Exercise 5, § 1

The notations of the preceding exercise are retained. Let $v\in \mathscr{L}^c(E)$.

a) The formal series det(1$-tv$)$\sum_{k\geqslant 0}t^kv^k$ in $\mathscr{L}(E)[[t]]$ has infinite radius of convergence.

b) Let $\lambda \in K$. The endomorphism $1_E-\lambda v$ of E is invertible if and only if det(1 $-\lambda v$)$\not = 0$.

c) Let $\lambda \in K$ such that det(1 $-\lambda v$) $= 0$. There exist unique closed subspaces $I_{\lambda}$ and $N_{\lambda}$ of E, stable under $v$, such that

(i) The space E is the topological direct sum of $I_{\lambda}$ and $N_{\lambda}$;

(ii) The endomorphism of $I_{\lambda}$ deduced from $1_E-\lambda v$ by passing to the subspaces is invertible ;

(iii) The endomorphism of $N_{\lambda}$ deduced from $1_E-\lambda v$ by passing to the subspaces is nilpotent.

d) Let $h\geqslant 1$ be the order of the zero $\lambda$ of det(1 $-tv$). One has dim(N$_{\lambda}$) $=h$.

The result of this exercise is to be compared with those of no$^o5$ of III, p. 77, which concern Banach spaces over $\mathbf{R}$ or $\mathbf{C}$.

In what follows, K denotes the field $\mathbf{R}$ or $\mathbf{C}$ and all topological vector spaces are over K.