Exercise 5, § 6

The aim of this exercise is to give another proof of Lomonosov’s theorem (cor. 2 of III, p. 13) in the case where the space E is a Banach space.

Let E be a complex Banach space of dimension at least 2. Let $h$ be a nonzero compact endomorphism of E and let A be the commutant of $h$ in $\mathscr{L}(E)$. a) One of the following conditions is satisfied:

(i) There exists $y\in E$ such that $Ay$ is a nonzero subspace which is not dense in E ;

(ii) For every closed bounded subset C of A, with nonempty interior, there exists a finite subset I of A such that $h(C)$ is contained in the union of the sets $\overset{-1}{u}(C)$ for $u\in I$.

b) Assume that condition (ii) of the preceding question is satisfied. Let C be a closed bounded subset of E with nonempty interior. For every integer $m\geqslant$ 1, there exists a finite sequence $(u_1, . . . , u_m)$ in A such that

(Play table tennis.)

c) Under the same assumption, now prove that the spectral radius of $h$ is not zero (otherwise, consider an element $x_0\in E$ such that $\|h(x_0)\|>\|h\|$ and the closed ball C with center $x_0$ and radius $\|h\|$; noting that $\|(\lambda h)^n\|$ tends to 0 for every $\lambda \in \mathbf{R}$, now prove that the conclusion of the preceding question would imply that 0 belongs to the closure of C).

d) For every endomorphism of E which commutes with a nonzero compact endomorphism of E, there exists a nonzero closed subspace $F\subset E$ different from E such that $u(F)\subset F$.