Metric Constrained Interpolation, Commutant Lifting and Systems, Softcover reprint of the original 1st ed. 1998
Operator Theory: Advances and Applications Series, Vol. 100

This book presents a unified approach for solving both stationary and nonstationary interpolation problems, in finite or infinite dimensions, based on the commutant lifting theorem from operator theory and the state space method from mathematical system theory. Initially the authors planned a number of papers treating nonstationary interpolation problems of Nevanlinna-Pick and Nehari type by reducing these nonstationary problems to stationary ones for operator-valued functions with operator arguments and using classical commutant lifting techniques. This reduction method required us to review and further develop the classical results for the stationary problems in this more general framework. Here the system theory turned out to be very useful for setting up the problems and for providing natural state space formulas for describing the solutions. In this way our work involved us in a much wider program than original planned. The final results of our efforts are presented here. The financial support in 1994 from the "NWO-stimulansprogramma" for the Thomas Stieltjes Institute for Mathematics in the Netherlands enabled us to start the research which lead to the present book. We also gratefully acknowledge the support from our home institutions: Indiana University at Bloomington, Purdue University at West Lafayette, Tel-Aviv University, and the Vrije Universiteit at Amsterdam. We warmly thank Dr. A.L. Sakhnovich for his carefully reading of a large part of the manuscript. Finally, Sharon Wise prepared very efficiently and with great care the troff file of this manuscript; we are grateful for her excellent typing.

A Interpolation And Time-Invariant Systems.- I. Interpolation Problems For Operator-Valued Functions.- II. Proofs Using The Commutant Lifting Theorem.- III. Time Invariant Systems.- IV. Central Commutant Lifting.- V. Central State Space Solutions.- VI. Parameterization Of Intertwining Liftings And Its Applications.- VII. Applications to Control Systems.- B Nonstationary Interpolation and Time-Varying Systems.- VIII. Nonstationary Interpolation Theorems.- IX. Nonstationary Systems and Point Evaluation.- X. Reduction Techniques: From Nonstationary to Stationary and Vice Versa.- XI. Proofs of the Nonstationary Interpolation Theorems by Reduction to the Stationary Case.- XII. A General Completion Theorem.- XIII. Applications of the Three Chains Completion Theorem to Interpolation.- XIV. Parameterization of All Solutions of the Three Chains Completion Problem.- Appendix on Factorization of Matrix-Valued Functions.- A.1. Square Outer Spectral Factorizations.- A.2. Inner-Outer Factorizations.- Notes to Appendix.- References.- List of Symbols.

This monograph combines the commutant lifting theorem for operator theory and the state space method from system theory to provide a unified approach for solving both stationary and nonstationary interpolation problems with norm constraints. Included are the operator-valued versions of the tangential Nevanlinna-Pick problem, the Hermite-Fejér problem, the Nehari problem, the Sarason problem, and the two-sided Nudelman problem, and their nonstationary analogues.