On the Irreducibility of a Class of Homogeneous Operators

Misra, Gadadhar; Roy, Subrata

doi:10.1007/978-3-7643-8137-0_3

Cited by 5 publications

(8 citation statements)

References 20 publications

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“…defines an unitary representation of G onto H K , provided U (H K ) ⊂ H K . We now recall following theorem about equivalent conditions for homogeneous operators from [7]. (b) The reproducing kernel K is quasi-invariant, that is, it transforms according to the rule K(z, w) = J g (z)K(gz, gw)J g (w) * , for all g ∈ G and z, w ∈ Ω for some continuous function J : G × Ω → GL(r, C) holomorphic in the second variable denoted by J(g, z) = J g (z).…”

Section: Homogeneity Of Multiplication Operators Of Quotient Modulesmentioning

confidence: 99%

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On irreducibility of a certain class of homogeneous operators obtained from quotient modules

Biswas¹,

Deb²,

Roy³

2022

Preprint

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Let Ω ⊂ C m be an open, connected and bounded set and A(Ω) be a function algebra of holomorphic functions on Ω. Suppose that Mq is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module M vanishing to order k along a smooth irreducible complex analytic set Z ⊂ Ω of codimension at least 2. In this article, we prove that the compression of the multiplication operators onto Mq is homogeneous with respect to a suitable subgroup of the automorphism group Aut(Ω) of Ω depending upon a subgroup G of Aut(Ω) whenever the tuple of multiplication operators on M is homogeneous with respect to G and both M as well as Mq are in the Cowen-Douglas class. We show that these compression of multiplication operators might be reducible even if the tuple of multiplication operators on M is irreducible by exhibiting a concrete example. Moreover, the irreducible components of these reducible operators are identified as Generalized Wilkins' operators.

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Section: Homogeneity Of Multiplication Operators Of Quotient Modulesmentioning

confidence: 99%

“…In Section 4, we discuss on irreducibility of the homogeneous operators obtained in Section 3. We point out from the example given in [7,Theorem 6.1] that the compression of the tuple of multiplication operators M onto H q , in general, is not irreducible. In this regard, we take Ω to be D m and consider the subspace…”

Section: Introductionmentioning

confidence: 98%

On irreducibility of a certain class of homogeneous operators obtained from quotient modules

Biswas¹,

Deb²,

Roy³

2022

Preprint

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“…The set {δ i > 0 : i = 1, 2, 3} satisfying the inequalities of Proposition 3.8 is nonempty. For instance, take δ 1 = 1, δ 2 = 2 and any δ 3 in the interval (9,12). Then {δ 1 , δ 2 , δ 3 } meets the requirement.…”

Section: It Will Be Convenient To Letmentioning

confidence: 99%

“…Thus the unitary equivalence class of the curvature at 0 determines the unitary equivalence class of these operators within the class of the generalised Wilkins operarotrs of rank k + 1 (cf. [12], [2, page 428]).…”

Section: Introductionmentioning

confidence: 99%

The curvature invariant for a class of homogeneous operators

Misra

Roy

2009

Proceedings of the London Mathematical Society

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For an operator T in the class Bn(Ω), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle ET determine the unitary equivalence class of the operator T . In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T ∈ Bn(Ω). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in Bn(D). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc D are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

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“…However, such a classification appears to be intractable. On the other hand, using the jet construction of [11], we produce many examples of homogeneous operators [3,17]. Unfortunately, this construction does not give a complete list of homogeneous operators in B n (D) except in the case of n = 1, 2.…”

mentioning

confidence: 97%