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

In this note we present the metric approximation property for weighted mixed-norm L-w((p1,...pn)) and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on R-n and functions of the harmonic oscillator in terms of global symbols.