Upper triangular operator matrices and stability of their various spectra

TL;DR

This paper investigates the spectra of upper triangular operator matrices, characterizing various invertibility types and providing conditions for spectral stability across arbitrary dimensions and spaces.

Contribution

It generalizes previous results by analyzing spectra stability for upper triangular operator matrices of any size in Hilbert or Banach spaces without separability assumptions.

Findings

Characterizations for different invertibility spectra types.

Sufficient conditions for spectral stability.

Generalization to arbitrary dimensions and spaces.

Abstract

Denote by $T_n^d(A)$ an upper triangular operator matrix of dimension $n$ whose diagonal entries $D_i$ are known, where $A=(A_{ij})_{1\leq i<j\leq n}$ is an unknown tuple of operators. This article is aimed at investigation of defect spectrum $\mathcal{D}^{\sigma_*}=\bigcup\limits_{i=1}^n\sigma_*(D_i)\setminus\sigma_*(T_n^d(A))$ , where $\sigma_*$ is a spectrum corresponding to various types of invertibility: (left, right) invertibility, (left, right) Fredholm invertibility, left Weyl invertibility, right Weyl invertibility. We give characterizations for each of the previous types, and provide some sufficent conditions for the stability of certain spectrum (case $\mathcal{D}^{\sigma_*}=\emptyset$). Our main results hold for an arbitrary dimension $n\geq2$ in arbitrary Hilbert or Banach spaces without assuming separability, thus generalizing results from \cite{WU}, \cite{WU2}. Hence, we complete a trilogy to previous work \cite{SARAJLIJA2}, \cite{SARAJLIJA3} of the same author, whose goal was to explore basic invertibility properties of $T_n^d(A)$ that are studied in Fredholm theory. We also retrieve a result from \cite{BAI} in the case $n=2$, and we provide a precise form of the well known 'filling in holes' result from \cite{HAN}.