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 [Formula: see text]-normal and [Formula: see text]-[Formula: see text]-normal operators, where [Formula: see text] 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 [Formula: see text]-[Formula: see text]-normal operators.