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Inverse Problems for First-Order Discrete Systems

We study inverse problems associated to first-order discrete systems in the rational case. We show in particular that every rational function strictly positive on the unit circle is the spectral function of such a system. Formulas for the coefficients of the system are given in terms of realizations of the spectral function or in terms of a realization of a spectral factor. The inverse problems associated to the scattering function and to the reflection coefficient function are also studied. An important role in the arguments is played by the state space method. We obtain formulas which are very similar to the formulas we have obtained earlier in the continuous case in our study of inverse problems associated to canonical differential expressions.
Daniel Alpay, Israel Gohberg

Stability of Dynamical Systems via Semidefinite Programming

In this paper, we study stability of nonlinear dynamical systems by searching for Lyapunov functions of the form \( \Lambda (x) = \sum\limits_{i = 1}^m {\alpha _i x_i } + \frac{1} {2}\sum\limits_{i = 1}^m {\lambda _i x_1^2 } ,\lambda _i > 0,i = 1, \ldots ,m \), 0, i = 1,..., m, respectively x T Ax, where A is a positive definite real matrix. Our search for Lyapunov functions is based on interior point algorithms for solving certain positive definite programming problems and is applicable for non-polynomial systems not considered by similar methods earlier.
Mihály Bakonyi, Kazumi N. Stovall

Ranks of Hadamard Matrices and Equivalence of Sylvester—Hadamard and Pseudo-Noise Matrices

In this paper we obtain several results on the rank properties of Hadamard matrices (including Sylvester-Hadamard matrices) as well as generalized Hadamard matrices. These results are used to show that the classes of (generalized) Sylvester-Hadamard matrices and of (generalized) pseudo-noise matrices are equivalent, i.e., they can be obtained from each other by means of row/column permutations.
Tom Bella, Vadim Olshevsky, Lev Sakhnovich

Image of a Jacobi Field

Consider the two Hilbert spaces H and T. Let K +: H → T- be a bounded operator. Consider a measure ρ on H -. Denote by ρK the image of the measure ρ under K +. This paper aims to study the measure ρK assuming ρ to be the spectral measure of a Jacobi field. We present a family of operators whose spectral measure equals ρK. We state an analogue of the Wiener-Itô decomposition for ρK. Finally, we illustrate our constructions by offering a few examples and exploring a relatively transparent special case.
Yurij M. Berezansky, Artem D. Pulemyotov

The Higher Order Carathéodory—Julia Theorem and Related Boundary Interpolation Problems

The higher order analogue of the classical Carathéodory-Julia theorem on boundary angular derivatives has been obtained in [7]. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result.
Vladimir Bolotnikov, Alexander Kheifets

A Generalization to Ordered Groups of a Kreĭn Theorem

We give an extension result for positive definite operator-valued Toeplitz-Krein-Cotlar triplets defined on an interval of an ordered group. When the triplet is positive definite and measurable we give a representation result.
Ramón Bruzual, Marisela Domínguez

A Fast QR Algorithm for Companion Matrices

It has been shown in [4, 5, 6, 31] that the Hessenberg iterates of a companion matrix under the QR iterations have low off-diagonal rank structures. Such invariant rank structures were exploited therein to design fast QR iteration algorithms for finding eigenvalues of companion matrices. These algorithms require only O(n) storage and run in O(n2) time where n is the dimensiosn of the matrix. In this paper, we propose a new O(n2) complexity QR algorithm for real companion matrices by representing the matrices in the iterations in their sequentially semi-separable (SSS) forms [9, 10]. The bulge chasing is done on the SSS form QR factors of the Hessenberg iterates. Both double shift and single shift versions are provided. Deflation and balancing are also discussed. Numerical results are presented to illustrate both high efficiency and numerical robustness of the new QR algorithm.
Shiv Chandrasekaran, Ming Gu, Jianlin Xia, Jiang Zhu

The Numerical Range of a Class of Self-adjoint Operator Functions

The structure of the numerical range and root zones of a class of operator functions, arising from one or two parameter polynomial operator pencils of waveguide type is studied. We construct a general model of such kind of operator pencils. In frame of this model theorems on distribution of roots and eigenvalues in some parts of root zones are proved. It is shown that, in general the numerical range and root zones are not connected but some connected parts of root zones are determined. It is proved that root zones, under some natural additional conditions which are satisfied for most of waveguide type multi-parameter spectral problems, are non-separated, i.e., they overlap.
Nurhan Çolakoglu

A Perturbative Analysis of the Reduction into Diagonal-plus-semiseparable Form of Symmetric Matrices

It is known that any symmetric matrix can be transformed by an explicitly computable orthogonal transformation into diagonal-plus-semiseparable form, with prescribed diagonal term. In this paper, we present perturbation bounds for such transformations, under the condition that the diagonal term is close to (part of) the spectrum of the given matrix. As an application, we provide new iterative schemes for the simultaneous refinement of the eigenvalues of a symmetric matrix, having quadratic convergence.
Dario Fasino

The Eigenstructure of Complex Symmetric Operators

We discuss several algebraic and analytic aspects of the eigenstructure (si.e., eigenvalues, eigenvectors, and generalized eigenvectors) of complex symmetric operators. In particular, we examine the relationship between the bilinear form [x,y] = <x, Cy> induced by a conjugation C on a complex Hilbert space H and the eigenstructure of a bounded linear operator T: H → H which is C-symmetric (T = CT*C).
Stephan Ramon Garcia

Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices with Symbols in Hölder-Zygmund Spaces

We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.
Alexei Yu. Karlovich

On an Eigenvalue Problem for Some Nonlinear Transformations of Multi-dimensional Arrays

It is shown that certain transformations of multi-dimensional arrays posses unique positive solutions. These transformations are composed of linear components defined in terms of Stieltjes matrices, and semi-linear components similar to uku 3. In particular, the analysis of the linear components extends some results of the Perron-Frobenius theory to multi-dimensional arrays.
Sawinder P. Kaur, Israel Koltracht

On Embedding of the Bratteli Diagram into a Surface

We study C*-algebras O λ which arise in dynamics of the interval exchange transformations and measured foliations on compact surfaces. Using Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli diagrams of such algebras and measured foliations. This approach allows us to apply K-theory of operator algebras to prove a strict ergodicity criterion and Keane’s conjecture for the interval exchange transformations.
Igor V. Nikolaev

Superfast Inversion of Two-Level Toeplitz Matrices Using Newton Iteration and Tensor-Displacement Structure

A fast approximate inversion algorithm is proposed for two-level Toeplitz matrices (block Toeplitz matrices with Toeplitz blocks). It applies to matrices that can be sufficiently accurately approximated by matrices of low Kronecker rank and involves a new class of tensor-displacement-rank structured (TDS) matrices. The complexity depends on the prescribed accuracy and typically is o(n) for matrices of order n.
Vadim Olshevsky, Ivan Oseledets, Eugene Tyrtyshnikov

On Generalized Numerical Ranges of Quadratic Operators

It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel and composition operators.
Leiba Rodman, Ilya M. Spitkovsky

Inverse Problems for Canonical Differential Equations with Singularities

The inverse problem for canonical differential equations is investigated for Hamiltonians with singularities. The usual notion of a spectral function is not adequate in this generality, and it is replaced by a more general notion of spectral data. The method of operator identities is used to describe a solution of the inverse problem in this setting. The solution is explicitly computable in many cases, and a number of examples are constructed.
James Rovnyak, Lev A. Sakhnovich

On Triangular Factorization of Positive Operators

We investigate the problem of the triangular factorization of positive operators in a Hilbert space. We prove that broad classes of operators can be factorized.
Lev A. Sakhnovich

Solutions for the H ∞(Dn) Corona Problem Belonging to exp(L1/2n-1

For a countable number of input functions in H (D n ), we find explicit analytic solutions belonging to the Orlicz-type space, \( \exp (L^{\tfrac{1} {{2n - 1}}} ) \). Note that \( H^\infty (D^n ) - BMO(D^n ) \subsetneqq \exp (L^{\tfrac{1} {{2n - 1}}} ) \subsetneqq \cap _1^\infty H^p (D^n ) \)
Tavan T. Trent

A Matrix and its Inverse: Revisiting Minimal Rank Completions

We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address a generic minimal rank problem that was proposed by David Ingerman and Gilbert Strang.
Hugo J. Woerdeman
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