Proposition 2

Let E be a Banach space and $u$ an endomorphism of E. Let $E_1$ and $E_2$ be closed subspaces of E invariant under $u$ such that $E = E_1\oplus E_2$. Suppose that the endomorphisms $u_1$ and $u_2$ of $E_1$ and $E_2$ induced by $u$ have disjoint spectra $H_1=$ Sp($u_1$) and $H_2$ = Sp($u_2$). Then Sp($u$) $= H_1\cup H_2$ and $e_{H_1}(u)$ is the projector with image $E_1$ and kernel $E_2$. In particular, one has $E_{H_1}= E_1$ and $E_{H_2}= E_2$.

One has Sp($u$) $= H_1\cup H_2($I, p. 128, Lemma 1). Since the sets $H_1$ and $H_2$ are compact, they are open and closed in Sp($u$). For every holomorphic function $f$ in a neighbourhood of Sp($u$), the endomorphism $f(u)$ leaves $E_1$ and $E_2$ invariant and coincides in $E_1$ with $f(u_1)$ and in $E_2$ with $f(u_2)$ (loc. cit.). Take in particular for $f$ the germ of holomorphic function $f_{H_1}$ which is equal to 1 in a neighbourhood of $H_1$ and 0 in a neighbourhood of $H_2$ (cf. No. 12 of I, p. 81); then $f_{H_1}(u_1)$ is the identity mapping of $E_1$ since $f_{H_1}$ = 1 in a neighbourhood of $H_1=$ Sp($u_1$), and $f_{H_1}(u_2)$ is zero since $f_{H_1}= 0$ in a neighbourhood of $H_2$. Hence $e_{H_1}(u) =f_{H_1}(u)$ is the projector with image $E_1$ and kernel $E_2$, and therefore one has $E_1= E_{H_1}$ and $E_2= E_{H_2}$.

Let E be a Banach space and $u$ an endomorphism of E. Let $(H_i)_{i\in I}$ be a finite family of open and closed subsets of Sp($u$), pairwise disjoint, and let H be their union. Relations (15) and (16) of I, p. 81 imply the following assertions :

a) The family of projectors $(e_{H_i}(u))_{i\in I}$ is orthogonal (that is to say, $e_{H_i}(u)e_{H_j}(u) = 0$ for all $(i, j)$ in $I^2$ such that $i\not=j$, cf. A, II, p. 18, Def. 7) and its sum is $e_H(u)$;

b) The vector space $E_H$ is the topological direct sum of the family $(E_{H_i})_{i\in I}$;

c) For every $j\in I$, one has $E_{Sp(u)-H_j}= E_{Sp(u)-H}\oplus \bigoplus_{i\not=j}E_{H_i}$;

d) The vector space $E_{Sp(u)-H}$ is the intersection of the family $(E_{Sp(u)-H_i})_{i\in I}$.

When H = Sp($u$), the decomposition as topological direct sum $\bigoplus_{i\in I}E_{H_i}$ of E is called the spectral decomposition of E associated with $u$ and with the finite partition $(H_i)_{i\in I}$ of Sp($u$).

Let $f$ be an element of $\mathscr{O}$(Sp($u$)). The spectrum of the endomorphism $f(u)$ of E is the image under $f$ of the spectrum of $u($I, p. 75, Prop. 8). Let L be a subset open and closed in Sp($f(u)$). The set H of elements $\lambda \in$ Sp($u$) such that $f(\lambda )$ belongs to L is open and closed in Sp($u$), and one has $e_L(f(u)) =e_H(u)$ since $f_L\circ f=f_H$ in $\mathscr{O}$(Sp($u$)) (loc. cit.).