Invertibility of nonautonomous linear difference operators in the space of bounded functions on Z
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.
scientific article; zbMATH DE number 3958041