On the range of the elementary operator <i>X</i> → <i>AXA</i> – <i>X</i>

Abstract

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hubert space H into itself. Given A ∈ L(H), we define the elementary operator ∆A: L(H) → L(H) by ∆A(X) = AXA — X. In this paper, we initiate the study of the class of operator A for which $\overline {R(\Delta _A )} = \overline {R(\Delta _{A*} ),} $, where $\overline {R(\Delta _A )} $ denotes the norm closure of the range of ∆A-We call such operators quasiadjoint. We give a characterization and some basic results concerning this class of operator.