| Title:
|
Schatten class generalized Toeplitz operators on the Bergman space (English) |
| Author:
|
Xu, Chunxu |
| Author:
|
Yu, Tao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
1173-1188 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mu $ be a finite positive measure on the unit disk and let $j\geq 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{(j)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{(j)}$ to be in the Schatten $p$-class for $1\leq p<\infty $ on the Bergman space $A^{2}$, and then give a sufficient condition for $T_{\mu }^{(j)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^{2}$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D, {\rm d}A)$ and $1<p<\infty $, we define the generalized Toeplitz operator $T_{\varphi }^{(j)}$ on the Bergman space $A^p$ and characterize the compactness of the finite sum of operators of the form $T_{\varphi _1}^{(j)}\cdots T_{\varphi _n}^{(j)}$. (English) |
| Keyword:
|
generalized Toeplitz operator |
| Keyword:
|
Schatten class |
| Keyword:
|
compactness |
| Keyword:
|
Bergman space |
| Keyword:
|
Berezin transform |
| MSC:
|
47B10 |
| MSC:
|
47B35 |
| idZBL:
|
Zbl 07442483 |
| idMR:
|
MR4339120 |
| DOI:
|
10.21136/CMJ.2021.0336-20 |
| . |
| Date available:
|
2021-11-08T16:05:43Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149247 |
| . |
| Reference:
|
[1] Arazy, J., Fisher, S. D., Peetre, J.: Hankel operators on weighted Bergman spaces.Am. J. Math. 110 (1988), 989-1053. Zbl 0669.47017, MR 0970119, 10.2307/2374685 |
| Reference:
|
[2] Engliš, M.: Toeplitz operators and group representations.J. Fourier Anal. Appl. 13 (2007), 243-265. Zbl 1128.47029, MR 2334609, 10.1007/s00041-006-6009-x |
| Reference:
|
[3] Luecking, D. H.: Trace ideal criteria for Toeplitz operators.J. Funct. Anal. 73 (1987), 345-368. Zbl 0618.47018, MR 0899655, 10.1016/0022-1236(87)90072-3 |
| Reference:
|
[4] Miao, J., Zheng, D.: Compact operators on Bergman spaces.Integral Equations Oper. Theory 48 (2004), 61-79. Zbl 1060.47036, MR 2029944, 10.1007/s00020-002-1176-x |
| Reference:
|
[5] Roman, S.: The formula of Faa di Bruno.Am. Math. Mon. 87 (1980), 805-809. Zbl 0513.05009, MR 0602839, 10.2307/2320788 |
| Reference:
|
[6] Simon, B.: Trace Ideals and Their Applications.London Mathematical Society Lecture Note Series 35. Cambridge University Press, Cambridge (1979). Zbl 0423.47001, MR 0541149, 10.1090/surv/120 |
| Reference:
|
[7] Suárez, D.: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space.Rev. Mat. Iberoam. 20 (2004), 563-610. Zbl 1057.32005, MR 2073132, 10.4171/RMI/401 |
| Reference:
|
[8] Suárez, D.: A generalization of Toeplitz operators on the Bergman space.J. Oper. Theory 73 (2015), 315-332. Zbl 1399.32010, MR 3346124, 10.7900/jot.2013nov28.2023 |
| Reference:
|
[9] Zhu, K.: Positive Toeplitz operators on the weighted Bergman spaces of bounded symmetric domains.J. Oper. Theory 20 (1988), 329-357. Zbl 0676.47016, MR 1004127 |
| Reference:
|
[10] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball.Graduate Texts in Mathematics 226. Springer, New York (2005). Zbl 1067.32005, MR 2115155, 10.1007/0-387-27539-8 |
| Reference:
|
[11] Zhu, K.: Operator Theory in Function Spaces.Mathematical Surveys and Monographs 138. American Mathematical Society, Providence (2007). Zbl 1123.47001, MR 2311536, 10.1090/surv/138 |
| Reference:
|
[12] Zhu, K.: Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball.New York J. Math. 13 (2007), 299-316. Zbl 1127.47029, MR 2357717 |
| . |
