Local Theory of the Fredholmness of Band-Dominated Operators with Slowly Oscillating Coefficients

Abstract

A band-dominated operator on an l ^p-space of vector-valued functions is known to be a Fredholm operator (in a generalized sense) if and only if all of its limit operators are invertible and if their inverses are uniformly bounded. We show that the limit operators approach is also compatible with the local Fredholmness of band-dominated operators with respect to localization over the maximal ideal space of the algebra of the slowly oscillating scalarvalued functions. A corollary of this result is that the uniform boundedness condition is redundant for band-dominated operators with slowly oscillating operator-valued coefficients.