Metric constrained interpolation, commutant lifting and systems
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 explain this a little more let us recall that the classical interpolation problems of Carathéodory and Nevanlinna-Pick were put under an operator theoretical umbrella by \textit{D. Sarason} [Trans. Am. Math. Soc. 127, 179-203 (1967; Zbl 0145.39303)]. A little later B. Sz.-Nagy and Foias proved the famous commutant lifting theorem (CLT) which can be viewed as a single framework out of which one can derive a host of interpolation results by merely arranging the background. Thus, finding the right conditions for the existence of interpolants (and even finding the interpolants themselves) becomes a matter of calculation. In recent years so-called nonstationary analogues were developed for the classical interpolation problems (in particular for the Carathéodory and Nevanlinna-Pick problems). Each such analogue was treated with various ad hoc methods because no analogue of the CLT was known to cover these generalizations. The authors of this book have recently found precisely such an analogue umbrella for the nonstationary interpolation problems. This result is the three chains theorem (TCT) which can be viewed as a vast generalization of the CLT. However, quite amazingly, the TCT has a very brief proof based on the CLT. Thus either the CLT or the TCT can be viewed as the main source result of linear interpolation theory, stationary and nonstationary alike. This circle of ideas, and its relation with nonstationary systems theory, is described in Part B. As in Part A, it should be noted that not all results in this part of the book appeared in article form before, so that even the specialist familiar with the journal literature will find new things here. This book brings the reader to the edge of current work in interpolation theory, and will be indispensable to researchers in the area. However, the book can also be used as an introduction to dilation theory and its applications, and as such will be of interest to graduate students. The material on system theory is also of current interest in that area.
interpolation problems of Carathéodory and Nevanlinna-Pick
scientific article; zbMATH DE number 1120689