COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL

doi:10.1007/s10473-025-0304-6

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 809-824.doi: 10.1007/s10473-025-0304-6

COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL

Chunyu DONG¹, Xin GUO^2,†, Qijian KANG³

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China;
3. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China

收稿日期:2023-08-23 修回日期:2024-05-23 出版日期:2025-05-25 发布日期:2025-09-30

COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL

Chunyu DONG¹, Xin GUO^2,†, Qijian KANG³

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China;
3. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China

Received:2023-08-23 Revised:2024-05-23 Online:2025-05-25 Published:2025-09-30
Contact: ^†Xin GUO, E-mail: xguo.math@zuel.edu.cn
About author:Chunyu DONG, E-mail: dongchunyu math@whu.edu.cn; Qijian KANG, E-mail: qjkang.math@whu.edu.cn
Supported by:
National Science Foundations of China (Grant No. 11771340, 12171373).

摘要/Abstract

摘要： Recently, Choe-Koo-Wang (J Funct Anal, 2020, 278: 108393) demonstrated the rigid phenomenon: The compact linear combination of composition operators under the Coefficient Non-cancellation Condition(CNC), implies that each difference is compact on the weighted Bergman space in the unit disk. Motivated by the subtle connection of composition operator theory on the weighted Bergman spaces, Korenblum spaces and bounded holomorphic function spaces, we first explore the rigid phenomenon which also holds on the Korenblum space over the unit ball. Furthermore, we discuss which difference of composition operators is compact when the compact combination of composition operators does not satisfy the condition (CNC) on Korenblum spaces and bounded holomorphic function spaces over the unit ball setting.

关键词: composition operator, linear combination, compact operator, weighted Bergman space, Korenblum space

Abstract: Recently, Choe-Koo-Wang (J Funct Anal, 2020, 278: 108393) demonstrated the rigid phenomenon: The compact linear combination of composition operators under the Coefficient Non-cancellation Condition(CNC), implies that each difference is compact on the weighted Bergman space in the unit disk. Motivated by the subtle connection of composition operator theory on the weighted Bergman spaces, Korenblum spaces and bounded holomorphic function spaces, we first explore the rigid phenomenon which also holds on the Korenblum space over the unit ball. Furthermore, we discuss which difference of composition operators is compact when the compact combination of composition operators does not satisfy the condition (CNC) on Korenblum spaces and bounded holomorphic function spaces over the unit ball setting.

Key words: composition operator, linear combination, compact operator, weighted Bergman space, Korenblum space

中图分类号:

32A35

Chunyu DONG, Xin GUO, Qijian KANG. COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL[J]. 数学物理学报(英文版), 2025, 45(3): 809-824.

Chunyu DONG, Xin GUO, Qijian KANG. COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL[J]. Acta mathematica scientia,Series B, 2025, 45(3): 809-824.

参考文献

[1] Choe B, Koo H, Park I. Compact differences of composition operators on the Bergman spaces over the ball. Potential Anal, 2014, 40: 81-102
[2] Choe B, Koo H, Wang M. Compact double differences of composition operators on the Bergman spaces. J Funct Anal, 2017, 272: 2273-2307
[3] Choe B, Koo H, Wang M. Compact linear combination of composition operators on Bergman spaces. J Funct Anal, 2020, 278: 108393
[4] Choe B, Koo H, Yang J. Compact double differences of composition operators on the Bergman spaces over the ball. Tohoku Math J, 2019, 71: 609-637
[5] Cowen C, MacCluer B. Composition Operators on Spaces of Analytic Functions. Boca Raton, FL: CRC Press, 1995
[6] Duren P, Weir R. The pseudohyperbolic metric and Bergman spaces in the ball. Trans Amer Math Soc, 2007, 359: 63-76
[7] El-Fallah O, Kellay K, Shabankhah M, Youssfi H. Level sets and composition operators on the Dirichlet space. J Funct Anal, 2011, 260: 1721-1733
[8] Galindo P, Laitila J, Lindström M. Essential norm estimates for composition operators on BMOA. J Funct Anal, 2013, 265: 629-643
[9] Gu C, Koo H. Compact difference of composition operators with smooth symbols on the unit ball. J Math Anal Appl, 2022, 506: 125555
[10] Guo X, Wang M. Linear combination of composition operators on Bergman and Korenblum spaces. Potential Anal, 2022, 57: 305-326
[11] Guo X, Wang M. Compact linear combinations of composition operators over the unit ball. J Operator Theory, 2022, 88: 59-82
[12] Halmos P. Measure Theory.New York: Springer-Verlag, 1974
[13] Hedenmalm H, Korenblum B, Zhu K.Theory of Bergman spaces. New York: Springer-Verlag, 2000
[14] Hyvärinen O, Kemppainen M, Lindström M, et al. The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integr Equ Oper Theory, 2012, 72: 151-157
[15] Izuchi K, Ohno S. Linear combinations of composition operators on $H^{\infty}$. J Math Anal Appl, 2008, 338: 820-839
[16] Koo H, Wang M. Joint Carleson measure and the difference of composition operators on $A_{\alpha^{p}(\mathbf{B}_{n})}$. J Math Anal Appl, 2014, 419: 1119-1142
[17] Koo H, Wang M. Cancellation properties of composition operators on Bergman spaces. J Math Anal Appl, 2015, 432(2): 1174-1182
[18] Kriete T, Moorhouse J. Linear relations in the Calkin algebra for composition operators. Trans Amer Math Soc, 2007, 359: 2915-2944
[19] Moorhouse J. Compact differences of composition operators. J Funct Anal, 2005, 219: 70-92
[20] Shapiro J.Composition Operators and Classical Function Theory. New York: Springer-Verlag, 1993
[21] Toews C. Topological components of the set of composition operators on $H^{\infty}\left(\mathbb{B}_N\right)$. Integr Equ Oper Theory, 2004, 48: 265-280
[22] Wang M, Guo X. Difference of composition operators on the Bergman spaces over the ball. Oper Matrices, 2018, 12: 1177-1197
[23] Wulan H, Zheng D, Zhu K. Compact composition operators on BMOA and the Bloch space. Proc Amer Math Soc, 2009, 137: 3861-3868
[24] Yao X. Linear combinations of composition operators on $H^{\infty}$ spaces over the unit ball and polydisk. J Inequal Appl, 2021, 1: Article 52
[25] Zhu K. Compact composition operators on Bergman spaces of the unit ball. Houston J Math, 2007, 33: 273-283

COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL

COMPACT LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON THE UNIT BALL

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