Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

TL;DR

This paper develops symbolic criteria for nuclear and r-nuclear operators on Lp-spaces over compact Lie groups, with applications to eigenvalue distribution and trace formulas, using matrix symbols on the non-commutative phase space.

Contribution

It introduces a symbolic approach to characterize nuclearity of operators on compact Lie groups, overcoming limitations of kernel-based criteria.

Findings

Provides criteria for nuclearity in terms of matrix symbols.

Applies criteria to eigenvalue distribution and trace formulas.

Offers a non-commutative phase space framework for analysis.

Abstract

Given a compact Lie group $G$, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on Lp-spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\hat{G}$, where $\hat{G}$ is the unitary dual of $G$.