New properties of some classes of Saphar type operators

Abstract

This paper explores additional properties of some classes of Saphar type operators, namely left Drazin invertible, essentially left Drazin invertible, right Drazin invertible, and essentially right Drazin invertible operators on Banach spaces, building upon the groundwork laid in \cite{GM} and \cite{ZS}. Specifically, we propose alternative definitions for these operators and characterize them with a new type of operator decomposition, providing a deeper understanding of their properties. Furthermore, we investigate their behavior under powers. The operators we study are distinguished from other operators bearing the same name in existing literature, see \cite{Aiena}, \cite{Q. Jiang}. By employing more refined definitions, we uncover a broader range of properties for these operators, setting them apart from their counterparts in the literature.