Ranges of Products of Operators

Abstract

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.