Upper triangular operator matrices and stability of their various spectra
Nikola Sarajlija
TL;DR
This paper studies the defect spectrum $\\mathcal{D}^{\\sigma_*}$ of upper triangular operator matrices $T_n^d(A)$ with diagonal blocks $D_i$, relating it to the spectra of the diagonals across left/right Weyl, Fredholm, and essential notions. The author develops generalized Weyl and Fredholm spectral relations for arbitrary Hilbert spaces by introducing explicit exceptional sets $\\Delta_1,\\Delta_2$ and proving stability criteria when $\\Delta_1\\cup\\Delta_2=\\emptyset$, including dual results for right spectra and separability-based equivalences. Key contributions include corrected generalizations of existing results to nonseparable spaces, a precise hole-filling description for $n=2$, and identification of diagonal-operator classes (e.g., compact or finite rank) that simplify analysis. Overall, the work extends Fredholm theory for block upper triangular matrices and provides robust spectral predictions from diagonal data in broad settings.
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 $\mathcal{D}^{σ_*}=\bigcup\limits_{i=1}^nσ_*(D_i)\setminusσ_*(T_n^d(A))$ , where $σ_*$ 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 $\mathcal{D}^{σ_*}=\emptyset$). 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}.