Metric Constrained Interpolation, Commutant Lifting and Systems

de C. Foias

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. The main results concern the existence of solutions, the explicit construction of the central solutions in state space form, the maximum entropy property of the central solutions, and state space parametrizations of all solutions. Direct connections between the various interpolation problems are displayed. Applications to H[infinity] control problems are presented. This monograph should appeal to a wide group of mathematicians and engineers. The material is self-contained and may be used for advanced graduate courses and seminars.