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

Abstract

If $\psi$ is analytic on the open unit disk $\mathbb{D}$ and $\phi$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{\psi ,\phi }$ is defined by $C_{\psi ,\phi }f(z)=\psi (z)f(\phi (z))$, 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 $C_{\psi ,\phi }$ is cohyponormal on $H^2(\beta )$, then $\psi$ never vanishes on $\mathbb{D}$ and $\phi$ is univalent, when $\psi \not\equiv 0$ and $\phi$ is not a constant function. Moreover, for $\psi =K_a$, where $|a|<1$, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{\psi ,\phi }$. After that, for $\phi$ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{\psi ,\phi }$, 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.

16 pages