Quasinormal and hyponormal weighted composition operators on $H^2$ and $A^2_{\alpha}$ with linear fractional compositional symbol

TL;DR

This paper characterizes quasinormal and hyponormal weighted composition operators with linear fractional symbols on Hardy and weighted Bergman spaces, revealing conditions for normality and providing new examples of hyponormal operators.

Contribution

It provides a complete characterization of quasinormal composition operators with linear fractional symbols and introduces new hyponormal weighted composition operators that are not quasinormal.

Findings

Quasinormal operators are necessarily normal in all known cases.

Several possibilities for hyponormal operators are eliminated.

New examples of hyponormal but not quasinormal weighted composition operators are constructed.

Abstract

In this paper, we study quasinormal and hyponormal composition operators \W with linear fractional compositional symbol $\ph$ on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on $H^{2}$ and $A_{\alpha}^{2}$ by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on $H^2$ which are not quasinormal.