Metric constrained interpolation, commutant lifting and systems
type: Item
instance of: scholarly article
title: Metric constrained interpolation, commutant lifting and systems
zbMATH Open document ID: 0923.47009
published in: Operator Theory: Advances and Applications
publication date: 1998-02-25T00:00:00Z
review text: The book under review presents a unified and comprehensive account of interpolation theory treated with tools from operator and system theories. The interdependence of the problems in these diverse theories is presented in great detail, and this book will become a standard reference in these areas. The text is divided into Part A dealing with stationary interpolation problems, and Part B dealing with nonstationary analogues. Part A has some overlap with the earlier monograph by \textit{C. Foias} and \textit{A. E. Frazho} [``The commutant lifting approach to interpolation problems'', Basel (1990; Zbl 0718.47010)] but the reader will find many new features. The first of these is the idea of stating interpolation results in which the variable is an operator rather than a complex number (see Chapter I). This may seem to be a pointless generalization, but in the hands of the authors it becomes a powerful tool for treating even the classical interpolation problems. For instance, systems of interpolation conditions for a complex variable can be written as an interpolation condition at a single (operator) point. This idea turns out to lead to ways of writing in closed, compact form the solutions of many problems, including many which are of interest in control theory. The second half of Part A is dedicated precisely to this goal, and here control theorists will find a wealth of material which did not appear in print before. Part B of the book is dedicated to two main ideas. The first one is that there is an intimate relationship between stationary and nonstationary interpolation problems. Even though nonstationary problems appear as a vast extension of the stationary ones, each nonstationary interpolation problem has an equivalent stationary formulation (in which the various linear spaces must of necessity be of infinite dimension). The second main idea is that there exists a fundamental result which is the source of most nonstationary interpolation theorems. In order to ex…
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 93A99
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 1120689
zbMATH Keywords: interpolation theory
zbMATH Keywords: stationary interpolation problems
zbMATH Keywords: nonstationary interpolation problems
zbMATH Keywords: interpolation problems of Carathéodory and Nevanlinna-Pick
zbMATH Keywords: commutant lifting theorem
zbMATH Keywords: three chains theorem
zbMATH Keywords: dilation theory
author: Ciprian Foias
author: Arthur E. Frazho
author: I. Gohberg
author: Marinus Kaashoek
reviewed by: H. Bercovici
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1120689