On quasinilpotent operators and the invariant subspace problem

TL;DR

This paper characterizes when quasinilpotent operators on infinite dimensional Banach spaces have invariant subspaces, linking it to specific rank-one perturbations and showing most such perturbations also have invariant subspaces.

Contribution

It provides a new characterization of invariant subspaces for quasinilpotent operators via rank-one perturbations and scalar parameters.

Findings

Invariant subspace existence is equivalent to certain rank-one perturbations being quasinilpotent.

Almost all perturbations of a fixed rank-one perturbation have invariant subspaces.

The results connect operator perturbations with the invariant subspace problem.

Abstract

We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $\alpha\in\mathbb{C}$, $\alpha\neq 0$, $\alpha\neq 1$, such that $T+F$ and $T+\alpha F$ are also quasinilpotent. We also prove that for any fixed rank-one operator $F$, almost all perturbations $T+\alpha F$ have invariant subspaces of infinite dimension and codimension.