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Linear combination of composition operators on \(H^\infty \) and the Bloch space

  • Yecheng Shi
  • Songxiao LiEmail author
Article
  • 8 Downloads

Abstract

Let \(\lambda _i (i=1,\ldots ,k)\) be nonzero complex scalars and \(\varphi _i (i=1,..,k)\) be analytic self-maps of the unit disk \(\mathbb {D}\). We show that the operator \(\sum _{i=1}^k\lambda _iC_{\varphi _i}\) is compact on the Bloch space \(\mathcal {B}\) if and only if
$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \lambda _1\varphi _1^n+\lambda _2\varphi _2^n+\cdots +\lambda _k\varphi _k^n\Vert _{\mathcal {B}}=0. \end{aligned}$$
We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.

Keywords

Composition operator Bloch space Linear combination Compact 

Mathematics Subject Classification

30H30 47B33 

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Notes

Acknowledgements

The authors thank the referee for his (or her) several helpful suggestions.

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLingnan Normal UniversityZhanjiangPeople’s Republic of China
  2. 2.Institute of Fundamental and Frontier SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

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