Proposition 1

a) The mapping $\alpha$ from $\mathscr{D}_B(E)$ into $\mathscr{B}(I)$ which associates with a diagonal endomorphism $u$ the family of the eigenvalues of $u$ is an isometric isomorphism of Banach algebras over K. If $K =\mathbf{C}$, it is a morphism of involutive algebras ;

b) For every bounded family $\lambda = (\lambda_i)_{i\in I}$, the family $(\lambda_ip_i)_{i\in I}$ is summable in the space $\mathscr{L}(E)$ endowed with the topology of simple convergence, and the sum of this family is the unique mapping $u\in \mathscr{D}_B(E)$ such that $\alpha (u) =\lambda$;

c) Let $u\in \mathscr{D}_B(E)$ and let $\lambda$ be the family of its eigenvalues relative to B. The spectrum of $u$ is the closure in K of the set of values of $\lambda$;

d) If K = $\mathbf{C}$, then the continuous functional calculus mapping of $\mathscr{C}$ (Sp($u$)) into $\mathscr{L}(E)$ associates with a continuous function $f\in \mathscr{C}$ (Sp($u$)) the endomorphism $\alpha (f\circ \lambda )$ in $\mathscr{D}_B(E)$.

By Lemma 1, the family of eigenvalues of an endomorphism diagonal in B is bounded and the mapping of $\mathscr{D}_B(E)$ into $\mathscr{B}(I)$ thus defined is a continuous isometric morphism of unital Banach algebras.

Let $i\in I$. For every $j\in I$, one has $\langle u^*(e_i)|e_j\rangle =\lambda_j\langle e_i|e_j\rangle =\langle \overline{\lambda_j}e_i|e_j\rangle$. It follows that $u^*(e_i) =\overline{\lambda}_ie_i$. The adjoint of $u$ is therefore the endomorphism diagonal in the basis B whose family of eigenvalues is $(\lambda_i)_{i\in I}$. Assertion a) follows.

Let $\lambda = (\lambda_i)_{i\in I}\in \mathscr{B}(I)$. For every $x\in E$, the family $(|\langle e_i|x\rangle |^2)_{i\in I}$ is summable, with sum $\|x\|^2$ (EVT, V, p. 22, prop. 5), whence, for every finite subset J of I:

$\|\sum_{i\in J}\lambda_ip_i(x)\|^2=\sum_{i\in J}|\lambda_i|^2|\langle e_i|x\rangle |^2\leqslant ($sup$_{i\in I}|\lambda_i|^2)(\sum_{j\in J}|\langle e_i|x\rangle |^2$

$\leqslant ($sup$_{i\in I}|\lambda_i|)^2\|x\|^2$.

Consequently, the family $(\lambda_ip_i)$ is summable in $\mathscr{L}(E)$ endowed with the topology of simple convergence. Its sum $u_{\lambda}$ satisfies $u_{\lambda}(e_i) =\lambda_ie_i$; it is therefore an endomorphism diagonal in the basis B, with eigenvalues $\lambda$. Assertion b) follows.

The last assertions result from a), from Example 3 of I, p. 17 and from Example 4 of I, p. 111.