| Title:
|
Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids (English) |
| Author:
|
Tytgat, Romaric |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
399-426 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Nous donnons des résultats théoriques sur l'idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\bar {f}$ tels que l'opérateur de Hankel $\smash {H_{\bar f}}$ sur l'espace de Bergman à poids soit dans l'idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten $\Cal {S}^{p}$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l'espace de Bergman. Nous parlons aussi des puissances de tels opérateurs. \endgraf \ehyph {\it Abstract}. In this paper, we give theoretical results on Macaev ideal and Dixmier trace. Then we give a characterization of antiholomorphic symbols $\bar {f}$ such that the Hankel operator $\smash {H_{\bar f}}$ on a Bergman weighted space is in an ideal of Macaev and we give the Dixmier trace. For this, we look at the behavior of Schatten's norms $\mathcal {S}^{p}$ when $p$ tends to $1$, using results of Engliš and Rochberg on Bergman space. We also give results on powers of such operators. (English) |
| Keyword:
|
Hankel operator |
| Keyword:
|
Dixmier trace |
| Keyword:
|
Bergman space |
| MSC:
|
32A36 |
| MSC:
|
32A37 |
| MSC:
|
47B10 |
| MSC:
|
47B35 |
| idZBL:
|
Zbl 06486956 |
| idMR:
|
MR3360436 |
| DOI:
|
10.1007/s10587-015-0185-2 |
| . |
| Date available:
|
2015-06-16T17:50:08Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144279 |
| . |
| 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] Arazy, J., Fisher, S. D., Peetre, J.: Möbius invariant function spaces.J. Reine Angew. Math. 363 (1985), 110-145. Zbl 0566.30042, MR 0814017 |
| Reference:
|
[3] Axler, S.: The Bergman space, the Bloch space, and commutators of multiplication operators.Duke Math. J. 53 (1986), 315-332. Zbl 0633.47014, MR 0850538 |
| Reference:
|
[4] Connes, A.: Noncommutative Geometry.Academic Press San Diego (1994). Zbl 0818.46076, MR 1303779 |
| Reference:
|
[5] Connes, A., Moscovici, H.: The local index formula in noncommutative geometry.Geom. Funct. Anal. 5 (1995), 174-243. Zbl 0960.46048, MR 1334867, 10.1007/BF01895667 |
| Reference:
|
[6] Engliš, M., Guo, K., Zhang, G.: Toeplitz and Hankel operators and Dixmier traces on the unit ball of $\mathbb C^{n}$.Proc. Am. Math. Soc. 137 (2009), 3669-3678. MR 2529873, 10.1090/S0002-9939-09-09331-9 |
| Reference:
|
[7] Engliš, M., Rochberg, R.: The Dixmier trace of Hankel operators on the Bergman space.J. Funct. Anal. 257 (2009), 1445-1479. Zbl 1185.47027, MR 2541276, 10.1016/j.jfa.2009.05.005 |
| Reference:
|
[8] Gohberg, I. C., Kreĭn, M. G.: Introduction to the Theory of Linear Nonselfadjoint Operators.Translations of Mathematical Monographs 18 American Mathematical Society, Providence (1969). Zbl 0181.13504, MR 0246142, 10.1090/mmono/018/01 |
| Reference:
|
[9] Li, S.-Y., Russo, B.: Hankel operators in the Dixmier class.C. R. Acad. Sci., Paris, Sér. I, Math. 325 (1997), 21-26. Zbl 0899.47019, MR 1461391, 10.1016/S0764-4442(97)83927-4 |
| Reference:
|
[10] Luecking, D. H.: Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk.J. Funct. Anal. 110 (1992), 247-271. Zbl 0773.47014, MR 1194989, 10.1016/0022-1236(92)90034-G |
| Reference:
|
[11] Meise, R., Vogt, D.: Introduction to Functional Analysis.Oxford Graduate Texts in Mathematics 2 Clarendon Press, Oxford (1997). Zbl 0924.46002, MR 1483073 |
| Reference:
|
[12] Pavlović, M.: Introduction to Function Spaces on the Disk.Posebna Izdanja Matematički Institut SANU, Belgrade (2004). Zbl 1107.30001, MR 2109650 |
| Reference:
|
[13] Peller, V.: Hankel Operators and Their Applications.Springer Monographs in Mathematics Springer, New York (2003). Zbl 1030.47002, MR 1949210 |
| Reference:
|
[14] Rudin, W.: Real an Complex Analysis.McGraw-Hill Series in Higher Mathematics McGraw-Hill Book Company, New York (1966). MR 0210528 |
| Reference:
|
[15] Seip, K., Youssfi, E. H.: Hankel operators on Fock spaces and related Bergman kernel estimates.J. Geom. Anal. 23 (2013), 170-201. Zbl 1275.47063, MR 3010276, 10.1007/s12220-011-9241-9 |
| Reference:
|
[16] Simon, B.: Trace Ideals and Their Applications.London Mathematical Society Lecture Note Series 35 Cambridge University Press, Cambridge (1979). Zbl 0423.47001, MR 0541149 |
| Reference:
|
[17] Tytgat, R.: Dixmier class of Hankel operators.J. Oper. Theory 72 (2014), 241-256 French. MR 3246989, 10.7900/jot.2012dec19.1987 |
| Reference:
|
[18] Zhu, K.: Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball.New York J. Math. (electronic only) 13 (2007), 299-316. Zbl 1127.47029, MR 2357717 |
| Reference:
|
[19] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball.Graduate Texts in Mathematics 226 Springer, New York (2005). Zbl 1067.32005, MR 2115155 |
| Reference:
|
[20] Zhu, K.: Analytic Besov spaces.J. Math. Anal. Appl. 157 (1991), 318-336. Zbl 0733.30026, MR 1112319, 10.1016/0022-247X(91)90091-D |
| Reference:
|
[21] Zhu, K.: Operator Theory in Function Spaces.Pure and Applied Mathematics 139 Marcel Dekker, New York (1990). Zbl 0706.47019, MR 1074007 |
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