Fredholm operators on abelian phase spaces
Robert Fulsche, Raffael Hagger
TL;DR
This work develops a general Fredholm theory for operators on $p$-coorbit spaces over abelian phase spaces $(\Xi,m)$ without requiring second-countability. By blending band-dominated operator techniques with quantum harmonic analysis, it characterizes compactness and Fredholmness via limit operators: an operator $A$ is compact exactly when $A\in\mathcal{C}_1^p$ and its limit operators vanish on the boundary, and $A$ is Fredholm precisely when every limit operator $\alpha_x(A)$ is invertible with uniformly bounded inverses. A key advance is the inclusion $\mathcal{C}_1^p(\varphi_0) \subseteq P_{\varphi_0}\mathrm{BDO}^p(\Xi)P_{\varphi_0}$, enabling a norm-localization analysis that ties the essential norm to boundary behavior. The results generalize spectral and stability analyses to broad, non-discrete, non-second-countable abelian phase spaces, leveraging both coorbit theory and limit-operator methods. Practical impact lies in providing robust criteria for Fredholmness and compactness in a wide class of function spaces arising in time-frequency analysis and quantum harmonic analysis.
Abstract
We study compactness and the Fredholm property for linear operators on coorbit spaces over locally compact abelian phase spaces. In contrast to previous works, we do not impose any countability assumptions on the underlying groups. Our results are achieved by merging tools from the theory of band-dominated operators with methods of quantum harmonic analysis, thereby achieving new results in both areas.