The structure of almost-invariant half-spaces for some operators (Q403323)

A closed subspace \(Y\) of an infinite-dimensional complex Banach space \(X\) is called a half-space if \(\dim Y=\infty\) and \(\text{codim} Y=\infty\). This half space is called almost-invariant for a linear bounded operator \(T\) acting on \(X\) if \(TY\subset Y+F\) for some finite-dimensional subspace \(F\subset X\). The minimal possible value of \(\dim F\) for \(F\) in the previous property is called the defect of \(Y\) with respect to \(T\). The authors study the connections between almost-invariant half-spaces and some properties of \(T\). We mention the following two results of the paper: every weakly compact (every quasinilpotent) operator admits an almost-invariant half-space of defect \(\leq1\).