Operator Theory

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.