This is an announcement for the paper "Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces" by Julio Delgado, Michael Ruzhansky and Baoxiang Wang. Abstract: In this paper we first prove the metric approximation property for weighted mixed-norm Lebesgue spaces. Then, using Gabor frame representation we show that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and r-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 46B26, 47B38, Secondary 47G10, 47B06, 42B35 Remarks: 20 pages Submitted from: m.ruzhansky@imperial.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4687 or http://arXiv.org/abs/1410.4687