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Magnetic Weyl Super Calculus: Schatten-class properties, commutator criterion, and complete positivity

Horia D. Cornean, Mikkel H. Thorn

Abstract

We combine our previous results on magnetic pseudo-differential operators for Hörmander symbols dominated by tempered weights [arXiv:2511.07184] with the magnetic Weyl super calculus of Lee and Lein [arXiv:2201.11487, arXiv:2405.19964]. This allows us to extend some previous results on the semi-super and super Moyal algebra, as well as to prove boundedness, compactness, and Schatten-class properties of super operators. Moreover, we prove a Beals-type commutator criterion for super operators and we also formulate sufficient conditions on super symbols in order to give rise to completely positive and trace preserving maps. For most of the proofs we use decompositions of operators and super operators based on Parseval frames of smoothing operators.

Magnetic Weyl Super Calculus: Schatten-class properties, commutator criterion, and complete positivity

Abstract

We combine our previous results on magnetic pseudo-differential operators for Hörmander symbols dominated by tempered weights [arXiv:2511.07184] with the magnetic Weyl super calculus of Lee and Lein [arXiv:2201.11487, arXiv:2405.19964]. This allows us to extend some previous results on the semi-super and super Moyal algebra, as well as to prove boundedness, compactness, and Schatten-class properties of super operators. Moreover, we prove a Beals-type commutator criterion for super operators and we also formulate sufficient conditions on super symbols in order to give rise to completely positive and trace preserving maps. For most of the proofs we use decompositions of operators and super operators based on Parseval frames of smoothing operators.

Paper Structure

This paper contains 26 sections, 17 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Outline:
  3. Preliminaries
  4. The Short Story
  5. The Detailed Detour
  6. Variable names:
  7. Operator spaces:
  8. Magnetic Weyl quantization:
  9. Magnetic super Weyl quantization:
  10. Hörmander classes:
  11. Frame Decompositions
  12. Matrix Representation of Super Operators
  13. Super Calculus of Hörmander Symbols
  14. Magnetic Semi-super Weyl Product
  15. Magnetic Super Weyl Product
  16. ...and 11 more sections

Key Result

Lemma 3.1

LeeLein2025 The family $(\mathcal{T}_{\tilde{\alpha},\tilde{\beta}}^A)_{\tilde{\alpha},\tilde{\beta}\in{\mathbb Z}^{2d}}$ defines a Parseval frame in $\mathcal{B}_2(L^2({\mathbb R}^d))$ andWe define inner products to be antilinear in the first entry and linear in the second. holds for every $S\in\mathcal{B}_2(L^2({\mathbb R}^d))$ with unconditional convergence.

Theorems & Definitions (45)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof : Proof of \ref{['i_matrix_rep']}
  • proof : Proof of \ref{['ii_matrix_rep']}
  • Remark 3.5
  • Lemma 4.1
  • ...and 35 more