Properties of $C$-normal operators (Research on preserver problems on Banach algebras and related topics)

Abstract

We study various properties of $C$-normal operators, i.e., $T*T$ = $CTT*C$ holds for a conjugation $C$ on $H$. Especially, we show that $T$ − λ$I$ is $C$-normal for all λ ∈ ℂ if and only if $T$ is a complex symmetric operator with the conjugation $C$. In addition, we prove that if $T$ is $C$-normal, then $T$ is normal ⇔ $T$ is quasinormal ⇔ $T$ is hyponormal ⇔ $T$ is $p$-hyponormal for 0 < $p$ ≤ 1. Finally, we investigate equivalent conditions so that Aluthge and Duggal transforms of $C$-normal operators to be $C$-normal operators.