Abstract

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ_A : L(H) → L(H) by Δ_A(X) = AXA − X. In this paper, we initiate the study of the class of operator A for which ¯R(¯Δ_A) = ¯R(¯Δ_A*), where ¯R(¯Δ_A) denotes the norm closure of the range Δ_A. We call such operators quasi-adjoint. We give a characterisation and some basic results concerning this class of operators.