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.)
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