Definition 1

Let $B = (e_i)_{i\in I}$ 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 $\lambda = (\lambda_i)_{i\in I}$ of elements of K such that $u(e_i) =\lambda_ie_i$ for all $i\in I$.

Let $B = (e_i)_{i\in I}$ be an orthonormal basis of E. We denote by $\mathscr{D}_B(E)$ the set of endomorphisms of E which are diagonal relative to B. It is a closed commutative unital subalgebra of $\mathscr{L}(E)$. Let $u\in \mathscr{D}_B(E)$. The family $\lambda = (\lambda_i)_{i\in I}$ such that $u(e_i) =\lambda_ie_i$ is uniquely determined by $u$ and by the basis B, and it is said that this is the family of eigenvalues of $u$ relative to B.

Suppose that $K =\mathbf{R}$ and identify E with a subspace of $E_{(\mathbf{C})}$. Let $B = (e_i)_{i\in I}$ be an orthonormal basis of E. Denote by $B_{(\mathbf{C})}$ the orthonormal basis $(1\otimes e_i)_{i\in I}$ of $E_{(\mathbf{C})}($cf. EVT, V, p. 29, cor. 1). The mapping $u\mapsto u_{(\mathbf{C})}$ induces an injective algebra morphism from $\mathscr{D}_B(E)$ into $\mathscr{D}_{B_{(\mathbf{C})}}(E_{(\mathbf{C})})$; its image is the set of $u\in \mathscr{D}_{B_{(\mathbf{C})}}(E_{(\mathbf{C})})$ such that $u(E)\subset E$, in other words the set of $u$ diagonal in the basis $B_{(\mathbf{C})}$ whose eigenvalues are real.

For every $i\in I$, we denote by $p_i$ the orthogonal projector of E with image $Ke_i$. We have $\|p_i\|= 1$ for every $i\in I$. The endomorphism $p_i$ is diagonal in the basis B, the family of its eigenvalues is the family $(\delta_{ij})_{j\in I}$ (Kronecker symbol, cf. A, II, p. 24).

Let $B'= (f_i)_{i\in I}$ be an orthonormal basis of E and $u: E\rightarrow E$ the isometric isomorphism such that $u(f_i) =e_i$. Then $\mathscr{D}_{B'}(E) =u^{-1}\mathscr{D}_B(E)u$.

We equip I with the discrete topology and denote by $\mathscr{B}(I) =\mathscr{C}_b(I; K)$ the unital Banach algebra of bounded functions on I with values in K (example 2 of I, p. 17) ; if $K =\mathbf{C}$, it is a star algebra (example 2 of I, p. 102).