Controlling almost-invariant halfspaces in both real and complex settings

TL;DR

This paper investigates the existence and control of almost-invariant halfspaces in both real and complex Banach spaces, focusing on finite-dimensional perturbations and extending previous results to real spaces.

Contribution

It extends the theory of AIHS to real Banach spaces and analyzes how finite-dimensional perturbations influence their existence.

Findings

AIHS existence depends on restrictions on perturbations

Results extend to real Banach spaces from complex cases

Provides conditions for controlling AIHS in various settings

Abstract

If $T$ is a bounded linear operator acting on an infinite-dimensional Banach space $X$, we say that a closed subspace $Y$ of $X$ of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under $T$ whenever $TY\subseteq Y+E$ for some finite-dimensional subspace $E$, or, equivalently, $(T+F)Y\subseteq Y$ for some finite-rank perturbation $F:X\to X$. We discuss the existence of AIHS's for various restrictions on $E$ and $F$ when $X$ is a complex Banach space. We also extend some of these and other results in the literature to the setting where $X$ is a real Banach space instead of a complex one.