Theorem 3

Let E be a separated locally convex space over $\mathbf{C}$ and A a subset of $\mathscr{L}(E)$. Make the following assumptions:

(i) There exists no closed vector subspace of E, distinct from $\{0\}$ and E, which is stable under A;

(ii) The set A contains a nonzero compact endomorphism. Then the commutant of A is reduced to the homotheties.

Replacing A by the unital subalgebra of $\mathscr{L}(E)$ generated by A, one is reduced to the case where A is a unital subalgebra of $\mathscr{L}(E)$. Let then $h$ be a nonzero compact endomorphism of E belonging to A.