A metric approach to limit operators (Q2826760)

An operator on the space \(\ell^p({\mathbb Z}^n)\) is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. \textit{V. Rabinovich} et al. proved in Theorem 2.2.1 of the monograph [Limit operators and their applications in operator theory. Operator Theory: Advances and Applications 150. Basel: Birkhäuser (2004; Zbl 1077.47002)] 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. Later, \textit{M. Lindner} and \textit{M. Seidel} [J. Funct. Anal. 267, No. 3, 901--917 (2014; Zbl 1292.47020)] proved that the condition on uniform boundedness is redundant in the above statement.NEWLINENEWLINEThe paper under review is devoted to the extension of the machinery of limit operators from \({\mathbb Z}^n\) to bounded geometry strongly discrete metric spaces \(X\). No group structure or action on the metric space \(X\) is assumed. It is shown that, if the metric space \(X\) has \textit{G.-L. Yu}'s property A [Invent. Math. 139, No. 1, 201--240 (2000; Zbl 0956.19004)] and \(E\) is a Banach space, then a rich band-dominated operator on \(\ell^p_E(X)\), \(1<p<\infty\), is \({\mathcal P}\)-Fredholm if and only if all its limit operators are invertible. It is also shown that the result is not true for metric spaces without property A.