Homogeneous Vector Bundles and Cowen-Douglas Operators

Abstract. In this paper we construct a class of homogeneous Hilbert modules over the disc algebra A(D) as quotients of certain natural modules over the function algebra A(D 2 ). These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class B k (D) and that they are mutually inequivalent, irreducible and homogeneous.

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“…[17], see also [22]. An alternative proof is obtained when we observe that Γ : Möb × D −→ C, where Γ ϕ −1 (z) =…”

Section: The Case Of the Tri-disc Dmentioning

confidence: 97%

“…This includes the calculation of the compression of the two operators, M 1 : f → z 1 f and M 2 : f → z 2 f for f ∈ M (α,β) , on the quotient module Q (block weighted shift operators) with respect to this orthonormal basis. These are homogeneous operators in the class B 2 (D) which were first discovered by Wilkins [22].…”

Section: Introductionmentioning

confidence: 99%

On the Irreducibility of a Class of Homogeneous Operators

Misra

Roy

2007

System Theory, the Schur Algorithm and Multidimensional Analysis

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“…For a complete discussion, we refer the reader to the survey paper [3]. D. Wilkins [28] has obtained a classification of all homogeneous Hilbert modules over the disc algebra which are in the class B k (D) for k > 1. However, he was able to give an explicit description of these modules only for rank 2.…”

Section: The Second Fundamental Formmentioning

confidence: 99%

Equivalence of quotient Hilbert modules–II

Douglas

Misra

2007

Trans. Amer. Math. Soc.

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Abstract.For any open, connected and bounded set Ω ⊆ C m , let A be a natural function algebra consisting of functions holomorphic on Ω. Let M be a Hilbert module over the algebra A and let M 0 ⊆ M be the submodule of functions vanishing to order k on a hypersurface Z ⊆ Ω. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module Q = M M 0 in the case of k = 2. In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.

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“…[7,2,3,1]). In this note a large family of examples is constructed and a step is made towards the description of all such operators in the Cowen -Douglas class of D (see [4]).…”

Section: Ou Construit Une Nouvelle Famille D'examples D'opérateurs Homentioning

confidence: 99%

New constructions of homogeneous operators

Korányi

Misra

2006

Comptes Rendus. Mathématique

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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 appartenantà la classe de CowenDouglas. Tout enétant 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.An operator T on a Hilbert space is said to be homogeneous if its spectrum is contained in the closure of the unit disc D in C and if g(T ) is unitarily equivalent to T for every element g of the holomorphic automorphism group G of D. There are general results about such operators, but relatively few examples are known (cf. [7,2,3,1]). In this note a large family of examples is constructed and a step is made towards the description of all such operators in the Cowen -Douglas class of D (see [4]). Construction of the examplesLetG denote the universal covering group of G. For λ > 0, let A (λ) denote the Hilbert space of holomorphic functions on D with reproducing kernel (1 − zw) −2λ . The well-known discrete series D + λ of unitary representation ofG acts on A (λ) by

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Homogeneous Vector Bundles and Cowen-Douglas Operators

Cited by 26 publications

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On the Irreducibility of a Class of Homogeneous Operators

On the Irreducibility of a Class of Homogeneous Operators

Equivalence of quotient Hilbert modules–II

New constructions of homogeneous operators

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