An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Sarkar, Jaydeb

doi:10.1007/978-3-0348-0692-3_59-1

Cited by 7 publications

(25 citation statements)

References 121 publications

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“…Now, we introduce the notion of isometric dilation of an n-tuple operators (cf. [15]). Let T and V be n-tuples of operators on Hilbert spaces H and K, respectively.…”

Section: Preliminariesmentioning

confidence: 99%

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Analytic model of doubly commuting contractions

Bhattacharyya¹,

Eswar²,

Sarkar³

2017

Operators and Matrices

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An n-tuple (n ≥ 2), T = (T 1 , . . . , T n ), of commuting bounded linear operators on a Hilbert space H is doubly commuting if T i T * j = T * j T i for all 1 ≤ i < j ≤ n. If in addition, each T i ∈ C •0 , then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple T can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H 2 D T * (D n ). As a consequence of the dilation theorem, we prove that there exists a closed subspace S T of the formwhere) is either a one variable inner function in z i , or the zero function.

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“…Now, we introduce the notion of isometric dilation of an n-tuple operators (cf. [15]). Let T and V be n-tuples of operators on Hilbert spaces H and K, respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…The proofs of the dilation theorem obtained in this way are quite different from any earlier proofs (cf. [11], [6], [4], [15]).…”

Section: Isometric Dilationmentioning

confidence: 99%

Analytic model of doubly commuting contractions

Bhattacharyya¹,

Eswar²,

Sarkar³

2017

Operators and Matrices

Self Cite

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“…Hence, UT U * = M Θ for some Θ(z) ∈ H ∞ (B(W(V ))) and moreover, M Θ ≤ 1 (see [27]). As in Proposition 3.1, we consider the contractions A T and A M Θ corresponding to the contractions T and M Θ , respectively.…”

Section: And Vjmentioning

confidence: 97%

“…, and the corresponding defect spaces are [27]). Thus, proving that T dilates to M z on H 2 (D T * ).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Pairs of Commuting Pure Contractions and Isometric dilation

Sarkar

2021

Preprint

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A particular case of the fundamental Sz.-Nagy-Foias functional model for a contraction states that a pure contraction always dilates to a pure isometry. We are interested in the similar question for pairs, more precisely: does a pair of commuting pure contractions always dilate to a pair of commuting pure isometries? The purpose of this article is to identify pairs of commuting pure contractions for which the above question has an affirmative answer. Our method is based on an explicit structure of isometric dilation for pure pairs of commuting contractions obtained in a recent work by Das, Sarkar and the author.

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“…, n simultaneously is called a shift-invariant subspace. The description of all shift-invariant subspaces for n > 1 is a central open problem of multivariable operator theory (see the recent surveys of Sarkar [13] and Yang [14]) which could therefore lead to a solution of the ISP.…”

Section: Introductionmentioning

confidence: 99%

Universal Toeplitz operators on the Hardy space over the polydisk

Ferreira,

Noor

2020

Preprint

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The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators T φ on the Hardy space H 2 (D n ) over the polydisk D n for n > 1 whose adjoints satisfy the Caradus criterion for universality, that is, when T * φ is surjective and has infinite dimensional kernel. In particular if φ in a non-constant inner function on D n , or a polynomial in the ring C[z 1 , . . . , zn] that has zeros in D n but is zero-free on T n , then T * φ is universal for H 2 (D n ). The analogs of these results for n = 1 are not true.

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An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Cited by 7 publications

References 121 publications

Analytic model of doubly commuting contractions

Analytic model of doubly commuting contractions

Pairs of Commuting Pure Contractions and Isometric dilation

Universal Toeplitz operators on the Hardy space over the polydisk

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