数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (3): 657-669.doi: 10.1007/s10473-021-0301-3

• 论文 •    下一篇

SEQUENCES OF POWERS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE

陈泳1, Kei Ji IZUCHI2, Kou Hei IZUCHI3, Young Joo LEE4   

  1. 1. Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China;
    2. Department of Mathematics, Niigata University, Niigata 950-2181, Japan;
    3. Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8511, Japan;
    4. Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
  • 收稿日期:2020-03-06 修回日期:2020-05-15 出版日期:2021-06-25 发布日期:2021-06-07
  • 通讯作者: Young Joo LEE E-mail:leeyj@chonnam.ac.kr
  • 作者简介:Yong CHEN,E-mail:ychen227@gmail.com,ychen@hznu.edu.cn;Kei Ji IZUCHI,E-mail:izuchi@m.sc.niigata-u.ac.jp;Kou Hei IZUCHI,E-mail:izuchi@yamaguchi-u.ac.jp
  • 基金资助:
    The first author was supported by NSFC (11771401) and the last author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2019R1I1A3A01041943).

SEQUENCES OF POWERS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE

Yong CHEN1, Kei Ji IZUCHI2, Kou Hei IZUCHI3, Young Joo LEE4   

  1. 1. Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China;
    2. Department of Mathematics, Niigata University, Niigata 950-2181, Japan;
    3. Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8511, Japan;
    4. Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
  • Received:2020-03-06 Revised:2020-05-15 Online:2021-06-25 Published:2021-06-07
  • Contact: Young Joo LEE E-mail:leeyj@chonnam.ac.kr
  • About author:Yong CHEN,E-mail:ychen227@gmail.com,ychen@hznu.edu.cn;Kei Ji IZUCHI,E-mail:izuchi@m.sc.niigata-u.ac.jp;Kou Hei IZUCHI,E-mail:izuchi@yamaguchi-u.ac.jp
  • Supported by:
    The first author was supported by NSFC (11771401) and the last author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2019R1I1A3A01041943).

摘要: We consider Toeplitz operators $T_u$ with symbol $u$ on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols $\varphi$ for which the sequence $T^{*k}_\varphi f$ or $T^{k}_\varphi f$ converges to 0 or $\infty$ as $k\to\infty$ in norm for every nonzero Bergman function $f$. Also, we characterize analytic symbols $\varphi$ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.

关键词: Bergman space, Toeplitz operator

Abstract: We consider Toeplitz operators $T_u$ with symbol $u$ on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols $\varphi$ for which the sequence $T^{*k}_\varphi f$ or $T^{k}_\varphi f$ converges to 0 or $\infty$ as $k\to\infty$ in norm for every nonzero Bergman function $f$. Also, we characterize analytic symbols $\varphi$ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.

Key words: Bergman space, Toeplitz operator

中图分类号: 

  • 47B35