Almost-invariant and essentially-invariant halfspaces

TL;DR

This paper investigates conditions under which operators on Banach spaces have almost-invariant or essentially-invariant half-spaces, demonstrating that all operators on infinite-dimensional spaces admit such structures and linking common AIHS to invariant half-spaces.

Contribution

It establishes that every operator on an infinite-dimensional complex Banach space admits an essentially-invariant half-space and connects common AIHS to invariant half-spaces in operator algebras.

Findings

Every operator on an infinite-dimensional Banach space has an essentially-invariant half-space.

Operators with a common AIHS in a closed algebra have a common invariant half-space.

Provides sufficient conditions for operators to have almost-invariant half-spaces.

Abstract

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well.