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Mathematical Physics, Mathematics, Functional Analysis
Abstract:
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 L2(Rd). 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 L2(B(L2(Rd))). 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.
