Interpolation Theory and Its Applications

Sakhnovich, Lev

doi:10.1007/978-94-009-0059-2

Cited by 100 publications

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“…Remark 2.3 Formula (2.23) is a particular case of a more general relation that follows [40,41] from (2.4) and (2.6):…”

Section: )mentioning

confidence: 99%

“…for all χ satisfying (3.4), and it was shown in [41] that the Weyl functions ϕ ∈ ∩ K<∞ N (w, K) have the characteristic property:…”

Section: )mentioning

confidence: 99%

“…For this purpose we shall need an equivalent representation of (1.3) in the zero curvature form [41] …”

Section: Matrix Toda Latticementioning

confidence: 99%

“…Interesting spectral results for the Toda lattice and its MTL and NTL generalizations have been obtained in [3,4,6,15,21,24,41,44,46] (see also references), though various related spectral problems are still to be solved. We shall construct a Bäcklund-Darboux transformation (BDT) for the DCS (1.1), MTL and NTL.…”

Section: Introductionmentioning

confidence: 99%

“…The Darboux matrix in this approach is presented as the transfer matrix function, which is a well known tool in system theory [22]. Transfer matrix function of the form w A (l) = I m − Π(2) * S −1 (A(1) − lI n ) −1 Π(1) (A(1)S − SA(2) = Π(1)Π(2) * ), (1.10) that we are going to use, was introduced by L. Sakhnovich in the context of his method of operator identities [40,41]. Representation (1.10) takes roots in the Livšic-Brodskii characteristic matrix function [27].…”

Section: Introductionmentioning

confidence: 99%

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Discrete canonical system and non‐Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

Sakhnovich

2007

Mathematische Nachrichten

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A version of the iterated Bäcklund-Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the system theory, is constructed for the discrete canonical system and Non-Abelian Toda lattice. Results on the transformations of the Weyl functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated.

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“…Remark 2.3 Formula (2.23) is a particular case of a more general relation that follows [40,41] from (2.4) and (2.6):…”

Section: )mentioning

confidence: 99%

“…for all χ satisfying (3.4), and it was shown in [41] that the Weyl functions ϕ ∈ ∩ K<∞ N (w, K) have the characteristic property:…”

Section: )mentioning

confidence: 99%

“…For this purpose we shall need an equivalent representation of (1.3) in the zero curvature form [41] …”

Section: Matrix Toda Latticementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discrete canonical system and non‐Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

Sakhnovich

2007

Mathematische Nachrichten

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Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

2002

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Let g(z, x) denote the diagonal Green's matrix of a self‐adjoint m × m matrix‐valued Schrödinger operator One of the principal results proven in this paper states that for a fixed x0 ∈ ℝ and all z ∈ ℂ+, g(z, x0) and g′(z, x0) uniquely determine the matrix‐valued m × m potential Q(x) for a.e. x ∈ ℝ. We also prove the following local version of this result. Let gj(z, x), j = 1, 2 be the diagonal Green's matrices of the self‐adjoint Schrödinger operators Suppose that for fixed a > 0 and x0 ∈ ℝ, for z inside a cone along the imaginary axis with vertex zero and opening angle less than π/2, excluding the real axis. Then Q1(x) = Q2(x) for a.e. x ∈ [x0 – a, x0 + a]. Analogous results are proved for matrix‐valued Jacobi and Dirac‐type operators.

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The matricial Schur problem in both nondegenerate and degenerate cases

Fritzsche

Kirstein

Lasarow

2009

Mathematische Nachrichten

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Key wordsThe principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]-[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D. Z. Arov and M. G. Kreȋn for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution. The main goal of this paper is to present an approach to the matricial version of the classical Schur problem (see [32] and [33]) in both nondegenerate and degenerate cases. Roughly speaking, our strategy is due to an adaptation of the authors' former investigations to the general (i.e. possibly degenerate) matricial Carathéodory problem (see [29]). Our method is essentially based on the first and second authors former investigations [23]-[27] on the central matrix-valued Schur function associated with a finite Schur sequence of complex matrices. In particular, we will make frequently use of the matrix ball description of the elements of matricial Schur sequences. The main results of this paper (see Theorems 1.1, 3.2, 3.6, and 4.2) contain descriptions of the solution set of a matricial Schur problem in terms of a linear fractional transformation, the generating matrix-valued function of which is a matrix polynomial. The canonical blocks of this matrix polynomial will be constructed with the aid of those quadruple of matrix polynomials which were used in [27] to derive right and left quotient representations of central matrix-valued Schur functions (see Theorem 2.4).A different approach to the degenerate matricial Schur problem was used in [11]-[16] of V. K. Dubovoj. His method is based on a combination of V. P. Potapov's fundamental matrix inequality approach with the technique of subspaces of type K which was created by V. K. Dubovoj to handle the specific of the matricial Schur problem *

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Interpolation Theory and Its Applications

Cited by 100 publications

References 0 publications

Discrete canonical system and non‐Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

Discrete canonical system and non‐Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

The matricial Schur problem in both nondegenerate and degenerate cases

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