Quasi-nilpotent Operators and Hyperinvariant Subspace Problem

Abstract

In this note, we strengthen some well-known results on the hyperinvariant subspace problem for quasi-nilpotent operators. We show that if T is a quasi-nilpotent quasi-affinity on a Hilbert space and there is a sequence $$\{x_n\}$$ of unit vectors, such that the closure of $$\{x_n : n\in \mathbb {N}\}$$ is compact and zero is not a weak limit of any subsequence of $$\{\frac{T^nT^{*n}x_n}{\Vert T^nT^{*n}x_n\Vert }\}$$ , then T has a nontrivial hyperinvariant subspace.