B-Weyl Operator Pencils: Structural Characterization, Perturbation Stability, and Examples

We investigate the properties of B-Weyl operator pencils of the form λM–T, where M is a bounded operator and T is a densely defined, closed, and unbounded operator on a Banach space. We introduce and characterize left and right B-Weyl operator pencils, establishing necessary and sufficient conditions for their existence. Stability results under finite-rank perturbations are derived, and connections with Drazin invertibility are explored. Additionally, we provide concrete examples, including differential operators, integral operators, and block operator matrices, to illustrate our theoretical developments. Our results extend classical B-Weyl operator theory to the case of operator pencils and contribute to the spectral analysis of unbounded operators. This work builds upon and refines the recent study by Bahloul 5, further developing the concept of unbounded operators and their spectral properties.