Insights into Some Sets of \(\varepsilon \)-Pseudo Fredholm Operators

One of the popular generalizations of the classical notion of essential spectra is the notion of pseudo essential spectra. The study around such notion has gained intensive interest in many research fields, particularly, in the theory of Fredholm operators. So, the novelty of our work is to develop a new sufficient criteria linked to the notion of \(\Phi \)-perturbation function (recently invested by M. Mebekhta in (J. Oper. Theory 51:3–18, 2004)) allowing us to derive some new spectral analysis of some sets of \(\varepsilon \)-pseudo Fredholm operators and their corresponding pseudo Weyl essential spectra. Moreover, our aim comes also to reach a new characterization in the setting of the so-called pseudo Weyl essential spectrum of a closed densely defined operator via the above approach of perturbation. Our results in this paper generalize and refine earlier work, particularly, the work done by S. Charfi et al. (Indag. Math. 28(3):670–679, 2017).