Integr. equ. oper. theory 53 (2005), 403–409 c  2005 Birkh¨auser Verlag Basel/Switzerland 0378-620X/030403-7, published online June 28, 2005 DOI 10.1007/s00020-004-1327-3 Integral Equations and Operator Theory On the Range of Elementary Operators Salah Mecheri To my wife 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, .., An) and B =(B1,B2, .., Bn) be n-tuples in B(H), we define the elementary operator EA,B : B(H) → B(H) by EA,B(X)= n i=1 Ai XBi . In this paper we initiate the study of some properties of the range of such operators. Mathematics Subject Classification (2000). Primary 47B47, 47A30, 47B20; Secondary 47B10. Keywords. Generalized derivation, elementary operators, trace class operators, finite rank operators. 1. Introduction Let B(H ) be the algebra of all bounded linear operators acting on a complex separable Hilbert space H . The generalized derivation operator δ A,B associated with (A, B), defined on B(H ) by δ A,B (X )= AX - XB was initially systematically studied by M.Rosenblum [31]. The properties of such operators have been much studied (see for example [1], [2], [4], [11], [32], [33] and [34]). If A = B, then δ A,A = δ A : B(H ) → B(H ) defined by δ A (X )= AX - XA, is called the inner derivation. The theory of derivations has been extensively dealt with in the literature (see, [7], [8], [10], [11], [16], [17], [18] and [19]). Let A =(A 1 ,A 2 , .., A n ) and B =(B 1 ,B 2 , .., B n ) be n-tuples in B(H ), we define the This work was supported by Ksu research center project No. Math/1422/10.