$C$-normal weighted composition operators on $H^2$

Abstract

A bounded linear operator $T$ on a separable complex Hilbert space $H$ is called $C$-normal if there is a conjugation $C$ on $H$ such that $C{T}^{\ast }TC=T{T}^{\ast }$. Let $\phi$ be a linear fractional self-map of $\mathbb{D}$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_{\phi }$ and weighted composition operator $W_{\psi ,\phi }$ to be $C$-normal with some conjugations $C$ and a function $\psi$.