Banach algebras of matrix: spectral analysis and stability

Fredholm theory relative to a Banach algebra has gained significant interest for decades from mathematicians and occupies an important place in spectral theory. So, in this paper, we are interested with the Fredholm theory relative to a Banach algebra in order to provide an abstract setting allowing to develop a new spectral analysis linked to matrix framework. Specifically, our interest aims at improving upon the foundation established by D. Kleinecke in [25] and R. Nagel in [30]. Our approach allows us to establish an interesting use of the before cited concept of Fredholm perturbation in Banach algebra and provide valuable insights into the analysis of some essential spectral relative to subalgebra of 2 × 2 block operator matrix field. The usefulness of our contribution is tested to an example of perturbed multiplication operator.