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

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.

Nulcearity for power series kernels on higher dimensions

“…Kernel criteria on L p spaces has been considered in [16], and the special case of L 2 in [12], [13], [17], [25] for the study of different spectral properties. Symbolic criteria to ensure the nuclearity of pseudodifferential operators on several kind of domains has been studied in [6], [7], [8], [9], [15], [22], [23]. The interesting case of the Fox-Li operator corresponds to a kernel of the form K(x, y) = e iω(x−y) 2 = e iωx 2 e −2iωxy e iωy 2 , where X = Y = [−1, 1] and ω is a positive real number.…”

Nuclearity for power series kernels in higher dimensions

“…Kernel criteria on L p spaces has been considered in [16], and the special case of L 2 in [12], [13], [17], [25] for the study of different spectral properties. Symbolic criteria to ensure the nuclearity of pseudodifferential operators on several kind of domains has been studied in [6], [7], [8], [9], [15], [22], [23]. The interesting case of the Fox-Li operator corresponds to a kernel of the form K(x, y) = e iω(x−y) 2 = e iωx 2 e −2iωxy e iωy 2 , where X = Y = [−1, 1] and ω is a positive real number.…”

L2-Lp estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

“…To extend pseudo-differential operators to other settings, one observes that the second R n in the Cartesian product R n × R n is the dual of the additive group R n . These observations allow us to extend the definition of pseudo-differential operators to other groups G, provided we have an explicit formula for the dual of G and an explicit Fourier inversion formula on G. Using this approach, the global theory of pseudodifferential operators on other classes of groups, such as S 1 , Z, affine groups, compact (Lie) groups, homogeneous spaces of compact (Lie) groups, Heisenberg groups, graded Lie groups, step two nilpotent Lie groups, and locally compact type I groups has been widely studied by several researchers [8,15,16,4,21,34,23,6,5].…”