Abstract. Let $H$ denote a complex Hilbert space and $B(H)$ denote the algebra of all bounded linear operators on $H$. In this paper, we study the class of pairs of operators $A,B\in B(H)$ that have the following property: $AT=TB$ implies $B^*T=TA^*$ for all $T\in C_{1}(H)$ (trace class operators). The main result is the equivalence between this character and the fact that the ultraweak closure of the range of a generalized derivation is closed under taking adjoints which is also equivalent to the generalized D-symmetric operators.

AMS Subject Classification (1991): 47B47, 47A30, 47B20, 47B10

Keyword(s): Generalized derivation Elementary operators, Trace class operators

Received April 14, 2005, and in final form March 7, 2006. (Registered under 5926/2009.)