$C$-normal weighted composition operators on $H^2$

TL;DR

This paper characterizes when certain composition and weighted composition operators on the Hardy space are $C$-normal, based on conjugations and the properties of their defining functions, advancing understanding of operator normality in functional analysis.

Contribution

It provides necessary and sufficient conditions for $C$-normality of composition and weighted composition operators on $H^2$, linking operator properties to function-theoretic conditions.

Findings

Characterization of $C$-normal composition operators with conjugation $C$.

Conditions for $C$-normal weighted composition operators with symbol $ ilde{ ext{phi}}$ and weight $ ilde{ ext{psi}}$.

Extension of $C$-normality concepts to linear fractional self-maps of the disk.

Abstract

A bounded linear operator $T$ on a separable complex Hilbert space $H$ is called $C$-normal if there is a conjugation $C$ on $H$ such that $ CT^\ast TC=TT^\ast$. Let $\varphi$ be a linear fractional self-map of $\mathbb{D}$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_\varphi$ and weighted composition operator $W_{\psi,\varphi}$ to be $C$-normal with some conjugations $C$ and a function $\psi$.