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The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications. (English) Zbl 1131.47021
Summary: Let $U^n$ be the unit polydisc of $\Bbb C^n$ $\phi=(\phi_1,\dots, \phi_n)$ be a holomorphic self-map of $U^n$, and ${\cal B}^p(U^n)$, ${\cal B}_0^p(U^n)$, and ${\cal B}_{0_*}^p(U^n)$ 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 estimates of the essential norms of bounded composition operators $C_\phi$ induced by $\phi$ between ${\cal B}^p(U^n)$ $({\cal B}_0^p(U^n)$ or ${\cal B}_{0_*}^p(U^n))$ and ${\cal B}^q(U^n)$ $({\cal B}_0^q(U^n)$ or ${\cal B}_{0_*}^q(U^n))$. As applications, some necessary and sufficient conditions for the (bounded) composition operators $C_\phi$ to be compact from ${\cal B}^p(U^n)$ $({\cal B}_0^p(U^n)$ or ${\cal B}_{0_*}^p(U^n))$ into ${\cal B}^q(U^n)$ $({\cal B}_0^q(U^n)$ or $B_{0_*}^q(U^n))$ are obtained.

47B33Composition operators
32A18Bloch functions, normal functions
46E15Banach spaces of continuous, differentiable or analytic functions
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