Which weighted composition operators are hyponormal on the Hardy and weighted Bergman spaces?

Abstract

In this paper, we study hyponormal weighed composition operators on the Hardy and weighted Bergman spaces. For functions $\psi \in A(\mathbb{D})$ which are not the zero function, we characterize all hyponormal compact weighted composition operators $C_{\psi,\varphi}$ on $H^{2}$ and $A^{2}_{\alpha}$. Next, we show that for $\varphi \in \mbox{LFT}(\mathbb{D})$, if $C_{\varphi}$ is hyponormal on $H^{2}$ or $A^{2}_{\alpha}$, then $\varphi(z)=\lambda z$, where $|\lambda| \leq 1$ or $\varphi$ is a hyperbolic non-automorphism with $\varphi(0)=0$ and such that $\varphi$ has another fixed point in $\partial \mathbb{D}$. After that, we find the essential spectral radius of $C_{\varphi}$ on $H^{2}$ and $A^{2}_{\alpha}$, when $\varphi$ has a Denjoy-Wolff point $\zeta \in \partial \mathbb{D}$. Finally, descriptions of spectral radii are provided for some hyponormal weighted composition operators on $H^{2}$ and $A^{2}_{\alpha}$.