On the Invertibility of the Sum of Operators

TL;DR

This paper investigates the conditions under which the sum of bounded and unbounded operators is invertible, providing characterizations for normal operators and implications for spectral properties.

Contribution

It offers new insights into the invertibility of operator sums, including a characterization for normal operators and a concise proof of self-adjointness with real spectra.

Findings

Characterization of invertibility for normal operators

Examples illustrating invertibility conditions

Short proof of self-adjointness for operators with real spectrum

Abstract

The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative point, we characterize invertibility for the class of normal operators. Also, we give a very short proof of the self-adjointness of a normal operator when the latter has a real spectrum.