Regular extensions and the solvability of operator equations

Abstract

Let T1 be a closed linear operator in a complex Banach space. In this paper we are concerned with the effect of changes in the complex parameter X on the solvability of the equation (T1-XI)x=y. By introducing the notion of a regular extension we are able to generalize a result of A. S. Markus [2]. In order to introduce extension terminology we consider T1 to be an extension of a closed linear operator To, denoted To CT1. We use D(T) to denote the domain of an operator T, R(T) for the range of T and N(T) for the null space of T. K(T) denotes the set of all elements y such that the equation Tnx = y is solvable for every positive integer n, a concept originally introduced by F. Riesz [3, p. 87]. We call a closed linear operator T a regular extension at X if l CT CTT1, R(T-XI) = R(T1 -XI) and T-XI has a bounded inverse. We call T a regular extension near Xo if for every X in some neighborhood of Xo, T is a regular extension at X. Let P be any connected component of the open set consisting of all complex numbers X such that there exists a regular extension near X.