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Sub-Bergman Hilbert spaces on the unit disk III

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

Published online by Cambridge University Press:  22 August 2023

Shuaibing Luo  and
Kehe Zhu Open the ORCID record for Kehe Zhu [Opens in a new window]
Shuaibing Luo*
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan 410082, China
Kehe Zhu
Affiliation:
Department of Mathematics and Statistics, State University of New York, Albany, NY 12222, USA e-mail: kzhu@albany.edu
*
e-mail: sluo@hnu.edu.cn
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Abstract

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$, we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $, respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$, written as $H(\varphi )$ and $H(\overline \varphi )$ and equipped with appropriate inner products, are called sub-Bergman spaces.

We prove the following three results in the paper: for $-1<\alpha \le 0$, the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$, we have $H(\varphi )=H(\overline \varphi )=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$, we have $D^2_\varphi (A^2_\alpha )= D^2_{\overline \varphi }(A^2_\alpha )=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible.

Keywords

Bergman spaceNevanlinna–Pick kernelToeplitz operatordefect operatorsub-Bergman spaces

MSC classification

Primary: 30H15: Nevanlinna class and Smirnov class 30H10: Hardy spaces 30H05: Bounded analytic functions 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Secondary: 30H25: Besov spaces and $Q_p$-spaces 30H30: Bloch spaces 47B07: Operators defined by compactness properties
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 5 , October 2024 , pp. 1520 - 1537
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

S. Luo was supported by the National Natural Science Foundation of China (Grant No. 12271149). K. Zhu was supported by the National Natural Science Foundation of China (Grant No. 12271328).

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