Showing papers by "Israel Gohberg published in 1993"
TL;DR: In this paper, a linear ordinary differential operator with bounded coefficients satisfying certain homogeneous initial conditions is shown to be invertible on Ln2(0, ∞) if and only if the underlying system of differential equations has a dichotomy.
Abstract: A linear ordinary differential operator with bounded coefficients satisfying certain homogeneous initial conditions is shown to be invertible onLn2(0, ∞) if and only if the underlying system of differential equations has a dichotomy. Moreover, in that case the operator is proved to be a direct sum of two infinitesimal generators ofC0-semigroups, one of which has support on the negative half-line and the other on the positive half-line. The effect of perturbations of the initial values on the dichotomy is also described.
52 citations
TL;DR: In this paper, a concise exposition of the local spectral theory of regular analytic matrix functions without any reference to elements from the global theory is given, without reference to the elements from global theory.
35 citations
TL;DR: In this paper, a generalization of the theory of Invertibility Symbol in Banach algebras is presented, which allows us to study the Fredholm Symbols of linear operators.
Abstract: This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.
34 citations
TL;DR: The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceLp(Γ,ρ) with an arbitrary weight ρ is studied in this article.
Abstract: The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceLp(Γ,ρ) with an arbitrary weight ρ is studied. The contour Γ consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.
25 citations
TL;DR: In this paper, a complete system of invariants and canonical form for normal operators acting in spaces with indefinite scalar product was introduced. But their invariants are not invariant for all normal operators.
Abstract: In our previous paper [2] a special class of normal operators acting in spaces with indefinite scalar product was introduced. The operators from this class are characterized by the property that in appropriate sip orthogonal bases their matrices have diagonal block-Toeplitz forms. In the present paper we find a complete system of invariants and canonical form for operators belonging to this class.
12 citations
Journal Article•
TL;DR: In this article, the authors develop results on interpolation of real rational matrix functions analogous to those proved in earlier work for rational matrices with complex coefficients. But their results are based on the approach exposed in [BGR3].
Abstract: We develop results on interpolation of real rational matrix functions analogous to those proved in earlier work for rational matrix functions with complex coefficients. Other relevant problems are studied as well; one of them: decide (in terms of the interpolation data) if a given interpolation problem admits a real rational matrix functions interpolants. The basic methodology is rooted in the approach exposed in [BGR3].
11 citations
TL;DR: In this paper, a discrete-time, time-varying analogue of the function theoretic, matricial Lagrange-Sylvester interpolation problem is introduced and solved.
Abstract: A discrete-time, time-varying analogue of the function theoretic, matricial Lagrange-Sylvester interpolation problem is introduced and solved. The set of all solutions is described via a linear fractional representation and a formula for a particular solution is given. The time-varying generalization of the bitangential matricial Nevanlinna-Pick interpolation problem is also considered. A time-varying version of the state space method plays an important role.
10 citations
TL;DR: In this article, it was shown that a rational matrix function W(λ) = I + U(λ), where U is strictly proper and has contractive values for all real λ, admits both left and right pseudocanonical factorization.
8 citations
TL;DR: In this article, a new version of the abstract band method is presented, which applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions.
Abstract: This paper presents a new version of the abstract band method. The new scheme applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions of the abstract band method.
7 citations
01 Jan 1993
TL;DR: In this paper, the Gelfand theory of commutative Banach algebras is introduced and analyzed, showing that piecewise continuous functions become continuous under the GELFand transformation.
Abstract: This chapter contains the Gelfand theory of commutative Banach algebras. The Gelfand spectrum and the Gelfand transform are introduced and analysed. The results are illustrated by various examples. In particular, it is explained in detail how under the Gelfand transformation piecewise continuous functions become continuous. Special attention is paid to finitely generated commutative Banach algebras and to the Banach algebra generated by a compact operator. The last two sections present applications to factorization of matrix functions and to Wiener-Hopf integral operators. The analysis starts with a study of multiplicative linear functionals.
6 citations
TL;DR: In this paper, the canonical factorization theorem for the symbol of a Toeplitz operator is generalized to a class of non-ToePlitz operators, which are described as input-output operators of time-varying linear systems.
TL;DR: In this paper, the band method for positive definite extension problems of non-band type is specified for the bordered case and a fractional type description of all positive extensions is given.
Abstract: The band method for positive definite extension problems of non-band type is specified for the bordered case. A fractional type description of all positive extensions is new. We illustrate the result for Fredholm integral operators. The notion of Schur complement plays a crucial role.
01 Jan 1993
TL;DR: In this paper, the authors considered the problem of bitangential interpolation for input-output operators of time-varying systems and proposed a generalization of the Nevanlinna pick problem.
Abstract: Editorial Introduction D. Alpay, P. Loubaton The tangential trigonometric moment problem on an interval and related topics 1. Introduction 2. Some lemmas on matrix-valued rational functions 3. The main result 4. The Nevanlinna Pick problem References. M. Bakonyi, V.G. Kaftal, G. Weiss, H.J. Woerdeman Maximum entropy and joint norm bounds for operator extensions 1. Introduction 2. A sharp bound in the 2 x 2 case 3. The maximum entropy method 4. An application to integral operators References. J.A. Ball, 1. Gohberg, M.A. Kaashoek Bitangential interpolation for input-output operators of time varying systems: the discrete time case 0. Introduction 1. Residue calculus and generalized point evaluation 2. Pairs of diagonal operators and homogeneous one-sided interpolation 3. Bitangential interpolation data set 4. Bitangential interpolation in geometric terms 5. Intermezzo about admissible Sylvester data sets 6. Construction of a particular solution 7. Parametrization of all solutions (without norm constraints) 8. Input-output operators of time-varying systems 9. Parametrization of all contractive input-output operators satisfying the bitangential interpolation conditions References. J.A.Ball, I.Gohberg, L. Rodman Two-sided tangential interpolation of real rational matrix functions 1. Introduction 2. Minimal realizations 3. Local data 4. Two-sided tangential interpolation: existence of real interpolants 5. Two-sided tangential interpolation with real-valued data: Description of interpolants 6. Degrees of interpolants 7. Generalized Nevanlinna-Pick interpolation for real rational matrix functions References. H. Du, C. Gu On the spectra of operator completion problems 1. Introduction 2. Case of finite dimensional spaces 3. Case of infinite dimensional spaces References. C. Foias, A.E. Frazho, W.S. Li The exact H2 estimate for the central H interpolant Introduction 1. An improved Kaftal-Larson-Weiss estimate 2. Some formulas for DB 3. The role of DA2II0 4. The four block problem 5. Optimal solutions References. A.E. Frazho, S.M. Kherat On mixed H2 - H tangential interpolation 1. Introduction 2. Formulas for the central solution 3. A state space approach 4. Applications of the H2-H tangential interpolation problem References. I. Gohberg, C. Gu On a completion problem for matrices 1. Introduction 2. Main theorems in the finite dimensional case 3. The full range case 4. The proof of the main theorems in the finite dimensional case 5. Infinite dimensional case References.
TL;DR: In this article, the authors solve a two-sided interpolation problem with arbitrary multiplicity for all-pass rational matrices and present explicit formulas for solutions with low McMillan degree.
TL;DR: In this paper, the following completion problem for square matrices is solved and analyzed: given a square matrix with unidentified last row, except the diagonal entry, describe the invariant polynomials of all possible completions of this matrix.
Abstract: In this paper the following completion problem for square matrices is solved and analyzed. Given a square matrix with unidentified last row, except the diagonal entry. Describe the invariant polynomials of all possible completions of this matrix.
01 Jan 1993
TL;DR: The first three sections of this chapter have an introductory character as discussed by the authors, and Section 2 contains a short introduction to Laurent operators and the first properties of block Toeplitz operators are derived.
Abstract: The first three sections of this chapter have an introductory character. Section 2 contains a short introduction to Laurent operators. In Section 3 the first properties of block Toeplitz operators are derived. Sections 4 and 5 develop the Fredholm theory of block Toeplitz operators defined by continuous functions.
01 Jan 1993
TL;DR: In this paper, the main elements of the theory of block shift operators are presented, which are among the simplest infinite dimensional operators and may serve as building blocks for more complicated operators.
Abstract: This chapter, which has an introductory character, contains the main elements of the theory of block shift operators. These operators are among the simplest infinite dimensional operators, and they may serve as building blocks for more complicated operators. In the first section block forward shifts are identified as pure isometries. In the second section it is shown that block backward shifts provide universal models for arbitrary operators. The third section describes the invariant subspaces of ℂ m -block forward shifts. Throughout this chapter all operators are Hilbert space operators.
01 Jan 1993
TL;DR: In this paper, the Banach algebras generated by Toeplitz operators defined by continuous and, more generally, by piecewise continuous functions are studied and analyzed.
Abstract: In this chapter the Banach algebras generated by Toeplitz operators defined by continuous and, more generally, by piecewise continuous functions are studied. These algebras are not commutative, but they contain the compact operators as a proper closed ideal and in the scalar case the corresponding quotient algebras turn out to be commutative. The Gelfand spectra and transforms of these quotient algebras are described and analyzed.