Wiener-Hopf operators with semi-almost periodic matrix symbols on weighted Lebesgue spaces
type: Item
instance of: scholarly article
title: Wiener-Hopf operators with semi-almost periodic matrix symbols on weighted Lebesgue spaces
zbMATH Open document ID: 1201.47031
author: D. Kharzeev
published in: Integral Equations and Operator Theory
publication date: 2009-10-19T00:00:00Z
review text: The paper is devoted to the Fredholm theory of Wiener-Hopf operators \[ W_\Phi : [L^p(\mathbb{R}_+, w)]^N \rightarrow [L^p(\mathbb{R}_+, w)]^N \] with semi-almost periodic \(N\times N\) matrix Fourier symbols \(\Phi\), on weighted Lebesgue spaces with exponent \(p \in (1, +\infty)\), and considering \(w\) in a subclass of the Muckenhoupt weights. A Fredholm criterion and a Fredholm index formula for this \(W_\Phi\) are obtained when considering the almost periodic representatives of \(\Phi\) at \(\pm\infty\) to admit an \(AP\) factorization, or when the almost periodic representatives of \(\Phi\) at \(\pm\infty\) have absolutely convergent Bohr-Fourier series and the weight \(w\) satisfies \[ \lim_{|t|\to\infty} {\underset{x,y\in [t,t+1]}{\text{sup\,ess}}} \big|\ln w(x)-\ln w(y)\big| = 0. \] In addition, for \(N=1\), a (semi-)Fredholm criterion for \(W_\Phi\) is given, and invertibility criteria for almost periodic and ``Wiener almost periodic'' symbols are also deduced. In view of obtaining these fundamental characterizations, several auxiliary results are deduced. Namely, an analogue of the Sarason representation of semi-almost periodic matrix functions, and the inverse closedness of the algebras of almost periodic and semi-almost periodic matrices are proved in the weighted framework under study.
reviewed by: L. P. Castro
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5617567
zbMATH Keywords: Wiener-Hopf operator
zbMATH Keywords: invertibility
zbMATH Keywords: Fredholm index
zbMATH Keywords: factorization
zbMATH Keywords: almost periodic matrix function
zbMATH Keywords: semi-almost periodic matrix function
zbMATH Keywords: weighted Lebesgue space
zbMATH Keywords: Fredholmness
MaRDI profile type: Publication
full work available at URL: s00020-008-1614-5
OpenAlex ID: W1989272687
DOI: 10.1007/S00020-008-1614-5
description: scientific article; zbMATH DE number 5617567