Intersections of various spectra of operator matrices

TL;DR

This paper extends the analysis of spectra of upper triangular operator matrices to general n×n cases, providing new characterizations of spectral perturbations in Banach spaces without separability assumptions.

Contribution

It generalizes spectral perturbation results from 2×2 to n×n operator matrices and extends findings to arbitrary Banach spaces using inner generalized inverses.

Findings

Characterization of spectral perturbations for n×n operator matrices

Extension of results to non-separable Banach spaces

Use of inner generalized inverse as an alternative to adjointness

Abstract

This paper is concerned with general $n\times n$ upper triangular operator matrices with given diagonal entries. We characterize perturbations of the left (right) essential spectrum, the essential spectrum, as well as the left (right) the Weyl spectrum of such operators, thus generalizing and improving some results of \cite{CAO}, \cite{CAO2}, \cite{SUNTHEORY}, \cite{SUN}, \cite{LIandDU}, \cite{ZHANG} from $n=2$ to case of $n\geq2$. We show that related results hold in arbitrary Banach spaces without assuming separability, thus extending and improving recent results from \cite{WU}, \cite{WU2}. For the lack of adjointness and orthogonality in Banach spaces, we use an adequate alternative: the concept of inner generalized inverse.