Generalizations of some classical theorems to D-normal operators on Hilbert spaces
М. Дана, R. Yousefi
Abstract
Abstract We say that a Drazin invertible operator T on Hilbert space is of class $[DN]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mo>]</mml:mo></mml:math> if $T^{D}T^{*} = T^{*}T^{D}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>T</mml:mi><mml:mi>D</mml:mi></mml:msup><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msup><mml:mi>T</mml:mi><mml:mi>D</mml:mi></mml:msup></mml:math> . 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 $(\beta)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>β</mml:mi><mml:mo>)</mml:mo></mml:math> . Finally, we generalize a very famous result on products of normal operators due to I. Kaplansky to D-normal matrices.