A fundamental result in pseudodifferential theory is the Calderón–Vaillancourt theorem, which states that a pseudodifferential operator defined from a Hörmander symbol of order 0 defines a bounded operator on \(L^2({\mathbb {R}}^d)\) . In this work we prove an analog for pseudodifferential super operator, i. e. operators acting on other operators, in the presence of magnetic fields. More precisely, we show that magnetic pseudodifferential super operators of order 0 define bounded operators on the space of Hilbert–Schmidt operators \({\mathfrak {L}}^2 \bigl ( {\mathcal {B}} \bigl ( L^2({\mathbb {R}}^d) \bigr ) \bigr )\) . Our proof is inspired by the recent work of Cornean et al. (J Fourier Anal Appl 30(21):1–21, 2024. https://doi.org/10.1007/s00041-024-10072-4) and rests on a characterization of magnetic pseudodifferential super operators in terms of their “matrix elements” computed with respect to a Parseval frame.
