Symmetric difference between pseudo B-Fredholm spectrum and spectra originated from fredholm theory

Abstract

In this paper, we continue the study of the pseudo B-Fredholm operators of Boasso, and the pseudo B-Weyl spectrum of Zariouh and Zguitti; in particular we find that the pseudo B-Weyl spectrum is empty whenever the pseudo B-Fredholm spectrum is, and look at the symmetric differences between the pseudo B-Weyl and other spectra.

A new characterization of Browder’s theorem

“…Also, T is a B-Weyl operator if and only if T X 1 is a Weyl operator and T X 2 is a nilpotent operator. More recently, B-Fredholm and B-Weyl operators were generalized to pseudo B-Fredholm and pseudo B-Weyl, see [13] [22][23] [25], precisely, T is a pseudo B-Fredholm operator, if there exists (X 1 , X 2 ) ∈ Red(T) such that T X 1 is a Fredholm operator and T X 2 is a quasi-nilpotent operator. T is said to be pseudo B-Weyl operator if there exists (X 1 , X 2 ) ∈ Red(T) such that T X 1 is a Weyl operator and T X 2 is a quasi-nilpotent operator.…”

Spectra Originated from Fredholm Theory and Browder's Theorem

“…Also, we will give serval necessary and sufficient conditions for T to have equality between the spectra originated from Fredholm theory and Drazin invertibility. In the same direction as our work [28], we will give conditions under which pseudo B-Fredholm and pseudo B-Weyl spectrum are stable under commuting Riesz perturbations. In section four, we will prove that we can perturb a pseudo B-Fredholm (resp.…”