On properties of C-normal operators

Abstract

A bounded linear operator T:H -> H is a C-normal operator if there exists a conjugation C on H such that [CT,(CT)*]=0 where [R,S]:=RS-SR. In this paper we study properties of C-normal operators. In particular, we prove that T-lambda is C-normal for all lambda is an element of C if and only if T is a complex symmetric operator with the conjugation C. Moreover, we show that if T is C-normal, then the following statements are equivalent; (i) T is normal, (ii) T is quasinormal, (iii) T is hyponormal, (iv) T is p-hyponormal for 0 < p <= 1. Finally, we consider operator transforms of C-normal operators.