On generalized Saphar operators on Banach spaces

A Saphar operator T on a Banach space X is one whose kernel is contained in its generalized range ⋂i=1∞R(Tn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcap \nolimits _{i=1}^{\infty }{\mathcal {R}}(T^{n})$$\end{document} and its range and kernel are closed complemented subspaces. The purpose of this paper is to introduce and study a new class of operators that subsumes the class of Saphar operators, namely, the class of generalized Saphar operators. It is shown that the operators introduced can be characterized in several ways, particularly by means of specific Kato decompositions. Generalized Saphar spectrum of an operator T is also introduced and proven to be a compact subset of the complex plane.