The purpose of this book is to describe the theory of matrix valued functions (mvf’s). The main topics treated in the book are J-contractive and J-inner mvf’s. The book also deals with the theory of meromorphic functions applied to mvf’s and with generalisation of certain theorems from functional analysis. After an introduction, the second chapter introduces a substantial number of definitions and transformations concerning various classes of complex matrices. The considered classes are constructed on the base of a self-adjoint and unitary matrix J (which is called a signature matrix). The Potapov–Ginsburg transform and various linear fractional transformations are described here for various classes of complex matrices. The first part of the third chapter presents a rich overview of definitions and basic properties of the classes of meromorphic scalar and matrix vector functions. The classes associated with the names of Nevanlinna, Smirnov, Schur, Carathéodory and Hardy are considered. The authors study, as is usual in algebra, representations and factorization of these classes and the concepts of minimal common multiple and maximal common divisor of a given set of mvf’s. The notions of denominators and scalar denominators are developed in this part.

In the next chapter, they develop ideas from the second part. It contains a lot of profound facts from the theory of J-contractive and J-inner mvf’s for the later applications. The fifth chapter starts with definitions of a positive kernel and the reproducing kernel Hilbert space (RKHS). The authors describe a number of RKHSs, which together with de Brange space are used in the next chapter for functional models of conservative systems. Interpolation problems are discussed in the seventh chapter. The generalised interpolation problems are discussed in the second part of this chapter. The special generalised Carathéodory interpolation problem is compared with the Krein helical extension problem. Darlington representations and related inverse problems for J-inner mvf’s, criteria for strong regularity and formulas for entropy functionals and their extremal values form the content of the concluding shorter chapters of the book. The whole book is well written with theoretical results of a high standard. It is necessary for the reader to be proficient in the theory of matrices, meromorphic functions and functional analysis. It can be recommended to scientists, engineers and students preparing their doctoral theses.