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The Range of the Cesàro Operator Acting on $H^{\infty }$

Part of: Special classes of linear operators Other generalizations Spaces and algebras of analytic functions

Published online by Cambridge University Press:  04 December 2019

Guanlong Bao ,
Hasi Wulan  and
Fangqin Ye
Guanlong Bao
Affiliation:
Department of Mathematics, Shantou University, Shantou515063, Guangdong, China Email: glbao@stu.edu.cnwulan@stu.edu.cn
Hasi Wulan
Affiliation:
Department of Mathematics, Shantou University, Shantou515063, Guangdong, China Email: glbao@stu.edu.cnwulan@stu.edu.cn
Fangqin Ye
Affiliation:
Business School, Shantou University, Shantou515063, Guangdong, China Email: fqye@stu.edu.cn
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Abstract

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

Keywords

The Cesàro operatorH∞Möbius invariant function spacesuperharmonic function

MSC classification

Primary: 47B38: Operators on function spaces (general)
Secondary: 30H25: Besov spaces and $Q_p$-spaces 31C25: Dirichlet spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 633 - 642
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The work was supported by NNSF of China (No. 11801347, No. 11720101003 and No. 11571217), NSF of Guangdong Province (No. 2018A030313512), Department of Education of Guangdong Province (No. 2017KQNCX078), Key projects of fundamental research in universities of Guangdong Province (No. 2018KZDXM034), and STU SRFT (No. NTF17020 and No. STF17005). F. Ye is the corresponding author.

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