A Magnetic Pseudodifferential Calculus for Operator-Valued and Equivariant Operator-Valued Symbols

TL;DR

This paper develops a comprehensive magnetic pseudodifferential calculus for operator-valued and equivariant operator-valued symbols, providing new criteria, resolvent constructions, and trace class conditions with applications in mathematical physics.

Contribution

It introduces a systematic, pedagogical framework for magnetic pseudodifferential operators with new Beals-type criteria and resolvent constructions, extending results to equivariant cases.

Findings

Established Beals-type commutator criteria.

Constructed Moyal resolvents for elliptic symbols.

Provided criteria for trace class operators.

Abstract

In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics, including in rigorous perturbation theory for slow-fast systems and perturbed periodic operators. Yet, a systematic treatise was hitherto missing. While many of the results can be found piecemeal in appendices and as sketches in other articles, this article does contain new results. For instance, we have established Beals-type commutator criteria for both cases, which then imply the existence of Moyal resolvents for (equivariant) selfadjoint-operator-valued, elliptic H\"ormander symbols and allows one to construct functional calculi. What is more, we give criteria on the function under which a magnetic pseudodifferential operator is (locally) trace class. Our aims for this article are three-fold: (1) Create a single, solid work that colleagues can refer to. (2) Be pedagogical and precise. And (3) give a straightforward strategy for extending results from the operator-valued to the equivariant case, pointing out some caveats and pitfalls that need to be kept in mind.