Spectral analysis of difference operators with strongly continuous coefficients (Q1277436)

Let \(G\) be a locally compact abelian group, \(X\) be a Banach space and \({\mathcal F}(G,X)\) be the set of measurable functions with values in \(X.\) The author considers difference operators \(\mathcal D\) of the form \(({\mathcal D}x)(g)=\sum A_n (g)x(g+g_n),\) where \(x\in {\mathcal F}(G,X)\), \(g, g_n\in G\) and the \(A_n(g)\) are strongly continuous functions on \(G\) with values in \(\text{End} X.\) The structure of the inverse operator to \(\mathcal D\) is studied. In particular spectral properties of the operator of weighted translation, i.e. \(({\mathcal D}x)(g)=A(g)x(g+g_0)\) are examined. An application to first order linear differential operators is given.