On quasinilpotent operators and 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 $\alpha \in \mathbb{C}$, $\alpha \ne 0$, $\alpha \ne 1$, 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.

References in corpus (1)

• Almost invariant half-spaces for operators on Hilbert space. II: operator matrices

Cited by in corpus (1)

• A conditional proof of the ISP for quasinilpotent operators