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2007 | Book

Introduction Read chapter Read first chapter

Modern Operator Theory and Applications

The Igor Borisovich Simonenko Anniversary Volume

Editors: Yakob M. Erusalimsky, Israel Gohberg, Sergei M. Grudsky, Vladimir Rabinovich, Nikolai Vasilevski

Publisher: Birkhäuser Basel

Book Series : Operator Theory: Advances and Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

(jointly with V.P. Zakharyuta and V.I. Yudovich), including a calculation of the capacity of condensers of complex form and dielectric materials with complex structure. In 1967, at the age of 32, six years after he defended his Ph.D. thesis, I.B. - monenko defended his thesis for a degree of Doctor of Science. In this thesis, entitled “Operators of local type and some other problems of the theory of linear operators,” he sharply turned towards the wide usage of the general methods of functional analysis. In 1971 professor I.B. Simonenko became the head of the - merical Mathematics Chair. The following year this chair was split into two; I.B. became the head of one of them, the Chair of Algebra and Discrete Mathematics. The Chair of Algebra and Discrete Mathematics can be rightfully called the Chair of I.B. Simonenko. Here he worked together with his colleagues, students, and the students of his students. Here he fully developed his teaching talent. He lectured on “Algebra and geometry,” “Mathematical logic,” “Discrete mathem- ics,” and “Mathematical analysis”.

Table of Contents

Frontmatter
Introduction
Coefficients Averaging for Functional Operators Generated by Irrational Rotation
Abstract
The problem under consideration can be posed in the following manner: what is the procedure of changing the coefficients of a functional operator to the coefficients with a simpler behavior under which the principle properties of the initial operator are preserved?
In the paper we consider a number of variants of precise formulation of the problem for the model functional operators generated by an irrational rotation of the circle. In particular we obtain the description of all the possible changes of coefficients under which the spectrum of the operator preserves.
A. B. Antonevich
On the Verification of Linear Equations and the Identification of the Toeplitz-plus-Hankel Structure
Abstract
Testing whether a given matrix is a Toeplitz-plus-Hankel matrix amounts to the verification of a system of linear equations for the matrix entries. If the matrix dimension is large, we are forced to work with the computer and hence cannot check whether something is exactly zero. We provide bounds such that if a test quantity is smaller than the bound, then the system of linear equations may be accepted to be valid and the probability for erroneously accepting the validity of the system is smaller than a prescribed value.
Albrecht Böttcher, David Wenzel
Asymmetric Factorizations of Matrix Functions on the Real Line
Abstract
We indicate a criterion for some classes of continuous matrix functions on the real line with a jump at infinity to admit both, a classical right and an asymmetric factorization. It yields the existence of generalized inverses of matrix Wiener-Hopf plus Hankel operators and provides precise information about the asymptotic behavior of the factors at infinity and of the solutions to the corresponding equations at the origin.
L. P. Castro, R. Duduchava, F. -O. Speck
On the Structure of the Square of a C 0(1) Operator
Abstract
We use the structure theory for C 0 operators to determine when the square of a C 0(1) operator is irreducible and when its lattices of invariant and hyperinvariant subspaces coincide.
Ronald G. Douglas, Ciprian Foias
On the Connection Between the Indices of a Block Operator Matrix and of its Determinant
Abstract
We consider a finite block operator matrix \( \mathcal{A}\) in a Hilbert space. If the entries of \( \mathcal{A}\) commute modulo the compact operators, then \( \mathcal{A}\) is a Fredholm operator if and only if det \( \mathcal{A}\) is a Fredholm operator, but in general ind \( \mathcal{A}\) ≠ ind det \( \mathcal{A}\) . On the other hand, if the commutators of the entries of \( \mathcal{A}\) are trace class operators then ind \( \mathcal{A}\) = ind det \( \mathcal{A}\) . We obtain formulas for the difference ind \( \mathcal{A}\) — ind det \( \mathcal{A}\) provided the entries of \( \mathcal{A}\) commute modulo some von Neumann—Schatten ideal. Then we indicate some ideals larger than the ideal of trace class operators for which the mentioned statement about the equality ind \( \mathcal{A}\) = ind det \( \mathcal{A}\) remains true.
Israel Feldman, Nahum Krupnik, Alexander Markus
Quasi-commutativity of Entire Matrix Functions and the Continuous Analogue of the Resultant
Abstract
This paper is an addition to the paper [3], where it was proved that the theorem about the null space of the classical Sylvester resultant matrix also holds for its continuous analogue for entire matrix function provided that a certain so-called quasi-commutativity condition is fulfilled. In the present paper we show that this quasi-commutativity condition is not only sufficient but also necessary.
I. Gohberg, M. A. Kaashoek, L. Lerer
Double Barrier Options Under Lévy Processes
Abstract
In this paper the problem of determination of the no arbitrage price of double barrier options in the case of stock prices is modelled on Lévy processes is considered. Under the assumption of existence of the Equivalent Martingale Measure this problem is reduced to the convolution equation on a finite interval with symbol generated by the characteristic function of the Lévy process. We work out a theory of unique solvability of the getting equation and stability of the solution under relatively small perturbations.
Sergei M. Grudsky
A Local-trajectory Method and Isomorphism Theorems for Nonlocal C*-algebras
Abstract
A nonlocal version of the Allan-Douglas local principle applicable to nonlocal C*-algebras \( \mathcal{B}\) associated with C*-dynamical systems is elaborated. This local-trajectory method allows one to study the invertibility of elements b ε \( \mathcal{B}\) in terms of invertibility of their local representatives. Isomorphism theorems for nonlocal C*-algebras are established.
Yu. I. Karlovich
Boundedness in Lebesgue Spaces with Variable Exponent of the Cauchy Singular Operator on Carleson Curves
Abstract
We prove the boundedness of the singular integral operator S Γ in the spaces L p(·)(Γ, ρ) with variable exponent p(t) and power weight ρ on an arbitrary Carleson curve under the assumptions that p(t) satisfy the logcondition on Γ. The curve Γ may be finite or infinite.
We also prove that if the singular operator is bounded in the space L p(·)(Γ), then Γ is necessarily a Carleson curve. A necessary condition is also obtained for an arbitrary continuous coefficient.
Vakhtang Kokilashvili, Vakhtang Paatashvili, Stefan Samko
On the Averaging Method for the Problem of Heat Convection in the Field of Highly-Oscillating Forces
Abstract
In the paper have been proved two theorems an averaging of convection problem and on stability or instability its periodic solutions.
V. B. Levenshtam
Finite Sections of Band-dominated Operators with Almost Periodic Coefficients
Abstract
We consider the sequence of the finite sections R n AR n of a band-dominated operator A on l 2(ℤ) with almost periodic coefficients. Our main result says that if the compressions of A onto ℤ+ and ℤ− are invertible, then there is a distinguished subsequence of (R n AR n) which is stable. Moreover, this subsequence proves to be fractal, which allows us to establish the convergence in the Hausdorff metric of the singular values and pseudoeigenvalues of the finite section matrices.
Vladimir S. Rabinovich, Steffen Roch, Bernd Silbermann
On the Toeplitz Operators with Piecewise Continuous Symbols on the Bergman Space
Abstract
The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We clarify the geometric regularities of the behavior of the essential spectrum of Toeplitz operators in dependence on their crucial data: the angles between jump curves of symbols at a boundary point of discontinuity and on the limit values reached by a symbol at that boundary point. We show then that the curves supporting the symbol discontinuities, as well as the number of such curves meeting at a boundary point of discontinuity, do not play any essential role for the Toeplitz operator algebra studied. Thus we exclude the curves of symbol discontinuity from the symbol class definition leaving only the set of boundary points (where symbols may have discontinuity) and the type of the expected discontinuity. Finally we describe the C*-algebra generated by Toeplitz operators with such symbols.
N. Vasilevski
Asymptotics of a Class of Operator Determinants
Abstract
In previous work of C.A. Tracy and the author asymptotic formulas were derived for certain operator determinants whose interest lay in the fact that quotients of them gave solutions to the cylindrical Toda equations. In the present paper we consider a more general class of operators which retain some of the properties of those cited and we find analogous asymptotics for the determinants.
Harold Widom
Metadata
Title
Modern Operator Theory and Applications
Editors
Yakob M. Erusalimsky
Israel Gohberg
Sergei M. Grudsky
Vladimir Rabinovich
Nikolai Vasilevski
Copyright Year
2007
Publisher
Birkhäuser Basel
Electronic ISBN
978-3-7643-7737-3
Print ISBN
978-3-7643-7736-6
DOI
https://doi.org/10.1007/978-3-7643-7737-3

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