Exercise 7, § 1

The aim of this exercise is to give a more direct proof of Theorem 1 of IV, p. 149. Let E be a complex infinite-dimensional Hilbert space and $u$ a compact and normal endomorphism of E.

(a) Now prove that, for every nonzero complex number $\lambda$, the eigenspace Ker($u-\lambda 1_E$) is of finite dimension.

(b) Suppose $u$ is hermitian. Now prove that there exists $x_0\not = 0$ in E such that

$\frac{\|u(x_0)\|^2}{\|x_0\|^2}=$ sup$_{x\in E-\{0\}}\frac{\|u(x)\|^2}{\|x\|^2}$.

(c) Now prove that $x_0$ is an eigenvector of $u$ for an eigenvalue $\lambda$ such that $|\lambda |=\|u\|$.

(d) Now prove that there exists a countable set I, an orthonormal family $(e_i)_{i\in I}$ in E and a family $(\lambda_i)_{i\in I}$ of real numbers such that

for all $x\in E$.

(e) Extend the result to the case where $u$ is compact and normal. (Write $u=u_1+iu_2$ where $u_1$ and $u_2$ are compact, hermitian and permutable.)