Ersatz Commutant Lifting with Test Functions

McCullough, Scott; Sultanic, Saida

doi:10.1007/s11785-007-0022-1

Cited by 6 publications

(9 citation statements)

References 14 publications

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“…Note that Q diag (z) is the special case of this where each d k = 1. One can also see the ideas of [2] as influencing the test-function approach of Dritschel-McCullough and collaborators, originating in [36,37] with followup work in [59,39,50,22].…”

Section: Agler Decomposition and Transfer-function Realization For Ncmentioning

confidence: 99%

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Ball¹,

Marx²,

Vinnikov

2018

Operator Theory in Different Settings and Related Applications

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The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur-Agler-class theory to the free noncommutative function setting. This includes the positive-kernel-decomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur-Agler class coincides with the contractive-multiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.1991 Mathematics Subject Classification. 47B32; 47A60.

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Section: Agler Decomposition and Transfer-function Realization For Ncmentioning

confidence: 99%

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Ball¹,

Marx²,

Vinnikov

2018

Operator Theory in Different Settings and Related Applications

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“…The lurking isometry in turn generates the Ψ-unitary colligation. A good deal of effort is required to show that the resulting transfer function solves the problem and the argument given here is patterned after that in [29], which in turn borrowed from [11,14] and closely related to those in [3].…”

Section: The Factorization and Lurking Isometrymentioning

confidence: 99%

“…Ambrozie and Eschmeier [3] establish a related CLT for the unit ball in C n . In [29] we establish a generalization of these results to the case of a finite collection Ψ together with a distinguished reproducing kernel Hilbert space H 2 (k), unlocking the prior tight connection between the coordinate (test) functions {z 1 , . .…”

Section: Introductionmentioning

confidence: 98%

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Agler-Commutant Lifting on an Annulus

McCullough

Sultanic

2012

Integr. Equ. Oper. Theory

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This note presents a commutant lifting theorem (CLT) of Agler type for the annulus A. Here the relevant set of test functions are the minimal inner functions on A-those analytic functions on A which are unimodular on the boundary and have exactly two zeros in A-and the model space is determined by a distinguished member of the Sarason family of kernels over A. The ideas and constructions borrow freely from the CLT of Ball et al. (Indiana Univ Math J 48(2):653-675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) for the polydisc, and Ambrozie and Eschmeier (A commutant lifting theorem on analytic polyhedra. Topological algebras, their applications, and related topics, 83108, Banach Center Publications, vol 67. Polish Academy of Sciences, Warsaw, 2005) for the ball in C n , as well as generalizations of the de Branges-Rovnyak construction like found in Agler (On the representation of certain holomorphic functions defined on a polydisc. Topics in operator theory: Ernst D. Hellinger memorial volume, operator theory: advances and applications, vol 48. Birkhäuser, Basel, pp 47-66, 1990) and Ambrozie et al. (J Oper Theory 47(2):287-302, 2002). It offers a template for extending the result in McCullough and Sultanic (Complex Anal Oper Theory 1(4):581-620, 2007) to infinitely many test functions. Among the needed new ingredients is the formulation of the factorization implicit in the statement of the results in Ball et al. (Indiana Univ Math J 48(2):653-675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) and McCullough and Sultanic (Complex Anal Oper Theory 1(4):581-620, 2007) in terms of certain functional Hilbert spaces of Hilbert space valued functions. Mathematics Subject Classification (2010). Primary 47A20; Secondary 47A48, 47A57, 47B32.

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“…There is yet another generalized theory of Schur class (or, more precisely, Schur-Agler class) and Nevanlinna-Pick interpolation where one defines a Schur class starting from a family of test functions (see [41,43,59] and see also [6,Chapter 13] for an introduction to this approach). It is well appreciated that the Agler theory for the polydisk is an example for this theory (indeed, this is the motivating example); precise specification of what other examples can be covered, such as the higher rank graph algebras of [57] and the Hardy algebras associated with product decompositions along semigroups more general than Z (see [88]), is an ongoing area of investigation [15].…”

Section: Introductionmentioning

confidence: 99%

Multivariable Operator-valued Nevanlinna-Pick Interpolation: a Survey

Ball

Horst

2009

Operator Algebras, Operator Theory and Applications

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The theory of Nevanlinna-Pick and Carathéodory-Fejér interpolation for matrix-and operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S. Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock space built from a W * -correspondence E over a W * -algebra A and a * -representation σ of A. In this review we provide an exposition of the Muhly-Solel interpolation theory accessible to operator theorists, and explain more fully the connections with the already existing interpolation literature. The abstract point evaluation first introduced by Muhly-Solel leads to a tensor-product type functional calculus in the main examples. A second kind of point-evaluation for the W * -correspondence Hardy algebra, also introduced by Muhly and Solel, is here further investigated, and a Nevanlinna-Pick theorem in this setting is proved. It turns out that, when specified for examples, this alternative point-evaluation leads to an operator-argument functional calculus and corresponding Nevanlinna-Pick interpolation. We also discuss briefly several Nevanlinna-Pick interpolation results for Schur classes that do not fit into the Muhly-Solel W * -correspondence formalism.1991 Mathematics Subject Classification. Primary 47A57; Secondary 42A83, 47A48.

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Ersatz Commutant Lifting with Test Functions

Cited by 6 publications

References 14 publications

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Agler-Commutant Lifting on an Annulus

Multivariable Operator-valued Nevanlinna-Pick Interpolation: a Survey

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