In this note we present the metric approximation
property for weighted mixed-norm L
(p1,...,pn)
w 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.