On almost-invariant subspaces and approximate commutation

A closed subspace Y of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T ∈ S there exists a finite-dimensional subspace F T of X such that T Y ⊆ Y + F T . In this paper, we study the existence of almost-invariant subspaces for algebras of operators. We show, in particular, that if a closed algebra of operators on a Hilbert space has a non-trivial almost-invariant subspace then it has a non-trivial invariant subspace. 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 T P − P T has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T = M + F where M ∈ M and F is of finite rank.