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

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  • Duván Cardona
  • Vishvesh Kumar

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 Hörmander classes.

Suggested Citation

  • Duván Cardona & Vishvesh Kumar, 2021. "The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory," Mathematische Nachrichten, Wiley Blackwell, vol. 294(9), pages 1657-1683, September.
    • Handle: RePEc:bla:mathna:v:294:y:2021:i:9:p:1657-1683
    DOI: 10.1002/mana.201900040
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