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Composition operators on the Bloch space of several complex variables. (English) Zbl 0967.32007
Let $$\Omega$$ be a homogeneous bounded domain in $${\mathbb C}^n$$. For any $$\phi\in Hol(\Omega, \Omega)$$ and $$f\in Hol(\Omega)$$, denote $$C_\phi f:=f\circ \phi$$ and call $$C_\phi$$ the composition operator induced by $$\phi$$.
The authors obtain the necessary and sufficient conditions for $$C_\phi$$ to be compact on the Bloch space $$\beta(B_n)$$ or the little Bloch space $$\beta_0(B_n)$$.

##### MSC:
 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX) 47B38 Linear operators on function spaces (general) 30D45 Normal functions of one complex variable, normal families
##### Keywords:
Bloch space; composition operator
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##### References:
 [1] K Madigan, A Matheson. Compact Composition operators on the Bloch Space. Trans Amer Math Soc, 1995, 347: 2679-2687 · Zbl 0826.47023 · doi:10.2307/2154848 [2] R Timoney. Bloch Function in Several Complex Variables I. Bull London Math Soc, 1980, 12, 241-267 · Zbl 0428.32018 · doi:10.1112/blms/12.4.241 [3] R Timoney. Bloch Function in Several Complex Variables II. J Reine Angew Math, 1980, 319, 1-22 · Zbl 0425.32008 · doi:10.1515/crll.1980.319.1
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