Almost invariant half-spaces of operators on Banach spaces

TL;DR

This paper investigates a modified invariant subspace problem, proving that many operators, including certain weighted shift operators on Banach spaces, have almost invariant half-spaces.

Contribution

It introduces and affirms the existence of almost invariant half-spaces for a broad class of operators on Banach spaces, extending the invariant subspace theory.

Findings

Almost invariant half-spaces exist for quasinilpotent weighted shift operators.

The problem is affirmatively solved for a large class of operators including those on l_p and c_0.

Provides new insights into the structure of operators on Banach spaces.

Abstract

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on l_p (1 \le p < \infty) or c_0.