Magnetic Weyl Super Calculus: Schatten-class properties, commutator criterion, and complete positivity

TL;DR

This paper extends magnetic pseudo-differential operator theory, establishing Schatten-class properties, a commutator criterion, and conditions for complete positivity within the magnetic Weyl super calculus framework.

Contribution

It combines previous magnetic pseudo-differential results with the magnetic Weyl super calculus to prove boundedness, compactness, Schatten-class properties, and criteria for complete positivity.

Findings

Proved Schatten-class properties of super operators.

Established a Beals-type commutator criterion.

Formulated conditions for super symbols to generate completely positive maps.

Abstract

We combine our previous results on magnetic pseudo-differential operators for H\"ormander symbols dominated by tempered weights [arXiv:2511.07184] with the magnetic Weyl super calculus of Lee and Lein [arXiv:2201.11487, arXiv:2405.19964]. This allows us to extend some previous results on the semi-super and super Moyal algebra, as well as to prove boundedness, compactness, and Schatten-class properties of super operators. Moreover, we prove a Beals-type commutator criterion for super operators and we also formulate sufficient conditions on super symbols in order to give rise to completely positive and trace preserving maps. For most of the proofs we use decompositions of operators and super operators based on Parseval frames of smoothing operators.