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Keywords:
Hankel operator; Dixmier trace; Bergman space
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.
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