On Almost-Invariant Subspaces and Approximate Commutation

TL;DR

This paper investigates the existence and structure of almost-invariant subspaces for operators on Banach and Hilbert spaces, revealing conditions under which operators decompose into a maximal abelian algebra element plus finite-rank perturbations.

Contribution

It establishes new results on the existence of infinite-dimensional almost-invariant subspaces and characterizes operators nearly commuting with maximal abelian self-adjoint algebras.

Findings

Existence of infinite-dimensional almost-invariant subspaces for various operators.

Operators with finite-rank commutators with projections decompose into algebra elements plus finite-rank operators.

Structural characterization of operators with maximal commuting families of almost-invariant subspaces.

Abstract

A closed subspace of a Banach space $\cX$ is almost-invariant for a collection $\cS$ of bounded linear operators on $\cX$ if for each $T \in \cS$ there exists a finite-dimensional subspace $\cF_T$ of $\cX$ such that $T \cY \subseteq \cY + \cF_T$. In this paper, we study the existence of almost-invariant subspaces of infinite dimension and codimension for various classes of Banach and Hilbert space operators. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if $T$ is an operator on a separable Hilbert space and if $TP-PT$ has finite rank for all projections $P$ in a given maximal abelian self-adjoint algebra $\fM$ then $T=M+F$ where $M\in\fM$ and $F$ is of finite rank.