Connections between polynomially Riesz and Fredholm elements in Banach algebras with closed inessential ideals

The purpose of this paper is to establish some spectral analysis in Fredholm theory that incorporates the concept of polynomially Riesz elements perturbation. Furthermore, we use the results we have gathered to evaluate the impact of certain essential spectra of perturbed elements through this type of perturbation.This kind of perturbation allows us to reinterpret the structure of Fredholm elements in terms of polynomially Riesz elements and the ideal of Riesz operators in Banach algebras. Finally, we investigate the connection between Riesz and Fredholm elements by perturbing polynomially Riesz elements through a closed inessential ideal.