A Proof of ${\mathfrak{L}}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames

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 \emph{super} operator, \ie 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 ${\mathfrak{L}}^2\left(\mathcal{B}\right(L^2({\mathbb{R}}^d)\left)\right)$. Our proof is inspired by the recent work of Cornean, Helffer and Purice and rests on a characterization of magnetic pseudodifferential super operators in terms of their “matrix element” computed with respect to a Parseval frame.