Spectral properties for unbounded block operator matrices via polynomially Riesz perturbations

Abstract

As well-known, the perturbation theory of polynomially Riesz operators is an attractive way to characterize certain spectral analysis in Fredholm theory, it is also a tool of great significance in the matrix framework. The first aim of this paper is to find some new arguments of perturbations allowing us to provide some original left-right Fredholm properties of $3 \times 3$ unbounded block operator matrix form defined with maximal domain and to provide an amelioration and a continuation of the recent work invested by Abdmouleh, Khlif and Walha in [Spectral description of Fredholm operators via polynomially Riesz operators perturbation, Georgian Math. J, 29, 3, 317-333, (2022)] in the context of the spectral analysis in Fredholm theory of the last $3 \times 3$ block operator matrices. Our second goal is to express the incidence of some essential spectra of the before-cited model of operator matrices involving the theory of polynomially Riesz operators perturbation. Our approach tolerate us to present a new description in the theory of unbounded operator matrices via a new technique and a new arguments of perturbations coined as polynomially Riesz perturbations.