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Keywords:
Hankel operator; Dixmier trace; Bergman space
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.
References:
[1] Arazy, J., Fisher, S. D., Peetre, J.: Hankel operators on weighted Bergman spaces. Am. J. Math. 110 (1988), 989-1053. DOI 10.2307/2374685 | MR 0970119 | Zbl 0669.47017
[2] Arazy, J., Fisher, S. D., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363 (1985), 110-145. MR 0814017 | Zbl 0566.30042
[3] Axler, S.: The Bergman space, the Bloch space, and commutators of multiplication operators. Duke Math. J. 53 (1986), 315-332. MR 0850538 | Zbl 0633.47014
[4] Connes, A.: Noncommutative Geometry. Academic Press San Diego (1994). MR 1303779 | Zbl 0818.46076
[5] Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), 174-243. DOI 10.1007/BF01895667 | MR 1334867 | Zbl 0960.46048
[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. DOI 10.1090/S0002-9939-09-09331-9 | MR 2529873
[7] Engliš, M., Rochberg, R.: The Dixmier trace of Hankel operators on the Bergman space. J. Funct. Anal. 257 (2009), 1445-1479. DOI 10.1016/j.jfa.2009.05.005 | MR 2541276 | Zbl 1185.47027
[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). MR 0246142 | Zbl 0181.13504
[9] Li, S.-Y., Russo, B.: Hankel operators in the Dixmier class. C. R. Acad. Sci., Paris, Sér. I, Math. 325 (1997), 21-26. DOI 10.1016/S0764-4442(97)83927-4 | MR 1461391 | Zbl 0899.47019
[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. DOI 10.1016/0022-1236(92)90034-G | MR 1194989 | Zbl 0773.47014
[11] Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford Graduate Texts in Mathematics 2 Clarendon Press, Oxford (1997). MR 1483073 | Zbl 0924.46002
[12] Pavlović, M.: Introduction to Function Spaces on the Disk. Posebna Izdanja Matematički Institut SANU, Belgrade (2004). MR 2109650 | Zbl 1107.30001
[13] Peller, V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics Springer, New York (2003). MR 1949210 | Zbl 1030.47002
[14] Rudin, W.: Real an Complex Analysis. McGraw-Hill Series in Higher Mathematics McGraw-Hill Book Company, New York (1966). MR 0210528
[15] Seip, K., Youssfi, E. H.: Hankel operators on Fock spaces and related Bergman kernel estimates. J. Geom. Anal. 23 (2013), 170-201. DOI 10.1007/s12220-011-9241-9 | MR 3010276 | Zbl 1275.47063
[16] Simon, B.: Trace Ideals and Their Applications. London Mathematical Society Lecture Note Series 35 Cambridge University Press, Cambridge (1979). MR 0541149 | Zbl 0423.47001
[17] Tytgat, R.: Dixmier class of Hankel operators. J. Oper. Theory 72 (2014), 241-256 French. DOI 10.7900/jot.2012dec19.1987 | MR 3246989
[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. MR 2357717 | Zbl 1127.47029
[19] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226 Springer, New York (2005). MR 2115155 | Zbl 1067.32005
[20] Zhu, K.: Analytic Besov spaces. J. Math. Anal. Appl. 157 (1991), 318-336. DOI 10.1016/0022-247X(91)90091-D | MR 1112319 | Zbl 0733.30026
[21] Zhu, K.: Operator Theory in Function Spaces. Pure and Applied Mathematics 139 Marcel Dekker, New York (1990). MR 1074007 | Zbl 0706.47019
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