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A proof of L2-boundedness for magnetic pseudodifferential super operators via matrix representations with respect to parseval frames

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Lee,  Gihyun       
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.48550/arXiv.2405.19964
(Preprint)

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2405.19964.pdf
(Preprint), 323KB

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Citation

Lee, G., & Lein, M. (submitted). A proof of L2-boundedness for magnetic pseudodifferential super operators via matrix representations with respect to parseval frames.


Cite as: https://hdl.handle.net/21.11116/0000-000F-5BCB-7
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.