arXiv subjects

Hassane Zguitti

Publications and source records attributed to Hassane Zguitti.

4 recordsLinked to original sources

arXiv2024

New characterization of $(b,c)$-inverses through polarity

In this paper we introduce the notion of $(b,c)$-polar elements in an associative ring $R$. Necessary and sufficient conditions of an element $a\in R$ to be $(b,c)$-polar are investigated. We show that an element $a\in R$ is $(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the $(b,c)$-polarity is a generalization of the polarity along an element introduced by Song, Zhu and Mosić [14] if $b=c$, and the polarity introduced by Koliha and Patricio [10]. Further characterizations are obtained in the Banach space context.

math.RA↗

arXiv2023

On the closed generalized Drazin-Riesz invertible operators and $C_{0}$-semigroups

This paper is a continuation of our paper [Med. J. Math 19, Article number: 31 (2022)] in which we extended the notion of generalized Drazin-Riesz invertible operators to closed operators. We establish here, results relating the notion of closed generalized Drazin-Riesz invertibility with the theory of $C_{0}$-semigroups. Firstly, we generalize results obtained in the bounded case [1] to the context of closed operators. Secondly, we investigate when an infinitesimal generator $A$ of a given $C_{0}$-semigroup is closed generalized Drazin-Riesz invertible. An application to $C_{0}$-groups and abstract second order differential equations is proposed, and an example of a $C_{0}$-group with closed generalized Drazin-Riesz invertible infinitesimal generator is given.

math.FA↗

arXiv2019

A note on the common spectral properties for bounded linear operators

Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and $B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if $A(BA)^2=ABACA=ACABA=(AC)^2A,$ then $$σ_{*}(AC)\setminus\{0\}=σ_{*}(BA)\setminus\{0\}$$ where $σ_*$ runs over a large of spectra originated by regularities.

math.FA↗

arXiv2019

Further common local spectral properties for bounded linear operators

In this note, we study common local spectral properties for bounded linear operators $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $$A(BA)^2=ABACA=ACABA=(AC)^2A.$$ We prove that $AC$ and $BA$ share the single valued extension property, the Bishop property $(β)$, the property $(β_ε)$, the decomposition property $(δ)$ and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.

math.FA↗