A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur--Agler class
type: Item
instance of: scholarly article
title: A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur--Agler class
zbMATH Open document ID: 1015.47004
DOI: 10.1016/S0022-247X(02)00226-3
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: It is now well understood that the interpolation theory of Nevanlinna-Pick type in several complex variables faces serious obstructions. The optimal way to circumvent these difficulties was proposed by \textit{J. Agler} in [Oper. Theory, Adv. Appl. 48, 47-66 (1990; Zbl 0733.32002)]. He replaced the space of bounded analytic functions in a polydisk by a smaller class, nowadays called the Schur-Agler class, on which interpolation problems can be solved in terms of positivity conditions, exactly as in the single variable case. A great deal of energy was put, by many working mathematicians, into understanding this approach to function theory of several variables. The article under review proves the correct tangential interpolation results for the polydisk, with respect to the Schur-Agler class. The main tool is the realization of Schur-Agler functions as characteristic functions of certain operator colligations. The basic theorems and their proofs parallel the classical single variable situation. This very clearly written article is self-contained and accessible to non-experts. I recommend it to all mathematicians interested in modern operator theory.
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 47A48
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 32A70
zbMATH DE Number: 1866565
zbMATH Keywords: interpolation
zbMATH Keywords: Schur-Agler class
zbMATH Keywords: characteristic functions of operator colligations
author: Joseph A. Ball
author: Vladimir Bolotnikov
MaRDI profile type: Publication
cites work: Q3360462
cites work: Nevanlinna-Pick interpolation on the bidisk
cites work: Q5727877
cites work: Q4529415
cites work: Q3331764
cites work: Scattering Matrices in the Theory of Unitary Extension of Isometric Operators
cites work: A bitangential interpolation problem on the closed unit ball for multipliers of the Arveson space
cites work: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions
cites work: Errata to: Two-sided tangential interpolation of real rational matrix functions
cites work: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions
cites work: A commutant lifting theorem on the polydisc
cites work: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables
cites work: On degenerate interpolation, entropy and extremal problems for matrix Schur functions
cites work: On boundary interpolation for matrix valued Schur functions
cites work: Linear Matrix Inequalities in System and Control Theory
cites work: Lossless inverse scattering, digital filters, and estimation theory
cites work: Q3032797
cites work: Advances in Linear Matrix Inequality Methods in Control
cites work: Some adaptive control problems which convert to a "classical" problem in several complex variables
cites work: An Abstract Interpolation Problem and the Extension Theory of Hermitian Operators
cites work: Hamburger moment problem: Parseval equality and \(A\)-singularity
cites work: Q3840243
cites work: The Carathéodory-Julia theorem for matrix-functions
cites work: A multiple boundary interpolation problem for contracting matrix-valued functions in the unit circle
cites work: Interpolation of nevanlinna-pick type on the boundary of the bidisc
cites work: Nevanlinna-Pick interpolation with boundary data
Recommended article: Nevanlinna-Pick interpolation for Schur-Agler class functions on domains with matrix polynomial defining function in \(\mathbb C^n\)
Recommended article: Q5308964
Recommended article: Interpolation in semigroupoid algebras
Recommended article: Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions
Recommended article: Q3645519
Recommended article: Remarks on the operator-valued interpolation for multivariable bounded analytic functions
Recommended article: Characteristic function for polynomially contractive commuting tuples
Recommended article: Products of Nevanlinna-Pick kernels and operator colligations
Recommended article: Interpolation problems for Schur multipliers on the Drury--Arveson space: from Nevanlinna--Pick to abstract interpolation problem
Recommended article: A commutant lifting theorem on the polydisc
description: scientific article; zbMATH DE number 1866565