Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. For given A ∈ L(H) , we define the derivation δ A : L(H) −→ L(H) by δ A (X) = AX − XA. In this paper we establish the orthogonality of the range R(δ A) and the kernel ker(δ A) of a derivation δ A induced by a cyclic subnormal operator A , in the usual sense. We give a version of the Putnam-Fuglede theorem. We establish a short proof of the principal result of F. Wenying and J. Guoxing in [10]. Relatad results for P-symmetric operators are also given.
Abstract. Let H be a separable, infinite-dimensional, complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of all bounded linear operators on H. Let δ AB : L(H) → L(H) denote the generalized derivation δ AB (X) = AX − XB. This note will initiate a study on the class of pairs (A, B) such that R(δ AB ) = R(δ A * B * ).
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