Normal, cohyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces

If $\psi$ is analytic on the open unit disk $\mathbb{D}$ and $\varphi$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{\psi,\varphi}$ is defined by $C_{\psi,\varphi}f(z)=\psi(z)f (\varphi (z))$, when $f$ is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^{2}(\beta)$, we prove that if $C_{\psi,\varphi}$ is cohyponormal on $H^{2}(\beta)$, then $\psi$ never vanishes on $\mathbb{D}$ and $\varphi$ is univalent, when $\psi \not \equiv 0$ and $\varphi$ is not a constant function. Moreover, for $\psi=K_{a}$, where $|a| < 1$, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{\psi,\varphi}$. After that, for $\varphi $ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{\psi,\varphi}$, when $\psi \not \equiv 0$ and $\psi$ is analytic on $\overline{\mathbb{D}}$. Finally, we find all normal weighted composition operators which are bounded below.