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

## TL;DR

This paper studies the nuclear trace of vector-valued Fourier multipliers on the torus, characterising their nuclearity, and deriving index formulas for elliptic periodic pseudo-differential operators, advancing index theory.

## Contribution

It provides new characterisations of nuclearity for vector-valued Fourier multipliers and formulates index theorems for periodic elliptic pseudo-differential operators.

## Key findings

- Characterisation of nuclearity for vector-valued Fourier multipliers.
- Sharp sufficient conditions for nuclearity on the torus.
- Explicit index formulas for elliptic periodic pseudo-differential operators.

## Abstract

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