Almost Invariant Half-spaces of Algebras of Operators

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 ${TY\subseteq Y+F}$ for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra ${\mathfrak A}$ of operators acting on X. We show that if ${\mathfrak A}$ is norm closed then the dimensions of “errors” corresponding to operators in ${\mathfrak A}$ must be uniformly bounded. Also, if ${\mathfrak A}$ 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 ${\mathfrak A}$ has an invariant half-space.