This book is dedicated to the memory of an outstanding mathematician and personality, Vladimir Petrovich Potapov, who made important contributions to and exerted considerable influence in the areas of operator theory, complex analysis and their points of juncture. The book commences with insightful biographical material, and then presents a collection of papers on different aspects of operator theory and complex analysis covering those recent achievements of the Odessa-Kharkov school in which Potapov was very active. The papers deal with interrelated problems and methods. The main topics are the multiplicative structure of contractive matrix and operator functions, operators in spaces with indefinite scalar products, inverse problems for systems of differential equations, interpolation and approximation problems for operator and matrix functions. The book will appeal to a wide group of mathematicians and engineers, and much of the material can be used for advanced courses and seminars.
The influence of V.P. Potapov and M.G. Krein on my scientific work.- 1. My first dissertation.- 2. A tilt toward operator theory.- 3. The results of Potapov’s group in network theory.- 4. Darlington method in the general theory of passive systems.- 5. Regular j-inner matrix functions and related generalized bitangential problems.- References.- The development of some of V.P. Potapov’s ideas. The geometric theory of operators in spaces with indefinite metric.- References.- On the Potapov theory of multiplicative representations.- References.- An operator approach to the Potapov scheme for the solution of interpolation problems.- I. Potapov’s method of solution of interpolation problems.- 1. Some information from j-algebra.- 2. Nevanlinna-Pick problem.- 3. The Schur problem.- II. Operator identities and interpolation problems.- 1. Formulation of the problem.- 2. The fundamental matrix inequality.- 3. The transformed inequality.- 4. The solution of nondegenerate interpolation problems.- 5. Weyl discs.- 6. Degenerate interpolation problems and the method of regularization.- 7. Applications of the general theory.- References.- Description of a class of functions which admit an approximation by rational functions with preassigned poles I.- 2. The class PCNM of pseudocontinuable functions.- 3. The Smirnov class N*.- 4. The weighted space PCH??(I+,I-) of pseudocontinuable meromorphic functions with prescribed denominators.- 5. G. Ts. Tumarkin’s theorem on functions which admit weighted approximation by a sequence of rational functions with preassigned poles.- 6. Formulation of the main approximation theorem.- 7. A fundamental approximation Lemma.- References.- An analysis and extension of V.P. Potapov’s approach to problems with applications to the generalized bi-tangential Schur-Nevanlinna-Pick problem and J-inner-outer factorization.- 1. Potapov’s approach to the Nevanlinna-Pick problem.- 2. An analysis of Potapov’s approach and the AIP.- 3. The abstract interpolation problem.- 4. The AIP and unitary extensions of an isometry.- 5. The generalized bi-tangential Schur-Nevanlinna-Pick (SNP) problem.- 6. Inner-outer factorization of J-contractive matrix-functions.- References.- On the theory of inverse problems for the canonical differential equation.- References.- Addendum.- Some properties of linear-fractional transformations and the harmonic mean of matrix functions.- References.- Modification of V.P. Potapov’s scheme in the indefinite case.- 0. Introduction.- 1. Preliminaries.- 2. Basic propositions.- 3. Extensions of the operator S.- 4. Examples.- References.- Inverse problems for equations systems.- 1. Introduction.- 2. Existence theorems.- 3. Classical examples.- 4. Uniqueness theorems.- References.
