Upper triangular operator matrices and stability of their various spectra

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}.