On an Inverse Spectral Problem for a Convolution Integro-differential Operator

The operator of double differentiation perturbed by the composition of a Volterra convolution operator and the differentiation one on a finite interval with Dirichlet boundary conditions is considered. It is proved that the standard asymptotics is necessary and sufficient for an arbitrary sequence of complex numbers to be the spectrum of such an operator, which is determined uniquely. A constructive procedure for solving the inverse problem is given. Received: March 5, 2007.     

The Constants in the Asymptotic Formulas by Rambour and Seghier for Inverses of Toeplitz Matrices

Rambour and Seghier established two deep results on the first order asymptotics of the entries of the inverses of certain Toeplitz matrices. The determination of the constants in the leading terms of the asymptotics turns out to be nontrivial. We here show how these constants can be obtained from a well known identity by Duduchava and Roch.Submitted: October 20, 2003     

Formulas for the Inverses of Toeplitz Matrices with Polynomially Singular Symbols

We consider large finite Toeplitz matrices with symbols of the form (1– cos )^p f() where p is a natural number and f is a sufficiently smooth positive function. By employing techniques based on the use of predictor polynomials, we derive exact and asymptotic formulas for the entries of the inverses of these matrices. We show in particular that asymptotically the inverse matrix mimics the Green kernel of a boundary value problem for the differential operator Submitted: June 20, 2003     

Admissible observation operators. Semigroup criteria of admissibility

We investigate admissible observation operators. Semigroup criteria of admissibility are derived. We also discuss the recent Russell-Weiss necessity condition of exact observability.This work was done when the first author was on leave from: Institute of Automatics Academy of Mining and Metallurgy al. Mickiewicza 30/B1, rm.314 PL-30-059 Cracow, Poland     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Geometric properties of positive definite matrices cone with respect to the Thompson metric

We shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to Thompson metric.     

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

Characterization of Transfer Functions of Pritchard–Salamon or Other Realizations with a Bounded Input or Output Operator

We show that the transfer functions that have a (continuoustime) well-posed realization with a bounded input operator are exactly those that are strong-H² (plus constant feedthrough) over some right half-plane. The dual condition holds iff the transfer function has a realization with a bounded output operator. Both conditions hold iff the transfer function has a Pritchard–Salamon (PS) realization. A state-space variant of the PS result was proved already in [3], under the additional assumption that the weighting pattern (or impulse response) is a function (whose values are bounded operators). We illustrate by an example that this does not cover all PS systems, not even if the input and output spaces are separable.     

Inverse Problems for Sturm–Liouville Operators on Noncompact Trees

We study Sturm–Liouville differential operators on noncompact graphs without cycles (i.e., on trees) with standard matching conditions in internal vertices. First we establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings. Received: February 13, 2007.     

Einige Existenz- und Einschließungssätze für Fixpunkte expandierender Integraloperatoren

Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions.

     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Weighting operator patterns of Pritchard-Salamon realizations

In this paper a complete characterization is given of those operator kernels which appear as weighting patterns of Pritchard-Salamon realizations. The result is based on an extension of the standard shift realization to weightedL ₂-spaces of vector-valued functions.     

Non-self-adjoint Dirac-type Systems and Related Nonlinear Equations: Wave Functions, Solutions, and Explicit Formulas

A version of the B?cklund-Darboux transformation, where Darboux matrix takes the form of the transfer matrix function from the system theory, for the non-self-adjoint Dirac-type system is considered. Related nonlinear Schr?dinger equations (coupled and multi-component), self-induced transparency equation, and non-Abelian sine-Gordon equation are treated. Explicit formulas for the wave functions and solutions are obtained.     

Optimal Control in Heterogeneous Domain Decomposition Methods for Advection-Diffusion Equations

New domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first and second order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem, and the convergence of iterative processes is analyzed. The work was supported by the Russian Foundation for Basic Research (04-01-00615) and it was partly carried out while the first author was visiting the IACS at EPFL.     

A connection between Fourier and Schur multipliers

We construct a continuous functionf on the circle such thatf is a Fourier multiplier inl _p , 1<p<, but (x, y)=f(x–y) is not a Schur multiplier inS _p ,p2.