The Berkani's property and a note on some recent results

TL;DR

This paper advances the understanding of property (UW_{Π}) in relation to Weyl theorems, corrects recent false claims, and clarifies previous research errors in operator theory.

Contribution

It extends the study of property (UW_{Π}) and provides counterexamples to disprove recent false results, offering corrected versions and insights into related theorems.

Findings

Counterexamples show some recent results are false

Corrected versions of previously claimed theorems

Clarification of mistakes in related literature

Abstract

In this paper, we continue the study of property $(UW_{\Pi})$ introduced in \cite{berkani2}, in connection with other Weyl type theorems. Moreover, we give counterexamples to show that some recent results related to this property, which are announced and proved by P. Aiena and M. Kachad in \cite{aiena1} are false. Furthermore, we specify the mistakes committed in each of them and we give the correct versions. We also give a global note on the paper \cite{jayanthi}.