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

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 $CT^\ast TC=TT^\ast$. Let $φ$ be a linear fractional self-map of $\mathbb{D}$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_φ$ and weighted composition operator $W_{ψ,φ}$ to be $C$-normal with some conjugations $C$ and a function $ψ$.

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