INTERPOLATION OF OPERATORS
CONTENTS BASIC BOUNDARY INTERPOLATION FOR GENERALIZED SCHUR FUNCTIONS AND FACTORIZATION OF RATIONAL J-UNITARY MATRIX FUNCTIONS 1 INTRODUCTION 2 AUXILIARY STATEMENTS 3 THE BASIC INTERPOLATION PROBLEM AT ONE BOUNDARY POINT 4 MULTIPOINT BOUNDARY INTERPOLATION 5 J-UNITARY FACTORIZATION 6 A FACTORIZATION ALGORITHM REFERENCES DISCRETE ANALOGS OF CANONICAL SYSTEMS WITH PSEUDO-EXPONENTIAL POTENTIAL INVERSE PROBLEMS 1 INTRODUCTION 2 PRELIMINARIES 21 THE CHARACTERISTIC SPECTRAL FUNCTIONS 22 UNITARY SOLUTIONS OF THE NEHARI PROBLEM 23 UNIQUENESS THEOREM 3 INVERSE SCATTERING PROBLEM 31 INVERSE SCATTERING PROBLEM ASSOCIATED TO THE SPECTRAL FACTOR 32 INVERSE SCATTERING PROBLEM ASSOCIATED TO A BLASCHKE PRODUCT 4 OTHER INVERSE PROBLEMS 41 INVERSE PROBLEM ASSOCIATED TO THE RE?ECTION COE?CIENT FUNCTION 42 INVERSE PROBLEM ASSOCIATED TO THE WEYL COE?CIENT FUNCTION 43 INVERSE SPECTRAL PROBLEM 5 INVERSE PROBLEM ASSOCIATED TO THE ASYMPTOTIC EQUIVALENCE MATRIX FUNCTION 6 THE CASE OF TWO-SIDED ?RST-ORDER SYSTEMS 7 A NUMERICAL EXAMPLE 8 AN EXAMPLE OF A NON-STRICTLY PSEUDO-EXPONENTIAL SEQUENCE 9 JACOBI MATRICES BOUNDARY NEVANLINNA–PICK INTERPOLATION PROBLEMS FOR GENERALIZED SCHUR FUNCTIONS 1 INTRODUCTION 2 MAIN RESULTS 3 SOME PRELIMINARIES 4 FUNDAMENTAL MATRIX INEQUALITY 5 PARAMETERS AND INTERPOLATION CONDITIONS 6 NEGATIVE SQUARES OF THE FUNCTION W = T?[E] 7 THE DEGENERATE CASE 8 AN EXAMPLE A TRUNCATED MATRICIAL MOMENT PROBLEM ON A FINITE INTERVAL 0 INTRODUCTION AND PRELIMINARIES 1 THE MOMENT PROBLEM 2 MAIN ALGEBRAIC IDENTITIES 3 FROM THE MOMENT PROBLEM TO THE SYSTEM OF FUNDAMENTAL MATRIX INEQUALITIES OF POTAPOV-TYPE 4 FROM THE SYSTEM OF FUNDAMENTAL MATRIX INEQUALITIES TO THE MOMENT PROBLEM 5 NONNEGATIVE COLUMN PAIRS 6 DESCRIPTION OF THE SOLUTION SET IN THE POSITIVE DE?NITE CASE 7 A NECESSARY AND SU?CIENT CONDITION FOR THE EXISTENCE OF A SOLUTION OF THE MOMENT PROBLEM 8 APPENDIX: CERTAIN SUBCLASSES OF HOLOMORPHIC MATRIX-VALUED FUNCTIONS AND A GENERALIZATION OF STIELTJES’ INVERSION FORMULA ACKNOWLEDGEMENT SHIFT OPERATORS CONTAINED IN CONTRACTIONS, SCHUR PARAMETERS AND PSEUDOCONTINUABLE SCHUR FUNCTIONS 0 INTRODUCTION 1 SHIFTS CONTAINED IN CONTRACTIONS, UNITARY COLLIGATIONS AND CHARACTERISTIC OPERATOR FUNCTIONS 11 SHIFTS CONTAINED IN CONTRACTIONS AND UNITARY COLLIGATIONS 12 CHARACTERISTIC OPERATOR FUNCTIONS 13 NAIMARK DILATIONS 2 CONSTRUCTION OF A MODEL OF A UNITARY COLLIGATION VIA THE SCHUR PARAMETERS OF ITS COF IN THE SCALAR CASE 21 SCHUR ALGORITHM, SCHUR PARAMETERS 22 GENERAL FORM OF THE MODEL 23 SCHUR DETERMINANTS AND CONTRACTIVE OPERATORS COMPUTATION OF TN+1,N 24 SCHUR DETERMINANTS AND CONTRACTIVE OPERATORS AGAIN COMPUTATION OF GN 25 DESCRIPTION OF THE MODEL OF A UNITARY COLLIGATION IF ? J=0(1 ?|?J|2) CONVERGES 26 DESCRIPTION OF THE MODEL OF A UNITARY COLLIGATION IN THE CASE OF DIVERGENCE OF THE SERIES ? J=0 |?J|2 27 DESCRIPTION OF THE MODEL IN THE CASE THAT THE FUNCTION ? IS A ?NITE BLASCHKE PRODUCT 28 COMMENTS 3 A MODEL REPRESENTATION OF THE MAXIMAL SHIFT VT CONTAINED IN A CONTRACTION T 31 THE CONJUGATE CANONICAL BASIS 32 A MODEL REPRESENTATION OF THE MAXIMAL UNILATERAL SHIFT VT CONTAINED IN A CONTRACTION T 4 THE CONNECTION OF THE MAXIMAL SHIFTS VT AND VT ?WITH THE PSEUDOCONTINUABILITY OF THE CORRESPONDING COF ? 41 PSEUDOCONTINUABILITY OF SCHUR FUNCTIONS 42 ON SOME CONNECTIONS BETWEEN THE MAXIMAL SHIFTS VT AND VT ? AND THE PSEUDOCONTINUABILITY OF THE CORRESPONDING COF ? 5 SOME CRITERIA FOR THE PSEUDOCONTINUABILITY OF A SCHUR FUNCTION IN TERMS OF ITS SCHUR PARAMETERS 51 CONSTRUCTION OF A COUNTABLE CLOSED VECTOR SYSTEM IN HGF AND INVESTIGATION OF THE PROPERTIES OF THE SEQUENCE (?N)? N=1 OF GRAM DETERMINANTS OF THIS SYSTEM 52 SOME CRITERIA OF PSEUDOCONTINUABILITY OF SCHUR FUNCTIONS 53 ON SOME PROPERTIES OF THE SCHUR PARAMETER SEQUENCES OF PSEUDOCONTINUABLE SCHUR FUNCTIONS 54 THE STRUCTURE OF PURE ?-SEQUENCES OF RANK 0 OR 1 THE MATRICIAL CARATH´EODORY PROBLEM IN BOTH NONDEGENERATE AND DEGENERATE CASES 0 INTRODUCTION 1 PRELIMINARIES 2 ON PARTICULAR MATRIX POLYNOMIALS 3 DESCRIPTION OF THE SET CQ[D, (?J)N J=0] 4 RESOLVENT MATRICES WHICH ARE CONSTRUCTED RECURSIVELY 5 THE NONDEGENERATE CASE 6 THE CASE OF A UNIQUE SOLUTION A GOHBERG-HEINIG TYPE INVERSION FORMULA INVOLVING HANKEL OPERATORS 0 INTRODUCTION 1 THE INDICATOR 2 THE MAIN THEOREM FOR KERNEL FUNCTIONS OF STABLE EXPONENTIAL TYPE 3 PROOF OF THE MAIN THEOREM (GENERAL CASE)