On the range kernel orthogonality and p-symmetric operators

Abstract

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 δA : L(H) −→ L(H) by δA(X) = AX − XA . In this paper we establish the orthogonality of the range R(δA) and the kernel ker(δA) of a derivation δA 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. Mathematics subject classification (2000): 47B47, 47B10, 47A30.