Almost invariant half-spaces of algebras of operators

TL;DR

This paper investigates the structure of subspaces that are nearly invariant under all operators in a Banach algebra, establishing bounds on error dimensions and conditions for the existence of invariant half-spaces.

Contribution

It proves that norm-closed algebras have uniformly bounded error dimensions and that algebras generated by commuting operators with almost invariant half-spaces possess invariant half-spaces.

Findings

Bounded error dimensions for norm-closed algebras.

Existence of invariant half-spaces in certain algebras.

Conditions linking almost invariance to actual invariance.

Abstract

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY is a subspace of Y+F for some finite-dimensional ``error'' F. In this paper, we study subspaces that are almost invariant under every operator in an algebra A of operators acting on X. We show that if A is norm closed then the dimensions of ``errors'' corresponding to operators in A must be uniformly bounded. Also, if A is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then A has an invariant half-space.