Almost invertible operators

TL;DR

This paper characterizes when a bounded linear operator can be decomposed into an invertible part and a part with a countable spectrum, using advanced spectral set derivatives.

Contribution

It provides a novel spectral condition involving the Cantor-Bendixson derivative for such operator decompositions.

Findings

Operator decomposability characterized by spectral derivatives

Spectral condition involving the $ ext{acc}^{ ext{ω}_1}$ derivative

New criterion for almost invertible operators

Abstract

We prove that a bounded linear operator $T$ is a direct sum of an invertible operator and an operator with at most countable spectrum iff $0\notin\mbox{acc}^{\omega_{1}}\,\sigma(T),$ where $\omega_{1}$ is the smallest uncountable ordinal and $\mbox{acc}^{\omega_{1}}\,\sigma(T)$ is the $\omega_{1}$-th Cantor-Bendixson derivative of $\sigma(T).$