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

TL;DR

This paper characterizes hyponormal weighted composition operators on Hardy and weighted Bergman spaces, identifying conditions for hyponormality, spectral properties, and specific forms of the symbol functions involved.

Contribution

It provides a complete characterization of hyponormal weighted composition operators on these spaces, including spectral radius calculations and fixed point conditions.

Findings

Hyponormality characterized for compact weighted composition operators.

Hyponormal composition operators on LFTs are either rotations or hyperbolic non-automorphisms.

Spectral radii are explicitly computed for certain hyponormal operators.

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 $A^{2}_{\alpha}$.