Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

Romaric Tytgat

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 399-426
  • ISSN: 0011-4642

Abstract

top
Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes f ¯ tels que l’opérateur de Hankel H 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 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. Abstract. In this paper, we give theoretical results on Macaev ideal and Dixmier trace. Then we give a characterization of antiholomorphic symbols f ¯ such that the Hankel operator H 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 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.

How to cite

top

Tytgat, Romaric. "Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids." Czechoslovak Mathematical Journal 65.2 (2015): 399-426. <http://eudml.org/doc/270088>.

@article{Tytgat2015,
abstract = {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 $\mathcal \{S\}^\{p\}$ 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. 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.},
author = {Tytgat, Romaric},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hankel operator; Dixmier trace; Bergman space},
language = {eng},
number = {2},
pages = {399-426},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids},
url = {http://eudml.org/doc/270088},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Tytgat, Romaric
TI - Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 399
EP - 426
AB - 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 $\mathcal {S}^{p}$ 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. 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.
LA - eng
KW - Hankel operator; Dixmier trace; Bergman space
UR - http://eudml.org/doc/270088
ER -

References

top
  1. Arazy, J., Fisher, S. D., Peetre, J., 10.2307/2374685, Am. J. Math. 110 (1988), 989-1053. (1988) Zbl0669.47017MR0970119DOI10.2307/2374685
  2. Arazy, J., Fisher, S. D., Peetre, J., Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. (1985) Zbl0566.30042MR0814017
  3. Axler, S., The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315-332. (1986) Zbl0633.47014MR0850538
  4. Connes, A., Noncommutative Geometry, Academic Press San Diego (1994). (1994) Zbl0818.46076MR1303779
  5. Connes, A., Moscovici, H., 10.1007/BF01895667, Geom. Funct. Anal. 5 (1995), 174-243. (1995) Zbl0960.46048MR1334867DOI10.1007/BF01895667
  6. Engliš, M., Guo, K., Zhang, G., 10.1090/S0002-9939-09-09331-9, Proc. Am. Math. Soc. 137 (2009), 3669-3678. (2009) MR2529873DOI10.1090/S0002-9939-09-09331-9
  7. Engliš, M., Rochberg, R., 10.1016/j.jfa.2009.05.005, J. Funct. Anal. 257 (2009), 1445-1479. (2009) Zbl1185.47027MR2541276DOI10.1016/j.jfa.2009.05.005
  8. Gohberg, I. C., Kreĭn, M. G., 10.1090/mmono/018/01, Translations of Mathematical Monographs 18 American Mathematical Society, Providence (1969). (1969) Zbl0181.13504MR0246142DOI10.1090/mmono/018/01
  9. Li, S.-Y., Russo, B., 10.1016/S0764-4442(97)83927-4, C. R. Acad. Sci., Paris, Sér. I, Math. 325 (1997), 21-26. (1997) Zbl0899.47019MR1461391DOI10.1016/S0764-4442(97)83927-4
  10. Luecking, D. H., 10.1016/0022-1236(92)90034-G, J. Funct. Anal. 110 (1992), 247-271. (1992) Zbl0773.47014MR1194989DOI10.1016/0022-1236(92)90034-G
  11. Meise, R., Vogt, D., Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics 2 Clarendon Press, Oxford (1997). (1997) Zbl0924.46002MR1483073
  12. Pavlović, M., Introduction to Function Spaces on the Disk, Posebna Izdanja Matematički Institut SANU, Belgrade (2004). (2004) Zbl1107.30001MR2109650
  13. Peller, V., Hankel Operators and Their Applications, Springer Monographs in Mathematics Springer, New York (2003). (2003) Zbl1030.47002MR1949210
  14. Rudin, W., Real an Complex Analysis, McGraw-Hill Series in Higher Mathematics McGraw-Hill Book Company, New York (1966). (1966) MR0210528
  15. Seip, K., Youssfi, E. H., 10.1007/s12220-011-9241-9, J. Geom. Anal. 23 (2013), 170-201. (2013) Zbl1275.47063MR3010276DOI10.1007/s12220-011-9241-9
  16. Simon, B., Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series 35 Cambridge University Press, Cambridge (1979). (1979) Zbl0423.47001MR0541149
  17. Tytgat, R., 10.7900/jot.2012dec19.1987, J. Oper. Theory 72 (2014), 241-256 French. (2014) MR3246989DOI10.7900/jot.2012dec19.1987
  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. (2007) Zbl1127.47029MR2357717
  19. Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics 226 Springer, New York (2005). (2005) Zbl1067.32005MR2115155
  20. Zhu, K., 10.1016/0022-247X(91)90091-D, J. Math. Anal. Appl. 157 (1991), 318-336. (1991) Zbl0733.30026MR1112319DOI10.1016/0022-247X(91)90091-D
  21. Zhu, K., Operator Theory in Function Spaces, Pure and Applied Mathematics 139 Marcel Dekker, New York (1990). (1990) Zbl0706.47019MR1074007

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.