Fredholm operators on abelian phase spaces

We study the Fredholm property for linear operators on coorbit spaces over locally compact abelian phase spaces.

As a special case we consider operators on $L^2(G)$, where $G$ is an arbitrary locally compact abelian group.

Our approach therefore extends the existing theory for discrete spaces to the continuous setting and complements the study in our previous work where compactness was characterized in terms of limit operators.

Our results are again achieved by merging tools from the theory of band-dominated operators with methods of quantum harmonic analysis, thereby achieving new results in both areas.

We emphasize that our results are new (and maybe most interesting) even in the $L^2$-setting.