Generalizations of some classical theorems to D-normal operators on Hilbert spaces

Abstract

Abstract We say that a Drazin invertible operator T on Hilbert space is of class [ D N ] $[DN]$ if T D T ∗ = T ∗ T D $T^{D}T^{*} = T^{*}T^{D}$ . 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 ( β ) $(\beta)$ . Finally, we generalize a very famous result on products of normal operators due to I. Kaplansky to D-normal matrices.

Keywords

• Drazin inverse
• D-normal operator
• Fuglede–Putnam theorem
• Bishop property