Approximation properties of operator coorbit spaces and sparsity classes

Extensions of coorbit spaces for functions to operators have been introduced by two different groups in doelumcskr24 and köbaLOC25, where one is based on the coorbit theory of Feichtinger-Gröchening while the other is based on the theory of localized frames. We show that for certain Gabor g-frames the co-orbit spaces in köbaLOC25 conincide with the ones in doelumcskr24 and we refer to this class of operators as operator coorbit spaces. Based on the description of operator coorbit spaces in terms of Gabor g-frames we provide operator dictionaries for these spaces that allow us to define sparsity classes in this setting. We establish that these sparsity classes also coincide with the operator coorbit spaces, which holds, in particular for all Feichtinger operators, a nice class of mixed states. Numerical examples confirm the expected approximation quality by few terms for appropriately chosen operators.