On Almost-Invariant Subspaces and Approximate Commutation

2012/04/20 by Marcoux, Laurent W., Popov, Alexey I., Radjavi, Heydar
#47A15 #47A46 #47B07 #47L10 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1204.4621

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 ∈ \cS there exists a finite-dimensional subspace \cF_T of \cX such that T \cY ⊆ \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∈\fM and F is of finite rank.

Related

• Almost invariant half-spaces of algebras of operators
• Almost invariant half-spaces of operators on Banach spaces
• Controlling almost-invariant halfspaces in both real and complex settings
• Every operator has almost-invariant subspaces
• Invariant subspaces for commuting operators in a real Banach space