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# Invertibility at Infinity of Band-Dominated Operators on the Space of Essentially Bounded Functions

Publikationstyp

Journal Article

Publikationsdatum

2004

Sprache

English

Start Page

297

End Page

325

Citation

In: Operator Theoretical Methods and Applications to Mathematical Physics [297-325]

Publisher DOI

Publisher

Springer

he topic of this paper is band operators and the norm limits of such — so-called band-dominated operators, both classes acting on L∞(ℝn). Invertibility at infinity is closely related to Fredholmness. In fact, in the discrete case ℓp(ℤn), 1 ≤ p ≤ ∞, both properties coincide. For many applications, e.g., the question of applicability of certain approximation methods, in the situation at hand, Lp(ℝn), 1 ≤ p ≤ ∞, it has however proved to be useful to study invertibility at infinity rather than Fredholmness.

We will present a criterion for a band-dominated operator’s invertibility at infinity in terms of the invertibility of its limit operators. It is the same criterion that was found for ℓp(ℤn), 1 < p < ∞ in [21] and for the C*- algebra L 2 (ℝn in [22]. Our investigations concentrate on one of the most unvolved cases, being L ∞(ℝn). With the techniques presented here it is clear now how the remaining cases ℓ1, ℓ∞ and L p, (p≠2) have to be treated.

We will present a criterion for a band-dominated operator’s invertibility at infinity in terms of the invertibility of its limit operators. It is the same criterion that was found for ℓp(ℤn), 1 < p < ∞ in [21] and for the C*- algebra L 2 (ℝn in [22]. Our investigations concentrate on one of the most unvolved cases, being L ∞(ℝn). With the techniques presented here it is clear now how the remaining cases ℓ1, ℓ∞ and L p, (p≠2) have to be treated.