The \(C^{*}\)-algebra of singular integral operators with semi-almost periodic coefficients.
type: Item
instance of: scholarly article
title: The \(C^{*}\)-algebra of singular integral operators with semi-almost periodic coefficients.
zbMATH Open document ID: 1048.47027
DOI: 10.1016/S0022-1236(03)00126-5
published in: Journal of Functional Analysis
publication date: 2004-02-03T00:00:00Z
review text: Let \((Sf)(x)=\frac{1}{\pi i}\int_R \frac{f(t)}{t-x}\,dt\) be the Cauchy singular integral operator on the real line \(\mathbb{R}\) and \(P=\frac 12 (I+S), Q=\frac 12 (I-S)\) be the projections generated by \(S\). In the space \(L^2_N\), the authors consider singular integral operators of the form \(A=PaP+Q+cH_++H_-dI\), where \(a,b,d \in SAP_{N\times N}\) (algebra of semi-almost periodic matrix functions) and \(H_\pm\in H_\infty.\) Here \(H_\infty\) is the closed two-sided ideal of the \(C^\ast\)-algebra \(\text{alg}(S, [C(\overline {\mathbb{R}})_{N\times N}])\) that is generated by all commutators \(uS-SuI\) with \(u\in C(\overline {\mathbb{R}})\), where \(C(\overline {\mathbb{R}})\) consists of complex valued continuous functions on \(\mathbb{R}\) which have finite limits at \(-\infty\) and \(\infty\). The authors give a Fredholm criterion for the operator \(A\) and a formula for the calculation of \(\operatorname{ind} A.\) The case of more general operators of the form \(aP+bQ+\sum_{k=1}^m c_kH_{k,+}+\sum_{j=1}^n H_{j,-}d_jI\) is also discussed. These results are applied to Toeplitz-like operators.
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A68
zbMATH DE Number: 2037120
zbMATH Keywords: singular integral operator
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: almost periodic function
zbMATH Keywords: semi-almost periodic function
zbMATH Keywords: matrix function
zbMATH Keywords: \(C^\ast\)-algebra
zbMATH Keywords: \(C^\ast\)-dynamical system
author: A. Böttcher
author: Yuri I. Karlovich
author: Ilya M. Spitkovsky
reviewed by: N. Karapetiants
MaRDI profile type: Publication
cites work: Linear functional equations. Operator approach. Transl. from the Russian by V. Muzafarov a. A. Iacob
cites work: Q4277008
cites work: Q4424396
cites work: Carleson curves, Muckenhoupt weights, and Toeplitz operators
cites work: Toeplitz operators with semi-almost-periodic matrix symbols on Hardy spaces
cites work: Convolution operators and factorization of almost periodic matrix functions
cites work: Toeplitz Operators with Piecewise Quasisectorial Symbols
cites work: Q4208848
cites work: TRANSLATION OPERATORS ON THE HALF-LINE
cites work: Q4150220
cites work: Q3866607
cites work: On algebras generated by convolutions and discontinuous functions
cites work: Q3922907
cites work: Q5577878
cites work: Algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients
cites work: Q5644641
cites work: One-dimensional linear singular integral equations. I. Introduction. Transl. from the German by Bernd Luderer and Steffen Roch
cites work: On Completely Continuous Hankel Matrices
cites work: Q3471091
cites work: Q4292009
cites work: FACTORIZATION OF ALMOST PERIODIC MATRIX-VALUED FUNCTIONS AND THE NOETHER THEORY FOR CERTAIN CLASSES OF EQUATIONS OF CONVOLUTION TYPE
cites work: Factorization of measurable matrix functions. Ed. by Georg Heinig. Transl. from the Russian by Bernd Luderer. With a foreword by Bernd Silbermann. (Licensed ed.)
cites work: Q3997462
cites work: Q5748468
cites work: Approximation of Piecewise Continuous Functions by Quotients of Bounded Analytic Functions
cites work: Toeplitz operators with semialmost periodic symbols
cites work: SOME GENERAL QUESTIONS IN THE THEORY OF THE RIEMANN BOUNDARY PROBLEM
full work available at URL: s0022-1236(03)00126-5
OpenAlex ID: W2087136669
description: scientific article; zbMATH DE number 2037120