Showing papers by "Israel Gohberg published in 2000"

Traces and determinants of linear operators

Abstract: A general theory of tracestr D A and determinantsdet D (I+A) in normed algebrasD of operators acting in Banach spacesB is proposed. In this approach trace and determinant are defined as continuous extensions of the corresponding functionals from finite dimensional operators. We characterize the algebras for which such extensions exist and describe sets of possible values of traces and determinants for the same operator in different algebras. In spite of the fact that the extended traces and determinants may differ in different algebrasD, operatorI+A (withA ∈D) is invertible inB if and only ifdet D (I+A) does not vanish. Cramer's rule and formulas for the resolvent are obtained and they are expressed in different algebras by the same formulas viadet D (I+A) andtr D (A). A large set of examples and illustrations are also presented.

Topics in Interpolation Theory of Rational Matrix-valued Functions

Abstract: One of the basic interpolation problems from our point of view is the problem of building a scalar rational function if its poles and zeros with their multiplicities are given. If one assurnes that the function does not have a pole or a zero at infinity, the formula which solves this problem is (1) where Zl , " " Z/ are the given zeros with given multiplicates nl, " " n / and Wb" " W are the given p poles with given multiplicities ml, ...,m , and a is an arbitrary nonzero number. p An obvious necessary and sufficient condition for solvability of this simplest Interpolation pr- lern is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +...+n/ = ml +...+m ' p The second problem of interpolation in which we are interested is to build a rational matrix function via its zeros which on the imaginary line has modulus 1. In the case the function is scalar, the formula which solves this problem is a Blaschke product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o] = 1, and the zj's are the given zeros with given multiplicities mj. Here the necessary and sufficient condition for existence of such u(z) is that zp :f: - Zq for 1~ ]1, q~ n.

Convolution equations on finite intervals and factorization of matrix functions

Abstract: Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.

Direct and Inverse Scattering Problem for Canonical Systems with a Strictly Pseudo – Exponential Potential

Abstract: Explicit formulas are given for the solutions of the direct and inverse scattering problems for a canonical differential system with a strictly pseudo–exponential potential. The proofs are self–contained and employ state space techniques from mathematical system theory. The paper supplements an earlier paper of the first two authors where explicit formulas were given using Marchenko's approach, and an earlier paper of the last three authors where self–contained proofs were given for the corresponding direct and inverse spectral problems. Two types of factorizations of the scattering matrix function appear and connections between them are considered.

Canonical Systems on the Line with Rational Spectral Densities: Explicit Formulas

Abstract: Explicit formulas for the direct and inverse spectral problems for a canonical system on the full line with rational spectral density are obtained via a reduction to the half line case.

Connections between the Carathéodory-Toepliz and the Nehari extension problems: The discrete scalar case

Abstract: We study the connections between the Caratheodory-Toeplitz extension problem and the Nehari extension problem in the discrete scalar case. This is the discrete counterpart of our previous paper [4].

Differential operators and related topics

Abstract: The second of two volumes presenting the proceedings of the Mark Krein International Conference on Operator Theory and Applications, this volume consists of papers providing late-1990s research in operator theory and its applications and also includes a bibliography of Krein's life and work.

Trace Class and Hilbert-Schmidt Operators in Hilbert Space

Abstract: The study of traces and determinants of trace class and Hilbert-Schmidt operators on Hilbert space requires a familiarity with these classes of operators. This chapter discusses all the related concepts and theorems needed in this book. Included are properties of singular numbers of compact operators and Lidskii’s trace theorem. We then proceed to introduce the regularized determinants of Hilbert-Schmidt operators. The formulas for the traces of integral operators with continuous kernel and with smooth kernel are presented. The determinant of operator pencils also appears. The last section treats the Sp spaces which are generalizations of the algebra of Hilbert-Schmidt operators.

Orthogonal Matrix-valued Polynomials and Applications: Seminar on Operator Theory at the School of Mathematical Sciences, Tel Aviv University

Abstract: Bibliography of Mark Grigor'Evich Krein.- On Orthogonal Matrix Polynomials.- n-Orthonormal Operator Polynomials.- Extension of a Theorem of M. G. Krein on Orthogonal Polynomials for the Nonstationary Case.- Hermitian Block Toeplitz Matrices, Orthogonal Polynomials, Reproducing Kernel Pontryagin Spaces, Interpolation and Extension.- Matrix Generalizations of M. G. Krein Theorems on Orthogonal Polynomials.- Polynomials Orthogonal in an Indefinite Metric.