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 FT of X such that TY⊆Y+FT. 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 TP−PT 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.