Homogeneous operators and projective representations of the Möbius group: A survey

Bagchi, Bhaskar; Misra, Gadadhar

doi:10.1007/bf02829616

Cited by 23 publications

(47 citation statements)

References 16 publications

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“…In subsection 6.2, we show that the angle invariant, which together with the transverse and the tangential curvatures forms a complete set of unitary invariants for the quotient module Q can be replaced by the second fundamental form I for the inclusion E ⊆ J (2) E. In view of the equation (6.10), we have stated the theorem in terms of the restriction of the curvature. One of the disadvantages in using the angle as an invariant for the isomorphism class of the quotient module is that for it to make sense we must introduce normalized reproducing kernels (cf.…”

Section: The Second Fundamental Formmentioning

confidence: 99%

“…These applications involve homogeneity of the modules under the action of the Möbius group. The study of homogeneity for Hilbert modules over the algebra A(Ω) for a bounded symmetric domain Ω ⊆ C m was initiated in [24] and was further studied in [2]. However, in these papers, it was assumed that Ω is irreducible.…”

Section: The Second Fundamental Formmentioning

confidence: 99%

“…, it is enough to describe this map just for the orthonormal basis {e (1) p , e (2) p : p ≥ 0}. A simple calculation shows that…”

Section: Theorem 3 Let M Be a Hilbert Module Over The Bi-disc Algebramentioning

confidence: 99%

“…While two of the three invariants obtained there consist of coefficients of the curvature form for the jet bundle for E, the third, which we called the "angle", seemed to be not so familiar. In this note, we show that the angle invariant can be replaced by the second fundamental form corresponding to the inclusion of the bundle E in the jet bundle J (2) E. Our main goal, however, is to obtain invariants that are complete, computable and natural in complex geometry -for general k, not just in the case k = 2. We'll state our results in the following section after we introduce the necessary notation.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Second Fundamental Formmentioning

confidence: 99%

Section: The Second Fundamental Formmentioning

confidence: 99%

“…, it is enough to describe this map just for the orthonormal basis {e (1) p , e (2) p : p ≥ 0}. A simple calculation shows that…”

Section: Theorem 3 Let M Be a Hilbert Module Over The Bi-disc Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivalence of quotient Hilbert modules–II

Douglas

Misra

2007

Trans. Amer. Math. Soc.

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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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Scalar perturbations of the Sz.-Nagy—Foias characteristic function

Bagchi

Misra

2001

Recent Advances in Operator Theory and Related Topics

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Homogeneous operators and projective representations of the Möbius group: A survey

Cited by 23 publications

References 16 publications

Equivalence of quotient Hilbert modules–II

Equivalence of quotient Hilbert modules–II

New constructions of homogeneous operators

Scalar perturbations of the Sz.-Nagy—Foias characteristic function

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