J-contractive and J-inner matrix valued functions have a wide range of applications in mathematical analysis, mathematical physics, control engineering and theory of systems and networks. This book provides a comprehensive introduction to the theory of these functions with respect to the open upper half-plane, and a number of applications are also discussed. The first chapters develop the requisite background material from the geometry of finite dimensional spaces with an indefinite inner product, and the theory of the Nevanlinna class of matrix valued functions with bounded characteristic in the open upper half-plane (with attention to special subclasses). Subsequent chapters develop this theory to include associated pairs of inner matrix valued functions and reproducing kernel Hilbert spaces. Special attention is paid to the subclasses of regular and strongly regular J-inner matrix valued functions, which play an essential role in the study of the extension and interpolation problems.
Damir Z. Arov is Professor of Mathematical Analysis at South-Ukraine Pedagogical University in Odessa. He is the author of more than 100 research and survey papers and has been an invited speaker at numerous international conferences. Harry Dym is Professor Emeritus in the Department of Mathematics at the Weizmann Institute of Science in Israel. He has written more than 100 papers and this is his 5th book. Professor Dym is well know for his work in this area and his research interests are Operator Theory, Interpolation Theory, Inverse Problems and Classical Analysis.
Preface; 1. Introduction; 2. Algebraic preliminaries; 3. The Nevanlinna class of meromorphic mvf's; 4. J-contractive and J-inner matrix valued functions; 5. Reproducing kernel Hilbert spaces; 6. Generalized interpolation problems; 7. Generalized Krein extension problems; 8. Darlington representations and related inverse problems; 9. More criteria for strong regularity; 10. Formulas for entropy functionals; Bibliography.