Compact Composition Operators between Bloch Type Spaces in the Polydisk

Zhong-Shan Fang

doi:10.1155/2012/732709

Abstract

Wulan et al. (2009), Wulan et al. characterized the compactness of composition operators on the Bloch space in the unit disk by the <svg style="vertical-align:-0.1638pt;width:8.6625004px;" id="M1" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> </svg>th power of the induced analytic function. This paper will generalize the result to the Bloch type space in the polydisk.

Highlights

Let Dn be the polydisk of Cn with boundary ∂Dn
The goal of this paper is to extend the above result in the unit disk to the polydisk
Throughout this paper, let N be the set of the positive integers, I {i ∈ N : φi ∞ 1} and J {j ∈ N : φj ∞ < 1}

Summary

Introduction

Let Dn be the polydisk of Cn with boundary ∂Dn. The class of all holomorphic functions on the domain Dn will be denoted by H Dn. The boundedness and compactness of composition operators between several spaces of holomorphic functions have been studied extensively. We refer the interested readers to the books in 1–3. See 4–9 , as well as the related references therein. In 10 , Wulan et al obtained a new result about the compactness of the composition operators on the Bloch space in the unit disk. Let φ be an analytic self-map of the unit disk D. The goal of this paper is to extend the above result in the unit disk to the polydisk. Throughout this paper, let N be the set of the positive integers, I {i ∈ N : φi ∞ 1} and J {j ∈ N : φj ∞ < 1}

Some Lemmas

Main Theorems