A Note on the Range of a Derivation

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on into itself. Given A, B ∈ L(H), define generalized derivation δA, L(L(H)) by B(X) = AX - XB. An operator A is P-symmetric if AT TA implies AT* T* for T C1(H) (trace class operators). In this paper, we give generalization operators. We initiate study pairs (A, B) such that R(δA, W* W*, where denotes ultraweak closure range B. Such are called P-symmetric. establish characterization those Related properties also given.