Approximation property and nuclearity on mixed-norm Lp, modulation and Wiener amalgam spaces

In this paper, we first prove the metric approximation property for weighted mixed-norm L-p spaces. Using Gabor frame representation, this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and -nuclearity for operators acting on these spaces as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.