Essential Norms of Weighted Composition Operators between Hardy Spaces in the unit Ball

TL;DR

This paper estimates the essential norm of weighted composition operators between Hardy spaces on the unit ball, providing exact formulas in specific cases and conditions for compactness.

Contribution

It introduces new estimates for the essential norm of weighted composition operators on Hardy spaces and characterizes their compactness criteria.

Findings

Derived bounds for the essential norm of $W_{\psi,\phi}$.

Provided an exact formula for the essential norm when $p=\infty$, $q=2$.

Established necessary and sufficient conditions for compactness of $W_{\psi,\phi}$.

Abstract

Let $\phi(z)=(\phi_1(z),...,\phi_n(z))$ be a holomorphic self-map of $B_n$ and $\psi(z)$ a holomorphic function on $B_n$, and $H(B_n)$ the class of all holomorphic functions on $B_n$, where $B_n$ is the unit ball of $C^n$, the weight composition operator $W_{\psi,\phi}$ is defined by $W_{\psi,\phi}=\psi f(\phi)$ for $f\in H(B_n)$. In this paper we estimate the essential norm for the weighted composition operator $W_{\psi,\phi}$ acting from the Hardy space $H^p$ to $H^q$ ($0<p,q\leq \infty$). When $p=\infty$ and $q=2$, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necessary conditions for the bounded weighted composition operator to be compact from $H^p$ to $H^q$.