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In this paper we first prove the metric approximation property for weightedmixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representationthis implies that the same property holds in weighted modulation and Wieneramalgam spaces. As a consequence, Grothendieck's theory becomes applicable, andwe give criteria for nuclearity and $r$-nuclearity for operators acting onthese space as well as derive the corresponding trace formulae. Finally, weapply the notion of nuclearity to functions of the harmonic oscillator onmodulation spaces.

Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces