Generalized Fuglede-Putnam theorem for A-normal operators

In this paper, we use a new version of Fuglede-Putnam theorem to establish a relation between the classes of A-normal and n-A-normal operators, where A is a given non-negative operator, so that many other results related to these classes will be obtained. Moreover, we characterize linear transformations defined on a separable Hilbert space that preserve the given classes of operators. In the second strand of the paper, we present a type of generalizations of the considered Theorem for (alpha,beta)-A-normal operators.