On reducing submodules of Hilbert modules with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>-invariant kernels

Biswas, Shibananda; Ghosh, Gargi; Misra, Gadadhar; Roy, Subrata

doi:10.1016/j.jfa.2018.10.025

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…) is an n-tuple of weighted shifts where each Λ (anti) k is a weighted shift on the generalized directed semi-tree G (k) anti . Similarly, we define Λ (sym) k : 2…”

Section: Pictorial Representation Formentioning

confidence: 99%

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On analytic structure of weighted shifts on generalized directed semi-trees

Ghosh¹,

Hazra²

2022

Preprint

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Inspired by natural classes of examples, we define generalized directed semi-tree and construct weighted shifts on the generalized directed semi-trees. Given an n-tuple of directed directed semi-trees with certain properties, we associate an n-tuple of multiplication operators on a Hilbert space H 2 (β) of formal power series. Under certain conditions, H 2 (β) turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in C n and the n-tuple of multiplication operators on H 2 (β) is unitarily equivalent to an n-tuple of weighted shifts on the generalized directed semi-trees. Finally, we exhibit two classes of examples of n-tuple of operators which can be intrinsically identified as weighted shifts on generalized directed semi-trees.

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“…) is an n-tuple of weighted shifts where each Λ (anti) k is a weighted shift on the generalized directed semi-tree G (k) anti . Similarly, we define Λ (sym) k : 2…”

Section: Pictorial Representation Formentioning

confidence: 99%

“…(2) There exists a natural number m such that card(Chi(u)) ≤ m for every u ∈ V and Let X (V ) be the set of all generalized directed semi-trees on V with Property (1) and Property (2). For V = {(n, 0) : n ∈ N 0 }, the following graph yields an example of an element in X (V ).…”

Section: Introductionmentioning

confidence: 99%

On analytic structure of weighted shifts on generalized directed semi-trees

Ghosh¹,

Hazra²

2022

Preprint

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“…Now we describe a procedure developed in [4,3] to obtain joint reducing subspaces of M f acting on a Hilbert space…”

Section: Introductionmentioning

confidence: 99%

Multiplication operator on the Bergman space by a proper holomorphic map

Ghosh¹

2020

Preprint

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Suppose that f := (f1, . . . , f d ) : Ω1 → Ω2 is a proper holomorphic map between two bounded domains in C d . In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) M f = (M f 1 , . . . , M f d ) on the Bergman space A 2 (Ω1), say M. We further show that the restriction of (M f 1 , . . . , M f d ) to M is unitarily equivalent to Bergman operator on A 2 (Ω2).

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“…In the case of a commuting m-tuple of operators T in the Cowen-Douglas class B n (Ω), the existence of a spanning section was proved in [32]. Some examples of spanning sections are given in [8].…”

Section: Introductionmentioning

confidence: 99%

Operators in the Cowen-Douglas Class and Related Topics

Misra¹

2019

Handbook of Analytic Operator Theory

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Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although, the study of self adjoint operators goes back a few decades, the non-self adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like A (Ω), where Ω ⊆ C m is a bounded domain, consisting of complex valued functions which are holomorphic on some open set U containing Ω, the closure of Ω. The book [29] showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The books [31] and [14] provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.such that the vectors {γ 1 (w ), . . . , γ n (w )} are linearly independent, w ∈ U . They also show that such an operator T defines a holomorphic Hermitian vector bundle E T :This means, for any fixed but arbitrary point w 0 ∈ Ω, there exists a holomorphic map γ T of the form γ T (w ) = (γ 1 (w ), . . ., γ n (w )), (T −w )γ i (w ) = 0 in some open neighbourhood U of w 0 . It is called a holomorphic frame for the operator T . Finally, Cowen and Douglas also assume that the linear span of {γ 1 (w ), . . ., γ n (w ) : w ∈ Ω} is dense in H . Let B n (Ω) denote this class of operators.One of the striking results of Cowen and Douglas says that there is a one to one correspondence between the unitary equivalence class of the operators T and the (local) equivalence classes of the holomorphic Hermitian vector bundles E T determined by them. As a result of this correspondence set up by the Cowen-Douglas theorem, the invariants the vector bundle E T like the curvature, the second fundamental form, etc.

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On reducing submodules of Hilbert modules with Sn-invariant kernels

Cited by 7 publications

References 25 publications

On analytic structure of weighted shifts on generalized directed semi-trees

On analytic structure of weighted shifts on generalized directed semi-trees

Multiplication operator on the Bergman space by a proper holomorphic map

Operators in the Cowen-Douglas Class and Related Topics

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