Unveiling quasi inverses of linear pencils in Banach algebra

We investigate the essential spectrum of linear pencils in Banach algebras, particularly their behavior under ideal perturbations. Building upon the foundational work of J. Shapiro and M. Snow in [The Fredholm spectrum of the sum and product of two operators, Transactions of the American Mathematical Society, 191 (1974), 387-393] on the Fredholm spectrum in Banach spaces, this study introduces novel characterizations of quasi-inverses and their role in spectral analysis. By leveraging these characterizations, we derive conditions ensuring that the essential spectrum of a linear pencil is confined within a specific sector of the complex plane. Our findings establish a refined connection between Fredholm theory and the algebraic structure of Banach algebras, offering both theoretical advancements and geometric insights into spectral containment. These results extend existing frameworks and open avenues for exploring spectral properties in more general algebraic settings with applications in operator theory and differential equations.