Chain-finite operators and locally chain-finite operators

Abstract

We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that TkBTk =Tk and TkB=BTk. Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1).