Let B(H) be space of all bounded linear operators on a complex Hilbert space H, and S; T 2 B(H) be Drazin invertible. In this paper we investigate a necessary and sufficient condition for the D-normality of ST and TS. Also, we deduce a result relating the factors in a polar decomposition of S to the D-normality of ST and TS. Moreover, we generalize Fuglede-Putnam commutativity theorem for D-normal matrices. Finally, we generalize these results when the n-power D-normal operators are considered.
