Sufficiency of Favard's condition for a class of band-dominated operators on the axis (Q2472873)

Properties of band-dominated operators on the space \(\ell^\infty({\mathbb Z},U)\) of all bounded sequences with values in a Banach space \(U\) are studied. One can associate with an operator \(A\) on \(\ell^\infty({\mathbb Z},U)\) a family of limit operators, where each member of the family represents a part of the behavior of \(A\) at infinity. Relations between the following conditions are studied: (C1) all limit operators of \(A\) are injective, (C2) all limit operators of \(A\) are surjective, (C3) the inverses of the limit operators of \(A\) are uniformly bounded. It is shown that for a large class of band-dominated operators, the conditions (C1)--(C3) are equivalent to the condition (C1) commonly known as Favard's condition. As a corollary, for operators in the Wiener algebra, the authors describe the essential spectrum of \(A\) on \(\ell^p({\mathbb Z},U)\), where \(1\leq p\leq \infty\), as the union of point spectra of its limit operators considered as acting on \(\ell^\infty({\mathbb Z},U)\).