Gelfand-Shilov spaces and localization operators

Abstract

We use Komatsu's approach in the study of Gelfand-Shilov spaces
of ultradifferentiable functions in both quasianalytic and non-quasianalytic case.
In particular, we prove the kernel theorem for such spaces and study
the action of time-frequency representations on Gelfand-Shilov spaces
and their dual spaces of ultradistributions.
We apply the results to prove the trace class properties of
localization operators with ultradistributional symbols.
As a bridge between those results we prove and use the description of certain Gelfand-Shilov spaces
and their dual spaces
as projective and inductive limits of Feichtinger's modulation spaces. For the sake of completeness,
we review continuity and compactness properties of
localization operators on modulation spaces with polynomial weights,
which concerns the space of tempered distributions instead.