Article
| Title: | Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains (English) |
| Author: | He, Le |
| Author: | Tang, Yanyan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 4 |
| Year: | 2024 |
| Pages: | 1097-1112 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We consider a class of unbounded nonhyperbolic complete Reinhardt domains $$ D_{n,m,k}^{\mu ,p,s}:=\Big \{(z,w_1,\cdots ,w_m)\in \mathbb {C}^{n}\times \mathbb {C}^{k_1}\times \cdots \times \mathbb {C}^{k_m}\colon \frac {\| w_1\|^{2p_1}}{{\rm e}^{-\mu _1\| z\|^{s}}}+\cdots +\frac {\| w_m\|^{2p_m}}{{\rm e}^{-\mu _m\| z\|^{s}}}<1\Big \}, $$ where $s$, $p_1,\cdots ,p_m$, $\mu _1,\cdots ,\mu _m$ are positive real numbers and $n$, $k_1,\cdots ,k_m$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2(D_{n,m,k}^{\mu ,p,s})$, then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. (English) |
| Keyword: | unbounded complete Reinhardt domain |
| Keyword: | Hankel operator |
| Keyword: | Hilbert-Schmidt operator |
| MSC: | 32A36 |
| MSC: | 32Q02 |
| MSC: | 47B35 |
| DOI: | 10.21136/CMJ.2024.0067-24 |
| . | |
| Date available: | 2024-12-15T06:36:33Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152691 |
| . | |
| Reference: | [1] Arazy, J.: Boundedness and compactness of generalized Hankel operators on bounded symmetric domains.J. Funct. Anal. 137 (1996), 97-151. Zbl 0880.47015, MR 1383014, 10.1006/jfan.1996.0042 |
| Reference: | [2] Arazy, J., Fisher, S. D., Janson, S., Peetre, J.: An identity for reproducing kernels in a planar domain and Hilbert-Schmidt Hankel operators.J. Reine Angew. Math. 406 (1990), 179-199. Zbl 0686.47023, MR 1048240, 10.1515/crll.1990.406.179 |
| Reference: | [3] 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: | [4] Beberok, T., Göğüş, N. G.: Hilbert-Schmidt Hankel operators over semi-Reinhardt domains.Available at https://arxiv.org/abs/1604.07059v1 (2016), 9 pages. 10.48550/arXiv.1604.07059 |
| Reference: | [5] Bi, E., Feng, Z., Tu, Z.: Balanced metrics on the Fock-Bargmann-Hartogs domains.Ann. Global Anal. Geom. 49 (2016), 349-359. Zbl 1355.32004, MR 3510521, 10.1007/s10455-016-9495-3 |
| Reference: | [6] Bi, E., Tu, Z.: Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains.Pac. J. Math. 297 (2018), 277-297. Zbl 1410.32001, MR 3893429, 10.2140/pjm.2018.297.277 |
| Reference: | [7] Çelik, M., Zeytuncu, Y. E.: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complex ellipsoids.Integral Equations Oper. Theory 76 (2013), 589-599. Zbl 1288.47028, MR 3073947, 10.1007/s00020-013-2070-4 |
| Reference: | [8] Çelik, M., Zeytuncu, Y. E.: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains.Czech. Math. J. 67 (2017), 207-217. Zbl 1482.47050, MR 3633007, 10.21136/CMJ.2017.0471-15 |
| Reference: | [9] Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables.AMS/IP Studies in Advanced Mathematics 19. AMS, Providence (2001). Zbl 0963.32001, MR 1800297, 10.1090/amsip/019 |
| Reference: | [10] D'Angelo, J. P.: An explicit computation of the Bergman kernel function.J. Geom. Anal. 4 (1994), 23-34. Zbl 0794.32021, MR 1274136, 10.1007/BF02921591 |
| Reference: | [11] Haslinger, F., Lamel, B.: Spectral properties of the canonical solution operator to $\bar{\partial}$.J. Funct. Anal. 255 (2008), 13-24. Zbl 1169.32009, MR 2417807, 10.1016/j.jfa.2008.03.013 |
| Reference: | [12] Huo, Z.: The Bergman kernel on some Hartogs domains.J. Geom. Anal. 27 (2017), 271-299. Zbl 1367.32004, MR 3606552, 10.1007/s12220-016-9681-3 |
| Reference: | [13] Kim, H., Ninh, V. T., Yamamori, A.: The automorphism group of a certain unbounded non-hyperbolic domain.J. Math. Anal. Appl. 409 (2014), 637-642. Zbl 1307.32017, MR 3103183, 10.1016/j.jmaa.2013.07.007 |
| Reference: | [14] Kim, H., Yamamori, A.: The holomorphic automorphism groups of twisted Fock-Bargmann-Hartogs domains.Czech. Math. J. 68 (2018), 611-631. Zbl 1499.32042, MR 3851879, 10.21136/CMJ.2018.0551-16 |
| Reference: | [15] Krantz, S. G., Li, S.-Y., Rochberg, R.: The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces.Integral Equations Oper. Theory 28 (1997), 196-213. Zbl 0903.47019, MR 1451501, 10.1007/BF01191818 |
| Reference: | [16] Le, T.: Hilbert-Schmidt Hankel operators over complete Reinhardt domains.Integral Equations Oper. Theory 78 (2014), 515-522. Zbl 1318.47047, MR 3180876, 10.1007/s00020-013-2103-z |
| Reference: | [17] Li, H.: Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains.Proc. Am. Math. Soc. 119 (1993), 1211-1221. Zbl 0802.47022, MR 1169879, 10.1090/S0002-9939-1993-1169879-9 |
| Reference: | [18] Paris, R. B.: A uniform asymptotic expansion for the incomplete gamma function.J. Comput. Appl. Math. 148 (2002), 323-339. Zbl 1013.33002, MR 1936142, 10.1016/S0377-0427(02)00553-8 |
| Reference: | [19] Peloso, M. M.: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains.Ill. J. Math. 38 (1994), 223-249. Zbl 0812.47023, MR 1260841, 10.1215/ijm/1255986798 |
| Reference: | [20] Retherford, J. R.: Hilbert Space: Compact Operators and the Trace Theorem.London Mathematical Society Student Texts 27. Cambridge University Press, Cambridge (1993). Zbl 0783.47031, MR 1237405, 10.1017/CBO9781139172592 |
| Reference: | [21] Schneider, G.: A different proof for the non-existence of Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on the Bergman space.Aust. J. Math. Anal. Appl. 4 (2007), Article ID 1, 7 pages. Zbl 1220.47040, MR 2326997 |
| Reference: | [22] Temme, N. M.: Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters.Methods Appl. Anal. 3 (1996), 335-344. Zbl 0863.33002, MR 1421474, 10.4310/MAA.1996.v3.n3.a3 |
| Reference: | [23] Tu, Z., Wang, L.: Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains.J. Math. Anal. Appl. 419 (2014), 703-714. Zbl 1293.32002, MR 3225398, 10.1016/j.jmaa.2014.04.073 |
| Reference: | [24] Yamamori, A.: The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function.Complex Var. Elliptic Equ. 58 (2013), 783-793. Zbl 1272.32002, MR 3170660, 10.1080/17476933.2011.620098 |
| Reference: | [25] Zhu, K.: Hilbert-Schmidt Hankel operators on the Bergman space.Proc. Am. Math. Soc. 109 (1990), 721-730. Zbl 0731.47028, MR 1013987, 10.1090/S0002-9939-1990-1013987-7 |
| Reference: | [26] Zhu, K.: Schatten class Hankel operators on the Bergman space of the unit ball.Am. J. Math. 113 (1991), 147-167. Zbl 0734.47017, MR 1087805, 10.2307/2374825 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
