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

Abstract

In this note we present the metric approximation property for weighted mixed-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 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 Rn and functions of the harmonic oscillator in terms of global symbols.