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Zbl 1131.47021
Zhou, Zehua; Liu, Yan
The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications. (English)
[J] J. Inequal. Appl. 2006, Article ID 90742, 22 p. (2006). ISSN 1029-242X/e

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.

MSC 2000:: *47B33 Composition operators
32A18 Bloch functions etc.
46E15 Banach spaces of functions defined by smoothness properties

Keywords: little $p$-Bloch space; essential norms; bounded composition operators; compactness

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