A Beurling type theorem in weighted Bergman spaces

Joseph A. Ball; Vladimir Bolotnikov

doi:10.1016/j.crma.2013.06.004

Mathematical Analysis

A Beurling type theorem in weighted Bergman spaces

Presented by: Gilles Pisier

Joseph A. Ball ¹; Vladimir Bolotnikov ²

¹ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
² Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 433-436.

Abstract
Résumé

For the vector-valued Hardy space $H^{2} (U)$ and the standard weighted Bergman space $A_{n} (Y)$ with coefficient Hilbert spaces $U$ and $Y$ , we single out a class of contractive multipliers from $H^{2} (U)$ to $A_{n} (Y)$ which we call partially isometric multipliers. We then show that a closed subspace $M \subset A_{n} (Y)$ is invariant under the shift operator $S_{n} : f (z) \mapsto z f (z)$ if and only if $M = Φ \cdot H^{2} (U)$ for some partially isometric multiplier Φ from $H^{2} (U)$ to $A_{n} (Y)$ .

Soit $H^{2} (U)$ lʼespace de Hardy aux valeurs vectorielles et soit $A_{n} (Y)$ lʼespace de Bergman aux valeurs vectorielles et au poids ${(1 - {| z |}^{2})}^{n - 2}$ , où les espaces des coefficients $U$ et $Y$ sont des espaces de Hilbert. Nous considérons une classe de multiplicateurs contractifs de $H^{2} (U)$ dans $A_{n} (Y)$ , que nous appelons multiplicateurs isométriques partiels. Nous montrons quʼun sous-espace $M \subset A_{n} (Y)$ qui est invariant pour lʼoperateur $S_{n} : f (z) \mapsto z f (z)$ est inclus isometriquement dans $A_{n} (Y)$ si et seulement si $M = Φ \cdot H^{2} (U)$ pour un multiplicateur isométrique partiel Φ de $H^{2} (U)$ dans $A_{n} (Y)$ .

Received: 2013-03-25
Accepted: 2013-06-19
Published online: 2013-06-01

DOI: 10.1016/j.crma.2013.06.004

Author's affiliations:

Joseph A. Ball ¹; Vladimir Bolotnikov ²

¹ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
² Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

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     author = {Joseph A. Ball and Vladimir Bolotnikov},
     title = {A {Beurling} type theorem in weighted {Bergman} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {433--436},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     year = {2013},
     doi = {10.1016/j.crma.2013.06.004},
     language = {en},
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Joseph A. Ball; Vladimir Bolotnikov. A Beurling type theorem in weighted Bergman spaces. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 433-436. doi : 10.1016/j.crma.2013.06.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.06.004/

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