Proposition 5

Let E be a complete normable space and let $u$ be a compact endomorphism of E. Every element of Sp$_s(u)$ is an eigenvalue of finite spectral multiplicity. Moreover:

a) If E is finite-dimensional, then Sp($u$) $=$ Sp$_s(u)$;

b) If E is infinite-dimensional, then Sp$_s(u) =$ Sp($u$)$-\{0\}$ and Sp$_{\omega}(u) =\{0\}$;

c) If Sp$_s(u)$ is infinite, it is the set of values of a sequence $(\lambda_n)_{n\in\mathbf{N}}$ of nonzero complex numbers, pairwise distinct, such that the sequence $(|\lambda_n|)_{n\in\mathbf{N}}$ is decreasing and converges to 0 ;

d) If E is infinite-dimensional, then Sp($u$) $=$ Sp$'_{\mathscr{L}^c(E)}(u)$.

Every element of Sp$_s(u)$ is an eigenvalue of finite spectral multiplicity (prop. 1 of III, p. 83).

Assertion a) is elementary (III, p. 85, example 1). Suppose now that E is infinite-dimensional.

Let $\lambda \in$ Sp($u$)$-\{0\}$. Then $u-\lambda 1_E$ is a Riesz endomorphism of E (cor. 2 of prop. 2 of III, p. 75), hence $\lambda$ belongs to Sp$_s(u)$. If E is infinite-dimensional, then Sp$_{\omega}(u)$ is not empty (III, p. 87, th. 1, a)). Necessarily one has Sp$_{\omega}(u) =\{0\}$, whence b).

The set Sp$_s(u)$ is discrete and bounded. Moreover, one has Sp$_s(u) =$ Sp($u$)$\cap (\mathbf{C}-\{0\})$ by b), hence Sp$_s(u)$ is closed in $\mathbf{C}-\{0\}$. Assertion c) therefore follows from Lemma 2.

The spectrum of $u$ is denumerable, and its complement in $\mathbf{C}$ is therefore connected (lemma 1). By the cor. to prop. 6 of I, p. 28, applied to the unital subalgebra $\mathscr{L}^c(E)\oplus \mathbf{C}1_E$ of $\mathscr{L}$ (E), one concludes that Sp($u$) $=$ Sp$'_{\mathscr{L}^c(E)}(u)$.