Numerical exploration of B-Fredholm spectra in unbounded block operator matrices

This paper investigates the B-Fredholm spectral properties of unbounded block operator matrices defined on Banach spaces, motivated by their foundational role in spectral theory and their occurrence in physical models governed by evolution equations. Under a relaxed set of assumptions, the analysis provides a refined characterization of the B-essential spectra. The study introduces pivotal theorems that establish relationships between the spectra of the full operator matrix and its diagonal components, using Schur complement techniques. Numerical examples are included to illustrate the theoretical results, with explicit computations of resolvents and B-essential spectra for selected operator structures. The results contribute to advancing the spectral theory of unbounded operator matrices and open new directions that link abstract results to concrete applications within a numerical framework.