Interpolation in semigroupoid algebras

Dritschel, Michael A.; Marcantognini, Stefania; McCullough, Scott

doi:10.1515/crelle.2007.033

Cited by 22 publications

(39 citation statements)

References 33 publications

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“…(20) in the seventh. Now use linearity and the fact that the linear span of elements like F is dense in L 2 (μ) ⊗ 2 to finish the proof of the first part of Lemma 7.2.…”

Section: Ieotmentioning

confidence: 99%

“…Results going back to [5] and including [3,4,7,10,15,16,19,20] among others view the starting point for Agler-Pick interpolation as a collection of functions Ψ, called test functions. Roughly speaking one constructs an operator algebra whose norm is as large as possible subject to the condition that each ψ ∈ Ψ is contractive.…”

Section: Introductionmentioning

confidence: 99%

“…A sample of references include [8,12,16,25]. Of special relevance for this paper is the work of Ambrozie [10] and the subsequent articles [19,20], where the set of test functions Ψ is allowed to be infinite with a compact Hausdorff topology.…”

Section: Introductionmentioning

confidence: 99%

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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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“…(20) in the seventh. Now use linearity and the fact that the linear span of elements like F is dense in L 2 (μ) ⊗ 2 to finish the proof of the first part of Lemma 7.2.…”

Section: Ieotmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Agler-Commutant Lifting on an Annulus

McCullough

Sultanic

2012

Integr. Equ. Oper. Theory

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“…Results going back to [1] and including [2,5,6,10,11,14,15] among others view the starting point for Agler-Pick interpolation as a collection of functions Ψ. Roughly speaking one then constructs an operator algebra whose norm is as large as possible subject to the condition that each ψ ∈ Ψ is contractive.…”

Section: Introductionmentioning

confidence: 99%

“…The corresponding AglerSchur class, or Ψ-Agler-Schur class, is then the unit ball of this operator algebra and interpolation is within this class. While much of the work assumes and exploits to some extent analytic structure, it turns out that much goes through independent of analyticity [2,14]. Indeed, the reproducing kernel Hilbert spaces impose an ersatz analytic structure.…”

Section: Introductionmentioning

confidence: 99%

Ersatz Commutant Lifting with Test Functions

McCullough

Sultanic

2007

Complex anal.oper. theory

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This note presents a commutant lifting theorem (CLT) with initial data a finite set of (test) functions and a compatible reproducing kernel k on a set X. This covers the CLT of Ball, Li, Timotin, and Trent [9] for the polydisc, but in general no analyticity is required, rather statements and proofs use the language and techniques of reproducing kernel Hilbert spaces. Uniqueness of the de Branges-Rovnyak construction like found in Agler [1] and Ambrozie, Englis, and Müller [5] and an abstract Beurling Theorem in the present context are of independent interest. Mathematics Subject Classification (2000). 47A57 (Primary), 47A13, 47L75, 47B38, 46E22 (Secondary).

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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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Interpolation in semigroupoid algebras

Cited by 22 publications

References 33 publications

Agler-Commutant Lifting on an Annulus

Agler-Commutant Lifting on an Annulus

Ersatz Commutant Lifting with Test Functions

Multivariable Operator-valued Nevanlinna-Pick Interpolation: a Survey

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