Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (3): 657-669.doi: 10.1007/s10473-021-0301-3

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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
  • About author:Yong CHEN,,;Kei Ji IZUCHI,;Kou Hei IZUCHI,
  • 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).

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

CLC Number: 

  • 47B35