Every operator has almost-invariant subspaces

TL;DR

This paper demonstrates that most operators on certain Banach spaces can be slightly altered to have large invariant subspaces, with specific results for Hilbert and non-reflexive spaces.

Contribution

It introduces new perturbation techniques to ensure the existence of invariant subspaces for a broad class of operators on Banach and Hilbert spaces.

Findings

Operators on reflexive Banach spaces can be perturbed to have infinite-dimensional invariant subspaces.

In non-reflexive spaces, operators with boundary spectrum non-eigenvalues can be similarly perturbed.

Perturbations in Hilbert spaces can be made arbitrarily small in norm, improving previous results.

Abstract

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.