A Proof of $\mathfrak{L}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames
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| Format: | Preprint |
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2024
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| _version_ | 1866916269010190336 |
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| author | Lee, Gihyun Lein, Max |
| author_facet | Lee, Gihyun Lein, Max |
| contents | 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 $\mathfrak{L}^2 \bigl ( \mathcal{B} \bigl ( L^2(\mathbb{R}^d) \bigr ) \bigr )$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Proof of $\mathfrak{L}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames Lee, Gihyun Lein, Max Mathematical Physics Functional Analysis 35S05, 46B15, 47C15, 47L80 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 $\mathfrak{L}^2 \bigl ( \mathcal{B} \bigl ( L^2(\mathbb{R}^d) \bigr ) \bigr )$. 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. |
| title | A Proof of $\mathfrak{L}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames |
| topic | Mathematical Physics Functional Analysis 35S05, 46B15, 47C15, 47L80 |
| url | https://arxiv.org/abs/2405.19964 |