On the Range of Elementary Operators

Abstract.

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A _n ) and B = (B₁, B₂,.., B _n ) be n-tuples in B(H), we define the elementary operator $E_{A,B} : B(H) \mapsto B(H)$ by $E_{A,B} (X) = \Sigma _{i = 1}^n A_i X\,B_i.$ In this paper we initiate the study of some properties of the range of such operators.