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

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 $H^q$.