Abstract
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 (Oper. Matrices 12(2):465–487, 2018) studied several properties of this class. We prove the Fuglede–Putnam commutativity theorem for D-normal operators. Also, we show that T has the Bishop property (beta). Finally, we generalize a very famous result on products of normal operators due to I. Kaplansky to D-normal matrices.
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