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In this note we present the metric approximation property for weightedmixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces.As a consequence, this also implies the same property for modulation andWiener-Amalgam spaces. We then characterise nuclear operators on such spacesand state the corresponding Grothendieck-Lidskii trace formulae. We apply theobtained results to derive criteria for nuclearity and trace formulae forperiodic operators on $\mathbb R^n$ and functions of the harmonic oscillator interms of the global symbol.

Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces