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 \({\mathcal {T}}\), on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by \( \hbox {Acc} \sigma _{\textrm{d}}({\mathcal {T}}_\textbf{diag})\), to that of the complete descent spectrum, \( \hbox {Acc} \sigma _{\textrm{d}}({\mathcal {T}})\), involves removing specific subsets within \( \hbox {Acc} \sigma _{\textrm{d}}(A_1) \cap \hbox {Acc} \sigma _{\textrm{a}}(A_2) \cap \hbox {Acc} \sigma _{\textrm{a}}(A_3)\). 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 \(3 \times 3\) 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.