A family of homogeneous operators in the Cowen-Douglas class over the poly-disc

Prahllad Deb; Somnath Hazra

doi:10.4064/sm220630-10-1

A family of homogeneous operators in the Cowen-Douglas class over the poly-disc

Prahllad Deb, Somnath Hazra

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We construct a large family of positive definite kernels K : Dn × Dn → M(r,C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb×...×Möb (n times) of the bi-holomorphic automorphism group of Dn. The adjoint of the n-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space HK determined by K. We show that these n-tuples are irreducible, are in the Cowen-Douglas class Br(Dn) and are mutually pairwise unitarily inequivalent.

Original language	English
Pages (from-to)	65-84
Number of pages	20
Journal	Studia Mathematica
Volume	271
Issue number	1
DOIs	https://doi.org/10.4064/sm220630-10-1
State	Published - 1 Jan 2023

Keywords

Cowen-Douglas class
curvature
hermitian holomorphic homogeneous vector bundles
homogeneous operators

ASJC Scopus subject areas

General Mathematics

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10.4064/sm220630-10-1

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abstract = "We construct a large family of positive definite kernels K : Dn × Dn → M(r,C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup M{\"o}b×...×M{\"o}b (n times) of the bi-holomorphic automorphism group of Dn. The adjoint of the n-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space HK determined by K. We show that these n-tuples are irreducible, are in the Cowen-Douglas class Br(Dn) and are mutually pairwise unitarily inequivalent.",

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N2 - We construct a large family of positive definite kernels K : Dn × Dn → M(r,C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb×...×Möb (n times) of the bi-holomorphic automorphism group of Dn. The adjoint of the n-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space HK determined by K. We show that these n-tuples are irreducible, are in the Cowen-Douglas class Br(Dn) and are mutually pairwise unitarily inequivalent.

AB - We construct a large family of positive definite kernels K : Dn × Dn → M(r,C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb×...×Möb (n times) of the bi-holomorphic automorphism group of Dn. The adjoint of the n-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space HK determined by K. We show that these n-tuples are irreducible, are in the Cowen-Douglas class Br(Dn) and are mutually pairwise unitarily inequivalent.

KW - Cowen-Douglas class

KW - curvature

KW - hermitian holomorphic homogeneous vector bundles

KW - homogeneous operators

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SP - 65

EP - 84

JO - Studia Mathematica

JF - Studia Mathematica

IS - 1

ER -

A family of homogeneous operators in the Cowen-Douglas class over the poly-disc

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Keywords

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