Spectral analysis of operator matrices: limit point insights

This paper explores the potential of local spectral theory to investigate the limit point set of the descent spectrum of upper triangular operator matrices, denoted by T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document}, on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by Accσd(Tdiag)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hbox {Acc} \sigma _{\textrm{d}}({\mathcal {T}}_\textbf{diag})$$\end{document}, to that of the complete descent spectrum, Accσd(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hbox {Acc} \sigma _{\textrm{d}}({\mathcal {T}})$$\end{document}, involves removing specific subsets within Accσd(A1)∩Accσa(A2)∩Accσa(A3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hbox {Acc} \sigma _{\textrm{d}}(A_1) \cap \hbox {Acc} \sigma _{\textrm{a}}(A_2) \cap \hbox {Acc} \sigma _{\textrm{a}}(A_3)$$\end{document}. Additionally, we present sufficient conditions that ensure the limit points of the descent spectrum of the operator matrix encompass the combined limit points of its diagonal entry spectra. This significantly addresses a longstanding question posed by Campbell (Linear Multilinear Algebra 14:195–198, 1983) regarding the limit points for the descent spectrum of the last 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \times 3$$\end{document} operator matrix form. Specifically, Campbell inquired about developing new methods to analyze the spectral properties of such matrices without resorting to partitioning their entries, a challenge that has remained unresolved for decades. Our findings provide a comprehensive solution, illustrating that a deeper understanding of the spectral behavior can be achieved by considering the entire matrix structure collectively.