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 $\varphi$ be a linear fractional self-map of $\mathbb{ D}$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_\varphi$ and weighted composition operator $W_{ \psi,\varphi}$ to be $C$-normal with some conjugations $C$ and a function $\psi$.
Categories: math.CV math.FA