The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications

Abstract

Let $U^{n}$ be the unit polydisc of ${\Bbb C}^{n}$ and $\phi=(\phi_1, >..., \phi_n)$ a holomorphic self-map of $U^{n}.$ By ${\cal B}^p(U^{n})$, ${\cal B}^p_{0}(U^{n})$ and ${\cal B}^p_{0*}(U^{n})$ 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 $C_{\phi}$ induced by $\phi$ between ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) and ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As their applications, some necessary and sufficient conditions for the bounded composition operators $C_{\phi}$ to be compact from ${\cal B}^p(U^n)$ $({\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n))$ into ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) are obtained.