), which is also denoted by widehatG in A, VIII, p. 396. 039N
Remark 3 For π ∈ widehatG, we shall identify overlineπ with the class in widehatG of the conjugate representation of π. 039O
Remark 4 If π is an irreducible unitary representation of G in a finite-dimensional complex Hilbert space, its character χ_π depends only on the class of π. One may therefore speak of the s… 039P
Definition 9 Let G be a topological group and let π be an irreducible continuous representation of G. Let varrho be a continuous representation of G in a separated locally convex space E. The π… 039Q
Proposition 8 Let G be a topological group. Let varrho be a unitary representation of G in a complex Hilbert space E. 039R
Proposition 9 Let G be a topological group and let varrho be a unitary representation of G in a complex Hilbert space E. Let π be an irreducible unitary representation of G in a complex Hilbert… 039S
Proposition 10 Let G be a topological group. Let π be an irreducible unitary representation of G in a complex Hilbert space E_π and let varrho be a unitary representation of G in a complex Hilber… 039T
Corollary Let G be a topological group and varrho a unitary representation of G in a complex Hilbert space E. The Hilbert sum of the π-isotypical components of G for π ∈ widehatG is the grea… 039U
Definition 10 Let G be a topological group. Let varrho be a unitary representation of G in a complex Hilbert space E and π an irreducible unitary representation of G in a complex Hilbert space. 039V
Remark It is possible for a unitary representation varrho to be nonzero, but for all the isotypical components of varrho relative to all the irreducible representations of G to be zero (c… 039W

§ 2. Representations of locally compact groups

Proposition 1 Let H be a locally compact group. Let varrho₁ (resp. varrho₂) be a continuous representation of G (resp. of H) in a separated locally convex K-vector space E₁ (resp. E₂). Let F den… 03A4
Corollary Let varrho be a continuous representation of G in a separated locally convex K-vector space E. The contragredient representation brevevarrho is continuous when E' is endowed with t… 03A5
Remark The contragredient representation is not necessarily continuous when E' is endowed with the strong topology (cf. INT, VIII, p. 191, § 2, exercise 3, d)). One can show that it is so… 03A6
Lemma 1 Let varrho be a unitary representation of G in a Hilbert space E. The mapping ν ↦ varrho (ν ) from M¹(G) into L(E) is a unital morphism of involutive Banach algebras. 03A7
Definition 1 A bounded measure ν ∈ M¹(G) is said to be central if one has ε_g*ν =ν *ε_g for every g∈ G. 03A8
Proposition 2 Let varrho be a unitary representation of G in a complex Hilbert space E. Let 𝔉 be a basis of filter on M_c(G) converging to the measure εₑ for the topology of compact convergence… 03A9
Lemma 2 Let varrho₁ be a nonzero unitary representation of G isomorphic to a subrepresentation of varrho. There exists a measure ν ∈ M such that varrho₁(checkν*ν ) is a nonzero hermitian c… 03AA
Example Suppose that G is unimodular, for example that G is a real semisimple Lie group. Let Γ ⊂ G be a discrete subgroup such that the quotient X = Γ backslash G is compact. The group G a… 03AB
Lemma 3 Let p be a real number geqslant 1. The left (resp. right) regular representation of G in Lᵖ(G, μ) (resp. in Lᵖ(G, μ')) is faithful. 03AC
Remark The biregular representation of G in Lᵖ(G, μ) is not necessarily faithful; its kernel is the image of the center of G under the mapping g↦ (g, g) (exercise 4 of V, p. 487). 03AD
Lemma 4 Let p be a real number geqslant 1. For f∈ L¹(G), the linear mapping γ ⁽ᵖ⁾_G(f) coincides with the endomorphism of Lᵖ(G) defined by φ ↦ f*^μφ. 03AE
Remark Recall that, for every function f on G, one defines the function checkf on G by checkf(g) =f(g⁻¹) for every g∈ G (INT, VII, p. 12, § 1, No.ᵒ1, formula (12)). 03AF
Proposition 3 a) Let (fₙ)_n∈N be a sequence in F_πᵖ(G) such that the series 03AG
Corollary Let (fₙ)_n∈N be a Cauchy sequence in L_πᵖ(G) and let f∈ F_π(G) be such that fₙ(g) converges to f(g) almost everywhere modulo H. Then f∈ L_πᵖ(G) and (fₙ)_n∈N converges to f in L_πᵖ(… 03AH
Proposition 4 Let f∈ L_E¹(G). 03AI
Lemma 5 Let f∈ F_π(G) be a μ-measurable function equal to zero outside C· H such that the function |f| is ν-integrable on G/H. Let s be the E-valued function on G such that 03AJ
Proposition 5 Let p∈ [1,+∞ [. The space L_πᵖ(G) is the space of functions f∈ F_π(G) such that f is μ-measurable and the function |f| belong to Lᵖ(G/H). 03AK
Lemma 6 Let f∈ K_overlineπ(G). For every g∈ G, the function 03AL
Definition 2 One says that the unitary representation of G in the space L²_overlineπ(G) thus defined is the unitary representation of G induced by the representation π of H relative to κ. It is… 03AM
Remark 1 Let varrho be a unitary representation of H in a complex Hilbert space F and let u:π → varrho be an H-morphism. For every function f∈ K_overlineπ(G), let v(f) denote the function g… 03AN
Remark 2 Let κ ': G→ R₊^* be such that 03AO
Lemma 7 We have 03AP
Lemma 8 Let f₁∈ K_χ(G) and f₂∈ L_χ²(G). 03AQ
Definition 3 Let π be an irreducible unitary representation of G in a Hilbert space E. One says that π is square-integrable modulo the center if the function on G/C deduced from the function |f… 03AR
Proposition 6 Let π be an irreducible unitary representation of G in a Hilbert space E. Let χ be the central character of π. Then π is square-integrable modulo the center if and only if there ex… 03AS
Lemma 9 Let π be an irreducible unitary representation of G in a Hilbert space E which is square-integrable modulo the center. Let χ be the restriction to Z of the central character of π… 03AT
Proposition 7 Let π be an irreducible unitary representation of G in a Hilbert space E which is square-integrable modulo the center. Let χ be the restriction to Z of the central character of π. 03AU
Lemma 10 There exists a real number c >0 such that the linear mapping w=c^1/2v is isometric. 03AV
Corollary Let π be an irreducible unitary representation of G and let χ be the restriction to Z of its central character. The representation π is square-integrable modulo the center if and o… 03AW
Definition 4 Let Z be a closed subgroup of C such that C/Z is compact. Let π be an irreducible unitary representation of G which is square-integrable modulo the center. The unique real number c… 03AX
Proposition 8 (Orthogonality relations) Let π be an irreducible unitary representation of G in a Hilbert space E which is square-integrable modulo the center. One has 03AY
Proposition 9 Let π₁ and π₂ be nonisomorphic irreducible unitary representations of G in Hilbert spaces E₁ and E₂. Suppose that π₁ and π₂ are square-integrable modulo the center and that the res… 03AZ
Remark The orthogonality relations of Props. 8 and 9 generalize those of A, VIII, p. 399 (see also the case where G is compact in § 2 of V, p. 457). 03B0
Proposition 10 a) Let M be a measurable subset of widehatG. The space E_M is a subrepresentation of the representation γ _G; 03B1
Remark Let χ be a character of G. If G is not compact, the χ-isotypical component of the regular representation of G in L²(G) is trivial. If G is compact, the χ-isotypical component of th… 03B2
Lemma 11 Let u be a self-adjoint partial operator on E. For t∈ R, put varrho (t) =eⁱᵗᵘ∈ L(E). Then the mapping varrho is a unitary representation of R. 03B3
Lemma 12 Let varrho be a unitary representation of R in E. Let D be the set of elements x of E such that the mapping ψₓ:t↦ varrho (t)x is differentiable at 0. The mapping u of D into E give… 03B4
Definition 5 Let varrho be a unitary representation of R in E. The symmetric partial operator defined in Lemma 12 is called the infinitesimal generator of varrho. 03B5
Theorem 1 (Stone) The mapping σ which to a unitary representation varrho of R in E associates its infinitesimal generator u defines a bijection of the set of unitary representations of R in E onto t… 03B6
Lemma 13 Let varrho be a unitary representation of R in E and let u be its infinitesimal generator. 03B7
Lemma 14 Let u be a self-adjoint partial operator on E and let varrho (t) =eⁱᵗᵘ be the unitary representation of R defined by u. The infinitesimal generator of varrho is equal to u. 03B8
Lemma 15 Let varrho be a unitary representation of R in E. Then the infinitesimal generator u of varrho is self-adjoint and varrho (t) =eⁱᵗᵘ for all t∈ R. 03B9
Corollary Let varrho be a unitary representation of R in a Hilbert space E. There exist a locally compact space X, a positive measure μ on X and a continuous function g on X with real values… 03BA
Remark Suppose that u is an endomorphism of E. The unitary representation of R in E defined by varrho (t) =eⁱᵗᵘ then satisfies the inequality |varrho (t)-1_E|leqslant |t| |u| for every t∈… 03BB
Example Let varrho be the regular representation of R in L²(R). The infinitesimal generator of varrho is the closure of the differential operator with domain D(R) defined by f↦ -if'. 03BC

§ 3. Functions of positive type

Theorem 1 Let f∈ C(X× X). The following conditions are equivalent: 03BX
Definition 1 A function f∈ C(X× X) is said to be a universally positive kernel on X if it satisfies the equivalent conditions of Theorem 1. 03BY
Proposition 1 Let f be a universally positive kernel on X. Let (E₁, g₁) and (E₂, g₂) be Hilbert realizations of f. Suppose that (E₁, g₁) is a cyclic Hilbert realization. There then exists a uniq… 03BZ
Proposition 2 The set Noy₊(X) is a self-adjoint cone in the space C(X× X); it is stable under product and it is closed when C(X× X) is endowed with the topology of simple convergence. 03C0
Proposition 3 Let A be a unital involutive Banach algebra. Let a∈ A. The spectrum of a^* is the image Sp_A(a)^* of Sp_A(a) under complex conjugation. For every f∈ O(Sp_A(a)), one has f(a)^*=f^*(… 03C1
Lemma 1 Let D be the open unit disk in C. There exists a unique holomorphic function f defined on D such that f(z)²= 1-z for every z∈ D and f(0) = 1. One has f^*=f. 03C2
Definition 2 Let A be an involutive algebra. A linear form λ on A is said to be positive if λ (a^*a)∈ R₊ for every a∈ A. 03C3
Lemma 2 Let A be an involutive Banach algebra and λ a positive linear form on A. 03C4
Example Let X be a compact topological space and let A be the star-algebra C(X). Positive linear forms on A are identified with positive measures on X (INT, III, p. 52, § 1, nᵒ6, Theorem 1… 03C5
Lemma 3 Let A be a unital involutive Banach algebra admitting an approximate unit (I, p. 120, Definition 7). Let λ be a continuous positive linear form on A. 03C6
Proposition 4 Let A be an involutive Banach algebra. 03C7
Definition 3 Let A be an involutive Banach algebra. Let λ ∈ A'₊ be a continuous positive linear form on A. A Hilbertian realization of λ is by definition a triple (E, x, φ ) consisting of a Hil… 03C8
Proposition 5 Let A be an involutive Banach algebra admitting an approximate unit. Every continuous positive linear form on A admits a cyclic Hilbertian realization (E, x, φ ). If A is unital, t… 03C9
Proposition 6 Let A be an involutive Banach algebra and let λ be a continuous positive linear form on A. Let (E₁, x₁, φ₁) and (E₂, x₂, φ₂) be Hilbertian realizations of λ. Suppose that (E₁, x₁… 03CA
Remark Retain the notation of the proposition. It is possible that u(x₁) is different from x₂ (exercise 3, b) of V, p. 493). However, the triplet (E₂, u(x₁), φ₂) is also a Hilbert represe… 03CB
Proposition 7 Let A be a star algebra. Every positive linear form λ on A is continuous. 03CC
Proposition 8 Let A be a stellar algebra and a a nonzero element of A. There exists a positive linear form λ ∈ A'₊ such that λ (a^*a)>0. 03CD
Theorem 2 (Gelfand–Naimark) Let A be a stellar algebra. There exist a Hilbert space E and an isometric morphism of involutive algebras φ of A into L(E). If A is unital, there exists such a unital morphism. 03CE
Definition 4 A continuous function φ ∈ C(G) is said to be of positive type on G if the function f defined by f(g, h) =φ (g⁻¹h) on G× G is a universally positive kernel on G. 03CF
Example Let varrho be a unitary representation of G in a Hilbert space E and let x∈ E. Let φ be the diagonal matrix coefficient defined by φ (g) =⟨ x|varrho (g)x⟩ for all g∈ G; the functio… 03CG
Definition 5 Let φ be a function of positive type on G. A Hilbert realization of φ is a pair (varrho , x) where varrho is a unitary representation of G in a Hilbert space E and x∈ E, such that… 03CH
Proposition 9 Let φ ∈ C(G). The following conditions are equivalent: 03CI
Proposition 10 Let φ be a function of positive type on G and let (varrho₁, x₁) and (varrho₂, x₂) be Hilbertian realizations of φ, the Hilbertian representation (varrho₁, x₁) being cyclic. There e… 03CJ
Proposition 11 Let φ ∈ Pos₁(G) and let (varrho , x) be a cyclic Hilbertian realization of φ. Then varrho is an irreducible representation of G if and only if φ is an extreme point (EVT, II, p. 57… 03CK
Lemma 4 Let φ ∈ Pos(G). The function φ is bounded on G and one has |φ |_∞=φ (e). Moreover, one has 03CL
Remark The sets Pos(G) (resp. Pos₁(G) and Pos_leqslant 1(G)) are self-adjoint convex subsets of the involutive algebra C_b(G). They are closed in the space C_b(G) endowed with the topolog… 03CM
Proposition 12 Let E be a complex Hilbert space and widetildeπ an involutive algebra morphism from L¹(G) into L(E). If the representation widetildeπ is non-degenerate, then there exists a unique… 03CN
Proposition 13 Let φ ∈ L^∞(G). Denote by λ_φ the continuous linear form f↦ ⟨ f, φ ⟩ on L¹(G). Then φ is the class of a function of positive type on G if and only if λ_φ is a positive continuous l… 03CO
Corollary In L^∞(G) endowed with the weak topology, the set Pos(G) is closed and the set Pos_leqslant 1(G) is compact. 03CP
Proposition 14 The set of extreme points of Pos_leqslant 1(G) is equal to the union of the set of extreme points of Pos₁(G) and of the zero function. Moreover, the closed convex hull of the set o… 03CQ
Theorem 3 (Raikov) The weak topology on Pos₁(G) coincides with the topology of compact convergence. 03CR
Lemma 5 Let X be a locally compact topological space and ν a positive measure on X. On every bounded subset of C_b(X), the topology induced by the weak topology σ (L^∞(X),L¹(X)) is coarser… 03CS
Lemma 6 Let ψ ∈ L¹(G). Let B be a bounded subset of the Banach space L^∞(G). The mapping φ ↦ ψ *φ of B into C_b(G) is continuous when B is endowed with the weak topology and C_b(G) with th… 03CT
Lemma 7 Let ψ ∈ L¹(G) be such that ψ geqslant 0 and ∫ψ = 1. Let us denote by p the seminorm on C_b(G) defined by p(φ ) =|⟨ φ , ψ ⟩ | for all φ ∈ C_b(G). For every φ ∈ Pos₁(G), one has 03CU
Lemma 8 Let K be a topological field and let E and F be topological vector spaces over K. Let f be a mapping of E into F and X a subset of E. Let x∈ X. The mapping f|X is continuous at x i… 03CV
Theorem 4 (Gelfand–Raikov) For every xnot =e in G, there exists an irreducible unitary representation π of G such that π (x) is not the identity endomorphism of the space of π. 03CW
Corollary For every elements gnot =h in G, there exists an irreducible representation π ∈ widehatG and a matrix coefficient f of π such that f(g)not =f(h). In particular, the subalgebra Υ(G)… 03CX
Lemma 9 The Fourier transform is a continuous mapping of M¹(G) endowed with the weak topology into C_b(widehatG) endowed with the topology induced by the weak topology σ (L^∞(widehatG),L¹(… 03CY
Theorem 5 (Bochner) A continuous function φ on widehatG belongs to Pos(widehatG) if and only if there exists a bounded positive measure ν on G such that φ =F(ν ). 03CZ

§ 4. Representations of compact groups

Proposition 1 Let varrho be a finite-dimensional linear representation of G in a separated topological vector space E. There exists a scalar product q on E such that varrho is a unitary represen… 03DN
Remark We shall see later (cor. 4 of V, p. 466) that every unitary representation of G is semisimple. 03DO
Corollary Let varrho₁ and varrho₂ be finite-dimensional representations of G. The representations varrho₁ and varrho₂ are isomorphic if and only if their characters are equal. 03DP
Lemma 1 Every irreducible unitary representation of G is square-integrable. 03DQ
Proposition 2 Let π be an irreducible unitary representation of G in a Hilbert space E. The dimension of E is finite and equal to the formal degree of π. 03DR
Corollary 1 Let π be an irreducible unitary representation of G in a Hilbert space E. The character of π is a hermitian element of the involutive algebra L¹(G). 03DS
Corollary 2 a) Let π be an irreducible unitary representation of G in a Hilbert space E. One has 03DT
Corollary 3 Let π₁ and π₂ be irreducible representations of G. In the Hilbert space L²(G), one has ⟨ χ_π₁|χ_π₂⟩ = 1 if π₁ and π₂ are isomorphic and ⟨ χ_π₁|χ_π₂⟩ = 0 otherwise. In other words… 03DU
Corollary 4 a) Let varrho be a finite-dimensional continuous representation of G. One has 03DV
Corollary 5 Let π₁ and π₂ be irreducible unitary representations of G. One has π₁(overlineχ_π₂) = 0 if π₁ is not isomorphic to π₂, and π₁(overlineχ_π₁) is multiplication by 1/ dim(π₁). 03DW
Remark When G is finite, Schur’s orthogonality relation (resp. the orthogonality formula for characters) coincides with that of A, VIII, p. 399 (resp. that of A, VIII, p. 400, prop. 4). O… 03DX
Proposition 3 Suppose that G = G₁× G₂ where G₁ and G₂ are compact groups. The mapping b from widehatG₁× widehatG₂ into widehatG which to (π₁, π₂) associates the class of π₁boxtimes π₂ is a bijec… 03DY
Proposition 4 The space Θ(G) is dense in C(G) and in L²(G). 03DZ
Proposition 5 Let π ∈ widehatG. The space varrho _G(π ) is a subrepresentation of varrho _G which is isomorphic to overlineπboxtimes π. In particular, it is an irreducible representation of G× G… 03E0
Theorem 1 (Peter–Weyl) Let G be a compact topological group. The biregular representation varrho _G of G is the Hilbert sum of the subrepresentations varrho _G(π ) for π ∈ widehatG. 03E1
Corollary 1 For every π ∈ widehatG, the subrepresentation varrho _G(π ) is the (overlineπboxtimes π )-isotypical component of varrho _G. 03E2
Corollary 2 Let π₁ and π₂ be non-isomorphic irreducible representations of G. The (overlineπ₁boxtimes π₂)-isotypical component of varrho _G is zero. 03E3
Corollary 3 The right (resp. left) regular representation of G is isomorphic to the Hilbert sum 03E4
Corollary 4 Let π ∈ widehatG. The π-isotypical component of δ _G (resp. the overlineπ-isotypical component of γ _G) is equal to varrho _G(π ). 03E5
Remark 1 If G is finite, these statements correspond to the results of A, VIII, p. 398, remark. 03E6
Remark 2 Suppose that G is commutative. The set widehatG is identified with the dual group of G (V, p. 393, remark). For every χ ∈ widehatG, the space of varrho _G(χ ) is the 1-dimensional… 03E7
Remark 3 If there exists an integer ngeqslant 0 such that G is a compact subgroup of GL(Cⁿ), the identical representation of G in GL(Cⁿ) suffices to separate the points of G, and one can th… 03E8
Remark 4 The Peter–Weyl theorem implies in particular that the natural continuous homomorphism of G into ∏_π∈widehatGU(E_π) is injective. 03E9
Proposition 6 The following subspaces of L²(G) are equal: 03EA
Corollary Let E⊂ L²(G) be a finite-dimensional vector subspace defining a subrepresentation of γ _G(resp. of δ _G, of varrho _G). Then E is contained in C(G). 03EB
Proposition 7 Let varrho be a continuous representation of G in a separated locally convex quasi-complete space E. The sum of the finite-dimensional subrepresentations of E is dense in E. 03EC
Corollary 1 Let π be an irreducible continuous representation of G in a separated quasi-complete locally convex space E. The space E is finite-dimensional. 03ED
Corollary 2 Let varrho be a continuous representation of G in a separated quasi-complete locally convex space E and let π be an irreducible continuous representation of G. 03EE
Corollary 3 Let varrho be a continuous representation of G in a separated quasi-complete locally convex space E. The endomorphism 03EF
Corollary 4 Let varrho be a unitary representation of G in a Hilbert space E. The space E is the Hilbert sum of the π-isotypical components M_π(varrho ) for π ∈ widehatG. In particular, the re… 03EG
Proposition 8 One has 03EH
Lemma 2 The graph of the equivalence relation “ x∈ G and y∈ G and x is conjugate to y ” in G is closed. 03EI
Lemma 3 Let π be an irreducible unitary representation of G. The vector space L²(G^sharp)∩ varrho _G(π ) has dimension one and is generated by the character of π. 03EJ
Proposition 9 The family (χ_π)_π∈widehatG of the characters of the irreducible unitary representations of G is an orthonormal basis of L²(G^sharp). 03EK
Corollary 1 The vector space Θ (G^sharp) is dense in C(G^sharp) and in L²(G^sharp). 03EL
Corollary 2 a) The family of characters of the irreducible representations of G is a basis of Θ (G^sharp); 03EM
Corollary 3 Let H be a locally compact topological group such that G is a compact subgroup of H. Let varrho be a unitary representation of H in a Hilbert space E. If the π-isotypical component… 03EN
Definition 1 For every measure ν ∈ M¹(G), the element (π (ν ))_π∈widehatG of F_b(widehatG) is denoted by F_G(ν ). The map of M¹(G) into F_b(widehatG) thus defined is called the Fourier cotransf… 03EO
Lemma 4 Let π be an irreducible representation of G. The mapping f↦ π (f) defines by passing to the subspaces an isometric isomorphism of varrho _G(overlineπ) into End(E_π). 03EP
Theorem 2 The Fourier cotransformation of G induces by passing to the subspaces an isometric isomorphism of L²(G) onto F²(widehatG). Its inverse F_G associates with an element (u_π)_π∈wideha… 03EQ
Definition 2 The isometric isomorphism F_G of F²(widehatG) onto L²(G) inverse to the isomorphism induced by the Fourier cotransform is called the Fourier transform of G. The image of an element… 03ER
Remark 1 The Fourier transform of (u_π)_π∈widehatG∈ F²(widehatG) is therefore the class in L²(G) of the series 03ES
Remark 2 Let f∈ L²(G). One then has the Plancherel formula 03ET
Proposition 10 The Fourier cotransform is an injective morphism of involutive Banach algebras of M¹(G) into F_b(widehatG) which maps L¹(G) into F₀(widehatG). 03EU
Proposition 11 a) Let u= (u_π)_π∈widehatG be an element of F(widehatG). If the family (F_π(u_π))_π∈widehatG is uniformly summable in C(G), then its sum f is a continuous function on G whose Fouri… 03EV
Definition 3 Let varrho be a finite-dimensional unitary representation of G in a Hilbert space E. Let k be a positive integer. The absolute moment of order 2k of varrho is defined to be, and is… 03EW
Theorem 3 (Larsen’s Alternative) Assume that G is infinite. Let π be a faithful unitary representation of G in a Hilbert space E of finite dimension geqslant 2. 03EX
Lemma 5 Let π be a unitary representation of G in a Hilbert space E of finite dimension. Let (varrhoᵢ)_i∈ I be a family of non-zero unitary representations of G and (nᵢ)_i∈ I a family of i… 03EY
Lemma 6 Let q be a non-degenerate bilinear form on a finite-dimensional complex vector space E of dimension geqslant 3. 03EZ
Lemma 7 Let H₂ be a real Lie group and let H₁ be a closed subgroup of H₂. The complexified Lie algebra of H₁ identifies with a subrepresentation of the adjoint representation of H₁ on the… 03F0
Lemma 8 Let E be a finite-dimensional complex Hilbert space of dimension n. 03F1
Remark 1 The condition "G is infinite" is necessary in Theorem 3 (exercise 9 of V, p. 507). 03F2
Remark 2 Let n∈ N. For every unitary representation varrho of a compact group in a Hilbert space of dimension n, one has M₄(varrho )leqslant n²; equality is possible (exercise 15 of V, p. 5… 03F3
Remark 3 One can prove (cf. R. Guralnick and P.H. Tiep, Decomposition of small tensor powers and Larsen’s conjecture, Representation Theory 9 (2005), 138–208) that the condition that G be i… 03F4