An Affirmative Answer to a Core Issue on Limit Operators

Abstract

An operator on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. It is known that a rich band-dominated operator is $\mathcal{P}$-Fredholm (which is a generalization of the classical Fredholm property) if and only if all of its so-called limit operators are invertible and their inverses are uniformly bounded. We show that the condition on uniform boundedness is redundant in this statement.

References in corpus (1)

• Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices

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