On C-quasinormal operators

Abstract

An operator \(T\in {{\mathcal {L}}}({{\mathcal {H}}})\) is called conjugation-quasinormal if there is a conjugation C on an infinite-dimensional complex Hilbert space \({{\mathcal {H}}}\) such that \([CT,T^{*}T]=0\) where \([R,S]:=RS-SR\). In particular, if such a conjugation C is specified, an operator T is said to be C-quasinormal. In this paper, we study various properties of conjugation-quasinormal operators. Especially, we provide several characterizations of conjugation-quasinormal operators. Moreover, if T is a partial isometry, we show that T is C-hyponormal if and only if T is C-quasinormal. Finally, we consider conjugation-operator transforms.