The nuclear trace of periodic vector?valued pseudo?differential operators with applications to index theory

In this paper we investigate the nuclear trace of vector-valued Fourier multipliers on torus and its applications to index theory periodic pseudo-differential operators. First characterise nuclearity operators acting Bochner integrable functions. regards, consider discrete cases. We go address problem finding sharp sufficient conditions for torus. end our investigation with two formulae. First, express a multiplier in terms operator-valued symbol then use formula expressing certain elliptic belonging Hörmander classes.