New constructions of homogeneous operators

Adam Korányi; Gadadhar Misra

doi:10.1016/j.crma.2006.04.002

Functional Analysis

New constructions of homogeneous operators

Presented by: Paul Malliavin

Adam Korányi ¹; Gadadhar Misra ²

¹ Lehman College, Bronx, NY 10468, USA
² Indian Statistical Institute, R. V. College Post, Bangalore 560 059, India

Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 933-936.

Abstract
Résumé

New examples of homogeneous operators involving infinitely many parameters are constructed. They are realized on Hilbert spaces of holomorphic functions with reproducing kernels which are computed explicitly. All the examples are irreducible and belong to the Cowen–Douglas class. Even though the construction is completely explicit, it is based on certain facts about Hermitian holomorphic homogeneous vector bundles. These facts also make possible a description of all homogeneous Cowen–Douglas operators, in a somewhat less explicit way.

Ou construit une nouvelle famille d'examples d'opérateurs homogènes dépendant d'une infinité de paramètres. Les exemples sont réalisés sur des espaces de fonctions holomorphes possédant des noyaux reproduisants qu'on calcule explicitement. Les exemples sont tous des opérateurs irréductibles appartenant à la classe de Cowen–Douglas. Tout en étant complètement explicite, la construction est fondée sur certaines propriétés des fibrés vectoriels hermitiens holomorphes homogènes. Ces propriétés permettent aussi une description, un peu moins explicite, de tous les opérateurs homogènes de la classe de Cowen–Douglas.

Received: 2006-03-14
Accepted: 2006-03-30
Published online: 2006-05-11

DOI: 10.1016/j.crma.2006.04.002

Author's affiliations:

Adam Korányi ¹; Gadadhar Misra ²

¹ Lehman College, Bronx, NY 10468, USA
² Indian Statistical Institute, R. V. College Post, Bangalore 560 059, India

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Adam Korányi; Gadadhar Misra. New constructions of homogeneous operators. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 933-936. doi : 10.1016/j.crma.2006.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.002/

References
Cited by

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[3] B. Bagchi; G. Misra The homogeneous shifts, J. Funct. Anal., Volume 204 (2003), pp. 293-319

[4] M.J. Cowen; R.G. Douglas Complex geometry and operator theory, Acta Math., Volume 141 (1978), pp. 187-261

[5] R.E. Curto; N. Salinas Generalized Bergman kernels and the Cowen–Douglas theory, Amer. J. Math., Volume 106 (1984), pp. 447-488

[6] A.A. Kirillov Elements of the Theory of Representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin, 1976

[7] D.R. Wilkins Homogeneous vector bundles and Cowen–Douglas operators, Int. J. Math., Volume 4 (1993), pp. 503-520

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