Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables
type: Item
instance of: scholarly article
title: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables
zbMATH Open document ID: 0914.47020
author: Joseph A. Ball
author: Tavan T. Trent
published in: Journal of Functional Analysis
publication date: 1999-06-06T00:00:00Z
review text: Some progress of late in \(d\)-variable systems and interpolation theories was initiated by the remarkable paper of \textit{J. Agler} [Oper. Theory, Adv. Appl. 48, 47-66 (1990; Zbl 0733.32002)] where the generalized Schur class \({\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) was introduced and described in different terms. This class consists of all holomorphic functions \(W(z)=\sum_{t\in {\mathbb Z}^d_+}\widehat{W}_tz^t\) on the polydisk \({\mathbb D}^d\) (here \({\mathbb Z}^d_+=\{ t\in {\mathbb Z}^d| t_k\geq 0, k=1,\ldots ,d\}\), \(z^t={z_1}^{t_1}\cdots {z_d}^{t_d}\)), with values in \({\mathcal L}({\mathcal E}, {\mathcal E}_*)\) for Hilbert spaces \({\mathcal E}\) and \({\mathcal E}_*\), such that for any \(d\)-tuple \(T=(T_1,\ldots ,T_d)\) of commuting contractions on some Hilbert space \({\mathcal X}\) and any \(0<r<1\) the inequality \(| | W(rT)| | \leq 1\) holds, where \(W(rT)=W(rT_1,\ldots , rT_d)=\sum_{t\in{\mathbb Z}^d_+}\widehat{W}_t\otimes {(rT)}^t\in {\mathcal L}({\mathcal E}\otimes {\mathcal X}, {\mathcal E}_*\otimes {\mathcal X})\). J. Agler proved that \(W(z)\in {\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) if and only if there exist auxiliary Hilbert spaces \({\mathcal M}_k\) and holomorphic functions \(H_k(z)\) on \({\mathbb D}^d\), with values in \({\mathcal L}({\mathcal M}_k, {\mathcal E}_*)\), \(k=1,\ldots ,d\), such that (*) \(I- W(z)W(w)^*=\sum_{k=1}^d(1-z_k\bar{w}_k)H_k(z)H_k(w)^*\) \((z, w \in {\mathbb D}^d)\), and that \(W(z)\in {\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) if and only if there exists a unitary colligation \(U=({\mathcal H}={\mathcal H}_1\oplus\cdots\oplus {\mathcal H}_d, {\mathcal E}, {\mathcal E}_*; A, B, C, D)\) such that (**) \(W(z)=D+CZ(z){(I-AZ(z))}^{-1}B\) \((z\in {\mathbb D}^d)\), where \(Z(z)=\sum_{k=1}^dz_kP_k\) and \(P_k: {\mathcal H}\to {\mathcal H}_k\) is the orthogonal projection. Besides, Agler established a criterion for the solvability of the \(d\)-variable Nevanlinna-Pick problem, that is \(W(z^i)=w_i\)…
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 47A48
Mathematics Subject Classification ID: 93C35
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 1214062
zbMATH Keywords: unitary colligations
zbMATH Keywords: Nevanlinna-Pick interpolation
zbMATH Keywords: \(d\)-variable linear systems
zbMATH Keywords: generalized Schur class
zbMATH Keywords: linear input-state-output system of the Roesser type
zbMATH Keywords: functional model
zbMATH Keywords: Toeplitz corona theorem
reviewed by: Dmitry Kaliuzhnyi-Verbovetskyi
MaRDI profile type: Publication
full work available at URL: jfan.1998.3278
OpenAlex ID: W2018960677
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DOI: 10.1006/JFAN.1998.3278
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description: scientific article; zbMATH DE number 1214062