Schatten classes on noncommutative tori: Kernel conditions

TL;DR

This paper establishes criteria for integral kernels on quantum tori to ensure their associated operators belong to Schatten classes, using a noncommutative Schwartz kernel theorem to provide a practical test.

Contribution

It introduces new kernel conditions and a test for Schatten class membership of operators on noncommutative tori, advancing understanding in noncommutative analysis.

Findings

Criteria for Schatten class membership of integral operators on quantum tori

Development of a noncommutative Schwartz kernel theorem

A practical test for Schatten class properties of bounded operators

Abstract

In this note, we give criteria on noncommutative integral kernels ensuring that integral operators on quantum torus belong to Schatten classes. With the engagement of a noncommutative Schwartz' kernel theorem on the quantum torus, a specific test for Schatten class properties of bounded operators on the quantum torus is established.