The structure of almost-invariant half-spaces for some operators

Given a Banach space X   and an operator T∈L(X)T∈L(X), an almost-invariant half-space (AIHS) is a closed subspace Y   satisfying TY⊆Y+FTY⊆Y+F for some finite-dimensional subspace F   and dim⁡(Y)=dim⁡(X/Y)=∞dim⁡(Y)=dim⁡(X/Y)=∞. In this paper we study the connection between AIHS and noncompactness of some subsets of the unit ball, chains of T-invariant subspaces, and different parts of spectrum of T   and T⁎T⁎. We describe a simple template for such subspaces and show that if T is quasinilpotent or weakly compact then it admits an AIHS Y.