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.
