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.

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