Kernel and symbol criteria for Schatten classes and $r$-nuclearity on compact manifolds

TL;DR

This paper establishes criteria based on symbols and kernels for operators on compact manifolds to belong to Schatten classes, introduces nuclearity tests, and explores applications to $L^p$ spaces, with special focus on compact Lie groups.

Contribution

It provides new symbol and kernel criteria for Schatten class membership and nuclearity on compact manifolds and Lie groups, including trace formulas and applications to $L^p$ spaces.

Findings

Criteria for Schatten class membership based on symbols and kernels.

A nuclearity test and associated trace formulas are developed.

Applications to $r$-nuclearity on $L^p$ spaces are demonstrated.

Abstract

In this note we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^{p}$ spaces are also obtained.