Invertibility of nonautonomous linear difference operators in the space of bounded functions on Z
type: Item
instance of: scholarly article
title: Invertibility of nonautonomous linear difference operators in the space of bounded functions on Z
zbMATH Open document ID: 0595.47023
DOI: 10.1007/BF01157966
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let E be a Banach space, \({\mathfrak M}^ a \)space of E-valued functions bounded on Z (the integers) with the sup-norm, \({\mathfrak M}_{\omega}\) its subspace of \(\omega\)-periodic functions. For every \(n\in Z\), let A(n) be a totally continuous linear operator in E with \(\sup \{\| A(n)\|:\) \(n\in Z\}<\infty\) and consider the continuous linear operator \({\mathcal D}\) in \({\mathfrak M}\) defined by \[ ({\mathcal D}x)(n)=x(n+1)-A(n)x(n),\quad x\in {\mathfrak M},\quad n\in Z. \] Theorem: If \[ \overline{\lim}_{\omega \to \infty}\inf_{y\in S_{\omega}}\max_{n\in I_{\omega}}\| y(n+1)\quad -A(n)y(n)\|_ E>0 \] where \(S_{\omega}=\{y:\) \(\| y\|_{{\mathfrak M}_{2\omega}}=1\}\) and \(I_{\omega}=[-\omega,\omega [\cap Z\), then \({\mathcal D}\) possesses a continuous inverse. In the case of \({\mathcal D}\) being almost periodic, the sufficient condition above is also necessary; it is shown by an example that this condition is not necessary in general.
reviewed by: J.Durdil
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 39A70
zbMATH DE Number: 3958041
zbMATH Keywords: continuous inverse
zbMATH Keywords: almost periodic
author: V. E. Slyusarchuk
MaRDI profile type: Publication
cites work: Exponential dichotomy for solutions of discrete systems
cites work: Defect subspaces and generalized resolvents of an Hermitian operator in the space \(\Pi_\kappa\)
cites work: Q3286587
description: scientific article; zbMATH DE number 3958041