We say that a Drazin invertible operator $T$ on Hilbert space is of class $[DN]$ if $T^{D}T^* = T^{*}T^{D}.$ The authors in \cite{Dana} studied several properties of such class. We prove a Fuglede-Putnam commutativity theorem for D-normal operators. Also, we show that $T$ has the Bishop's property $(\beta)$. Finally, we generalize a very famous result on products of normal operators, due to I. Kaplansky to D-normal matrices.
