ON INVARIANT SUBSPACES FOR POWER-BOUNDED OPERATORS OF CLASS $C_{1\cdot}$

Abstract

We prove that if $T$ is a power-bounded operator of class $C_{*\cdot}$ on a Hilbert space which commutes with a nonzero quasinilpotent operator, then $T$ has a nontrivial invariant subspace. Connections with the questions of convergence of $T^n$ to $0$ in the strong operator topology and of cyclicity of power-bounded operators of class $C_{1\cdot}$ are discussed.