Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

Grinshpan, Anatolii; Kaliuzhnyi-Verbovetskyi, Dmitry S.; Vinnikov, Victor; Woerdeman, Hugo J.

doi:10.1016/j.jfa.2008.09.012

Cited by 43 publications

(43 citation statements)

References 27 publications

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“…We finally note that an analogue of the Agler-decomposition characterization was recently obtained in [29] for the operator-valued d-variable Schur class using the methods developed in the present paper (with a reference to its earlier preprint version), namely the formal de Branges-Rovnyak model for a scattering system associated with the given Schur-class function. (This result was later reproved in [37] using standard reproducing-kernel techniques, however only in the case of scalar-valued functions .…”

Section: So That S(z) Is Realized In the Formmentioning

confidence: 68%

Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

Ball

Kaliuzhnyi-Verbovetskyi

Sadosky³

et al. 2014

Complex Anal. Oper. Theory

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A Schur-class function in d variables is defined to be an analytic contractiveoperator valued function on the unit polydisk. Such a function is said to be in the Schur-Agler class if it is contractive when evaluated on any commutative d-tuple of strict contractions on a Hilbert space. It is known that the Schur-Agler class is a strictly proper subclass of the Schur class if the number of variables d is more than two. The Schur-Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone-Roesser) of conservative multidimensional linear system. Previous work of the authors identified the Schur-Agler class as those Schurclass functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax-Phillips multievolution scattering system having some additional geoCommunicated by Irene Sabadini.C. Sadosky passed away in December 2010. We lost a collaborator of many years, and a wonderful friend. Like many others, we miss her. J. A. Ball et al. metric structure. The present paper links this additional geometric scattering structure directly with a known reproducing-kernel characterization of the Schur-Agler class. We use extensively the technique of formal reproducing kernel Hilbert spaces that was previously introduced by the authors and that allows us to manipulate formal power series in several commuting variables and their inverses (e.g., Fourier series of elements of L 2 on a torus) in the same way as one manipulates analytic functions in the usual setting of reproducing kernel Hilbert spaces.

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Section: So That S(z) Is Realized In the Formmentioning

confidence: 68%

Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

Ball

Kaliuzhnyi-Verbovetskyi

Sadosky³

et al. 2014

Complex Anal. Oper. Theory

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“…The theorem is proved. As we have pointed out before, the above von Neumann inequality for tuples in T n p,q (H) is finer and conceptually different (under the finite rank assumption) from the one obtained by Grinshpan, Kaliuzhnyi-Verbovetskyi, Vinnikov and Woerdeman [18]. 5.…”

Section: Letmentioning

confidence: 72%

“…This also allows us to prove the von Neumann inequality for tuples in T n p,q (H) (that is, Statement 2 holds). In particular, in a larger context (see the examples in Subsection 2.3), we prove that the Grinshpan, Kaliuzhnyi-Verbovetskyi, Vinnikov and Woerdeman's n-tuples of operators [18] admit explicit isometric dilations and hence yield the von Neumann inequality. Our recipe even provides sharper results with new proofs of the results of Grinshpan, Kaliuzhnyi-Verbovetskyi, Vinnikov and Woerdeman.…”

Section: Introductionmentioning

confidence: 81%

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

Barik

Das

Haria

et al. 2019

Trans. Amer. Math. Soc.

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The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in C[z] or C[z 1 , z 2 ], respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for n-tuples, n ≥ 3, of commuting contractions. The goal of this paper is to provide a taste of isometric dilations, von Neumann inequality and a refined version of von Neumann inequality for a large class of n-tuples, n ≥ 3, of commuting contractions.Statement 1 (On isometric dilations): Let T ∈ T n (H). Then there exist a Hilbert space K(⊇ H) and an n-tuple of commuting isometries V ∈ T n (K) such that T dilates to V . Now any n-tuple of commuting isometries V ∈ T n (K) can be extended to an n-tuple of commuting unitaries, that is, there exist a Hilbert space L containing K and an n-tuple of commuting unitary operators U ∈ T n (L) which extends V [24]. Therefore, the celebrated von Neumann inequality is an immediate consequence of Statement 1 (cf. [24]):

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“…Nevertheless, Nevanlinna's result survives as a theorem about the representation of elements of L n . Other than the work in [11] very little is known about the representation of functions in P n for three or more variables.…”

Section: Theorem 13 a Function H Defined On π Belongs To P If And Omentioning

confidence: 99%

Nevanlinna representations in several variables

Agler

Tully‐Doyle

Young

2016

Journal of Functional Analysis

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We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalf-plane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We find four different representation formulae and we show that every function in the Loewner class has one of the four representations, corresponding precisely to four different growth conditions at infinity.

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Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

Cited by 43 publications

References 27 publications

Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

Nevanlinna representations in several variables

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