Theorem 4

Let E be a separated locally convex space and let $h$ be a compact endomorphism of E. The set Σ of the $\lambda \in K$ such that $1_E-\lambda h$ is not an automorphism of E is a countable, closed and discrete subset of the field K.

The set Σ is also the set of $\lambda \in K$ such that $1_E-\lambda h$ is not injective and the set of $\lambda \in K$ such that $1_E-\lambda h$ is not surjective.

There exist a Banach space G, a continuous linear mapping $p: E\rightarrow G$ and a compact linear mapping $q: G\rightarrow E$ such that $h=q\circ p($III, p. 5, prop. 5). The endomorphism $h'=p\circ q$ of G is compact, and for $1_E-\lambda h$ to be an automorphism of E it is necessary and sufficient that $1_G-\lambda h'$ be an automorphism of G (prop. 10 of III, p. 49, e)). It is therefore enough for us to prove the theorem when E is a Banach space, which we henceforth suppose.

Let $\lambda \in K$. Since the index of a Riesz endomorphism is zero, it amounts to the same thing to say that the mapping $1_E-\lambda h$ is an automorphism of E, or that it is injective, or again that it is surjective.

The set Σ is closed in K since the set of automorphisms of E is open in $\mathscr{L}(E)$. Let $\lambda$ be an element of Σ. We have $\lambda \not = 0$. Denote by N the nilspace of $h$ and by I its conilspace, and denote by $h_I$ and $h_N$ the endomorphisms of I and N which $h$ defines by passing to the subspaces. Then $1_I-\lambda h_I$ is an automorphism of I, and there exists a neighbourhood U of $\lambda$ in K such that $1_I-\mu h_I$ is an automorphism of I for every $\mu\in U$. The endomorphism $1_N-\lambda h_N$ of N is nilpotent; for every $\mu\not =\lambda$, we have

therefore $1_N-\mu h_N$ is an automorphism of N. Consequently, the set $U\cap \Sigma$ is reduced to the single element $\lambda$ and the set Σ is discrete. It is countable by the lemma 3.