The Essential Norm of Composition Operator between Generalized Bloch Spaces in Polydiscs and its Applications

Abstract

Let $U^n$ be the unit polydisc of ${\mathbb{C}}^n$ and $\varphi =({\varphi }_1,>...,{\varphi }_n)$ a holomorphic self-map of $U^n.$ By ${\mathcal{B}}^p(U^n)$, ${\mathcal{B}}_0^p(U^n)$ and ${\mathcal{B}}_{0\ast }^p(U^n)$ denote the $p$-Bloch space, Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc $U^n$ respectively, where $p,q>0$. This paper gives the estimates of the essential norms of bounded composition operators $C_{\varphi }$ induced by $\varphi$ between ${\mathcal{B}}^p(U^n)$ (${\mathcal{B}}_0^p(U^n)$ or ${\mathcal{B}}_{0\ast }^p(U^n)$) and ${\mathcal{B}}^q(U^n)$ (${\mathcal{B}}_0^q(U^n)$ or ${\mathcal{B}}_{0\ast }^q(U^n)$). As their applications, some necessary and sufficient conditions for the bounded composition operators $C_{\varphi }$ to be compact from ${\mathcal{B}}^p(U^n)$ $({\mathcal{B}}_0^p(U^n)$ or ${\mathcal{B}}_{0\ast }^p(U^n))$ into ${\mathcal{B}}^q(U^n)$ (${\mathcal{B}}_0^q(U^n)$ or ${\mathcal{B}}_{0\ast }^q(U^n)$) are obtained.

24 Pages

Cited by in corpus (4)

• Essential Norms of Weighted Composition Operators between Hardy Spaces in the unit Ball
• Difference of composition operators in the Polydiscs
• Composition operators in the Lipschitz Space of the Polydiscs
• Compact differences of composition operators from Bloch space to bounded holomorphic function space in the Polydisc