Invertibility of almost-periodic operators (Q1080141)

Let G be a locally compact abelian group and E a finite dimensional Banach space. A linear bounded operator T from \(L_ p(G,E)\), \(1\leq p\leq \infty\), or C(G,E) to itself is called almost periodic iff the set \(\{S_ gTS_{-g}:g\in G\), \((S_ gx)(t)=x(t+g)\}\) is relatively compact in the algebra of all linear bounded operators. Sufficient conditions for special types of almost periodic operators to be invertible are given. Only sketch of proves.