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

Gihyun Lee1 & Max Lein2

###### 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
$L^{2}(\mathbb{R}^{d})$. In this work we prove an analog for
pseudodifferential _super_ operator, i. e. 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
[CHP22] and rests on a characterization of magnetic pseudodifferential super
operators in terms of their “matrix elements” computed with respect to a
Parseval frame.

1 Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent
University Krijgslaan 281, S8 9000 Gent, Belgium<EMAIL_ADDRESS>
2 University of Postdam, Institute for Mathematics Campus Golm, Building 9,
Karl-Liebknecht-Str. 24-25, 14476 Potsdam OT Golm, Germany lein2@uni-
potsdam.de

Key words: Pseudodifferential operators, magnetic operators, $L^{2}$
boundedness, non-commutative $L^{p}$ spaces
MSC 2020: 35S05, 46B15, 47C15, 47L80

###### Contents

  1. 1 Introduction
    1. 1.1 Our method of choice: representing magnetic pseudodifferential super operators as “infinite matrices”
    2. 1.2 The difficulty in proving $\mathfrak{L}^{p}$ boundedness for $p\neq 2$
      1. 1.2.1 The lack of a simple characterization of trace-class and bounded operators in terms of their matrix elements
      2. 1.2.2 The state-of-the-art for some simple cases
  2. 2 Setting and fundamental definitions
    1. 2.1 The relevant Parseval frame
    2. 2.2 Magnetic Weyl calculus
    3. 2.3 Magnetic pseudodifferential super operator calculus
  3. 3 Abstract setting: characterization of (super) operators via their matrix elements
    1. 3.1 Matrix representation of operators
      1. 3.1.1 Hilbert-Schmidt operators
    2. 3.2 Matrix representation of super operators
    3. 3.3 Expressing products of super operators and operators in terms of their matrix elements
  4. 4 The main technical result: characterization of symbols of super $\Psi$DOs via their matrix elements
  5. 5 Boundedness criteria for magnetic pseudodifferential super operators
    1. 5.1 Boundedness of magnetic pseudodifferential superoperators
    2. 5.2 More advanced boundedness results

##  1 Introduction

This work is a continuation of the authors’ study of magnetic
pseudodifferential _super_ operators [LL22], and we aim to prove boundedness
criteria for them. An example of such a super operator would be the Liouville
super operator

$\displaystyle\hat{L}^{A}(\hat{\rho}^{A})=-\mathrm{i}\,\bigl{[}\mathrm{op}^{A}(h),\hat{\rho}^{A}\bigr{]}$

or a more general Lindblad super operator acting on some trace class operator
$\hat{\rho}^{A}\in\mathfrak{L}^{1}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$,
where $\mathrm{op}^{A}(h)$ is the magnetic Weyl quantization [MP04, Lei11,
IMP07] of a suitable function such as $h(x,\xi)\equiv h(X)=\xi^{2}+V(x)$.
Readers unfamiliar with _magnetic_ pseudodifferential theory can ignore the
presence of the magnetic vector potential $A$ for the moment; we will give the
relevant definitions in Section 2 below.

Our work [LL22] explains that such a Liouville super operator can be viewed as
the quantization $\hat{L}^{A}=\mathrm{Op}^{A}(L)$ with respect to the function

$\displaystyle L(X_{L},X_{R})=-\mathrm{i}\,\bigl{(}h(X_{L})-h(X_{R})\bigr{)},$
$\displaystyle X_{L},X_{R}\in\Xi.$

Both, left ($L$) and right ($R$) variables are elements of phase space
$\Xi:=T^{*}\mathbb{R}^{d}\cong\mathbb{R}^{d}\times\mathbb{R}^{d}$, which we
view as a cotangent bundle endowed with the magnetic symplectic form (see e.
g. [Lei11, Sections 2.1.1.1 and 3.6]). The indices $L$ and $R$ indicate
whether after quantization the operator will act from the left or the right.

That poses the question we wish to answer here: for generic functions $L$
belonging to some Hörmander class, under what conditions on $L$ does the
associated magnetic pseudodifferential _super_ operator define a bounded
operator

$\displaystyle\mathrm{Op}^{A}(L):\mathfrak{L}^{1}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\longrightarrow\mathfrak{L}^{1}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
(1.1)

between trace class operators? More generally, when is

$\displaystyle\mathrm{Op}^{A}(L)\in\mathcal{B}\Bigl{(}\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\Bigr{)}$
(1.2)

a bounded operator between $p$-Schatten classes where $1\leq p<\infty$? Or we
could pose the same question for the space of bounded operators
$\mathfrak{L}^{\infty}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}=\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$
to itself, which corresponds to $p=\infty$.

As the notation suggests, these operator spaces are particular cases of non-
commutative $\mathfrak{L}^{p}$ spaces [Ter81, Tak03], where the trace plays
the role of integration with respect to a measure. Since the
pseudodifferential context involves additional structures derived from twisted
crossed product $C^{*}$-algebras [MPR05], some subcommunities refer to non-
commutative $\mathfrak{L}^{p}$-spaces equipped with these extra structures as
_quantum_ $\mathfrak{L}^{p}$ spaces (cf. [AMS02, Section 3]). Likewise,
imposing additional regularity gives rise to e. g.  _non-commutative_ or
_quantum_ Sobolev spaces (see, e. g. [Laf22]).

Our main result answers the question of boundedness (1.2) for $p=2$, which
should be seen as a super operator analog of the Calderón-Vaillancourt theorem
from psuedodifferential theory (cf. e. g. [IMP07, Theorem 3.1]):

######  Theorem 1.1.

Suppose the components
$B_{jk}\in\mathcal{C}^{\infty}_{\mathrm{b}}(\mathbb{R}^{d})$ of the magnetic
field $B=\sum_{j,k=1}^{d}B_{jk}\,\mathrm{d}x_{j}\wedge\mathrm{d}x_{k}$ are
smooth, bounded and possess bounded derivatives to any order. Moreover, let
$A\in\mathcal{C}^{\infty}_{\mathrm{pol}}(\mathbb{R}^{d},\mathbb{R}^{d})$ be a
smooth, polynomially bounded vector potential with $\mathrm{d}A=B$.

Then any Hörmander symbol $L\in S^{m}_{\rho,0}(\Xi^{2})$ (cf. Definition 2.4)
of order $m\leq 0$ and type $(\rho,0)$ where $0\leq\rho\leq 1$ defines a
bounded operator

$\displaystyle\mathrm{Op}^{A}(L)\in\mathcal{B}\Bigl{(}\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\Bigr{)}.$

In fact, we are not only able to treat regular Hörmander classes, but also
Hörmander classes where we allow the growth in left and right momentum
variables $\xi_{L,R}$ to be different (cf. Theorem 5.3 and Corollary 5.6 for
details).

###  1.1 Our method of choice: representing magnetic pseudodifferential super
operators as “infinite matrices”

There are several established strategies to prove the boundedness of
pseudodifferential operators. One of them (e. g. implemented for magnetic
pseudodifferential operators as the proof of [IMP07, Theorem 3.1]) slices up
the operator into pieces that are localized in a grid cell, estimates each
localized operator with oscillatory integral techniques and controls their
sum, the original operator, via the Stein-Cotlar-Knapp Lemma. We could
alternatively follow the strategy implemented by Alinhac and Gérard (cf.
[AG07, pp. 29–31]) and Saint Raymond (cf. [Ray91, pp. 53–55]), and base our
proof on Schur’s test; the latter gives an explicit criterion for the
boundedness of operators in terms of their operator kernels.

However, we will choose a different path inspired by two recent works by
Cornean, Helffer and Purice [CHP18, CHP22]. The idea is to characterize
magnetic pseudodifferential (super) operators through their “matrix elements”
with respect to a Gabor frame, an approach which was pioneered by Cordero,
Heil, Gröchenig and others (cf. e. g. [HG99, Grö06, GR08]). A frame is a
countable set of vectors that forms an “overcomplete basis” of a separable
Hilbert space (cf. Section 2.1 and references therein for a proper
mathematical definition). Operators are therefore uniquely defined by their
matrix elements. The modifier “Gabor” refers to the fact that in this case the
frame is generated from a single vector and indexed by a lattice vector. The
Gabor frame Cornean et al. use is also a tight, normalized (or _Parseval_)
frame of $L^{2}(\mathbb{R}^{d})$, i. e. a frame that in many respects acts
like an orthonormal basis.

There are multiple advantages to this approach: first of all, one only needs
to use oscillatory integral techniques one time, namely in [CHP22, Theorem
3.1] and our analogs, Theorem 4.1 and Corollary 4.2. These results give a one-
to-one characterization of magnetic pseudodifferential _super_ operators
associated to a Hörmander symbol and their matrix elements. Proofs of
subsequent theorems then only involve matrix elements, and a common theme is
to impose conditions that ensure the existence of certain infinite sums over
the matrix elements’ indices. Conceptually and technically, this is a great
simplification. Moreover, it avoids the tedium of proving the existence of
certain oscillatory integrals and estimating relevant seminorms — which is
further exacerbated when additional magnetic phase factors and variables are
present (see e. g. [LL22, Appendix B]).

The second main advantage is the ease of adaptability: one of the authors used
this matrix element point of view to extend commutator criteria to magnetic
pseudodifferential operators associated to operator-valued and equivariant
operator-valued symbols (specifically Theorems 3.4.23 and 4.3.1 in [DLS22]). A
lot of more advanced results crucially depend on the given Fréchet topology.
And because the Fréchet spaces involved are usually _not_ nuclear, it is not
possible to _easily_ adapt existing proofs in a straightforward and
transparent manner. In fact, this is true here, too: the relevant Hörmander
classes (cf. Definition 2.5) are over
$\Xi^{2}=\Xi\times\Xi:=T^{*}\mathbb{R}^{d}\times T^{*}\mathbb{R}^{d}$, and it
would be tempting to split

“$S^{m_{L},m_{R}}_{\rho,\delta}(\Xi^{2})\cong
S^{m_{L}}_{\rho,\delta}(\Xi)\otimes S^{m_{R}}_{\rho,\delta}(\Xi)$” (incorrect
equation)

as a tensor product, and then invoke [CHP22, Theorem 3.1] to obtain a proof of
Theorem 4.1. Unfortunately, _the above equation is false._ Hörmander symbol
classes are non-nuclear Fréchet spaces [Wit97], and consequently, the tensor
product on the right is ill-defined: there exist _at least_ two topologies
with respect to which we can complete the algebraic tensor product. And
neither of these two gives the space on the left (cf. e. g. Theorem 3.2,
Remark 3.3, and Proposition 4.4 in [Wit97]). Yet at the end of the day, the
result _is_ as if we could treat the Hörmander class on the left as a tensor
product. The characterization via matrix elements _does_ neatly extend to
these more general cases as [DLS22] and the present work show.

###  1.2 The difficulty in proving $\mathfrak{L}^{p}$ boundedness for $p\neq
2$

We framed our work by looking at Liouville and Lindblad super operators, which
typically act on (dense subspaces of)
$\mathfrak{L}^{1}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.
Unfortunately, we had to exclude the highly desirable cases $p=1$ (trace-class
operators) and $p=\infty$ (bounded operators). For the benefit of the
interested reader, we will explain a few of the reasons and put our results
into context with the state-of-the-art. We intend to return to this subject in
the future.

####  1.2.1 The lack of a simple characterization of trace-class and bounded
operators in terms of their matrix elements

The key ingredient in our approach is to represent super operators as well as
the operators they act on as “infinite matrices”. Each of the matrix elements
is defined with respect to a Parseval frame
$\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ for a lattice $\Gamma$, and
properties such as
$\hat{g}\in\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}$ should
translate to properties of

$\displaystyle\hat{g}_{\alpha,\beta}:=\bigl{\langle}\mathcal{G}_{\alpha},\hat{g}\,\mathcal{G}_{\beta}\bigr{\rangle},$

where e. g. $\mathcal{H}=L^{2}(\mathbb{R}^{d})$ and $\alpha$ and $\beta$ take
values in some countably infinite index set $\Gamma$. The application of a
super operator $\hat{F}$ on an operator $\hat{g}$ can now be described as a
matrix product111The position of the indices to be summed over may seem
unusual, but we will explain our choice in Section 3.

$\displaystyle(\hat{F}\hat{g})_{\alpha,\beta}=\sum_{\alpha^{\prime},\beta^{\prime}\in\Gamma}\hat{F}_{\alpha,\beta^{\prime},\alpha^{\prime},\beta}\,\hat{g}_{\beta^{\prime},\alpha^{\prime}}.$

Hence, proving boundedness of the super operator
$\hat{F}\in\mathcal{B}\bigl{(}\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\bigr{)}$
is reduced to arguments ensuring the existence of the above sum in a specific
sense. However, that requires a characterization of the matrix elements of
$p$-Schatten class operators.

When $p=2$ we may identify the Hilbert space of Hilbert-Schmidt operators on
$\mathcal{H}$ with the Hilbert space
$\mathcal{H}\otimes\mathcal{H}\cong\ell^{2}(\Gamma)\otimes\ell^{2}(\Gamma)$ in
a canonical fashion. Unfortunately, for all other values of $p$ we are not
aware of such a simple characterization. Specifically, results for $p=1$ and
$p=\infty$ are particularly desirable, not only because they are the most
relevant in applications, but also because they form the two end points in a
Riesz-Thorin interpolation argument (cf. e. g. Theorems 2.9 and 2.10 as well
as Remark 1 on p. 23 in [Sim05] or [GK69, Theorem 13.1]).

For $p<\infty$ the best results we are aware of are Theorems A and B in
[BKZ15]. Simply put, they characterize the cases $2\leq p<\infty$ and $0<p\leq
2$, respectively, in terms of the $p$-summability of
$\lVert\hat{g}\mathcal{G}_{\alpha}\rVert$. Unfortunately, Theorem B which
covers trace-class operators states that we need $p$-summability with respect
to _some_ frame; that is, the frame will in general depend on the operator
under consideration.

The case $p=\infty$ has to be treated separately. Fortunately, if all we care
about is proving the boundedness of magnetic pseudodifferential _super_
operators, we do not need a one-to-one characterization of bounded operators
in terms of their matrix elements. Still, such a characterization exists and
can be found in e. g. Chapter 1 of the recent monograph [dMMR21], specifically
Lemmas 1.3 and 1.4 as well as Crone’s Theorem 1.24. Unfortunately, none of
these conditions translate to simple conditions on the matrix elements. What
is more, the existing results mentioned below indicate that looking at

$\displaystyle\mathrm{Op}^{A}(F):\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\longrightarrow\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$

might not be the way to go for the cases $p=1$ and $p=\infty$.

####  1.2.2 The state-of-the-art for some simple cases

There _is_ a partial result in this direction, e. g. [GPJP21]. The space
$\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$ is a special case of
quantum Euclidean spaces studied in [GPJP21] (cf. loc. cit. Remark 1.3).
Therefore, the pseudodifferential theory developed in [GPJP21] pertains to
super operators that multiply with a pseudodifferential operator _from one
side_ , i. e. (non-magnetic) super operators of the form

$\displaystyle\mathrm{Op}^{A=0}\bigl{(}f_{L}\otimes
1\bigr{)}=\mathrm{op}^{A=0}(f_{L})\otimes\mathds{1}:\hat{g}\mapsto\mathrm{op}^{A=0}(f_{L})\,\hat{g}$

when we view them as operators acting from the left.222As González-Pérez,
Junge and Parcet have noted in [GPJP21, Remark 2.5], we could have
equivalently considered operators acting from the right. The two cases are
connected via the adjoint operation. The calculus for magnetic
pseudodifferential super operators we have developed in [LL22] applies to much
more general super operators, which need _not_ be “product super operators” of
the form

$\displaystyle\mathrm{Op}^{A}\bigl{(}f_{L}\otimes
f_{R}\bigr{)}:\hat{g}^{A}\mapsto\mathrm{op}^{A}(f_{L})\,\hat{g}^{A}\,\mathrm{op}^{A}(f_{R})$

and incorporate magnetic fields in a covariant fashion. Still, this simplified
case can give us guidance on what results we should expect to hold true. The
proof of $L^{p}$-boundedness of pseudodifferential operators on
$\mathbb{R}^{d}$ for $1<p<\infty$ can be found in [Fef73, Ste93]. González-
Pérez, Junge and Parcet extended this $\mathfrak{L}^{p}$-boundedness result to
the setting of quantum Euclidean spaces [GPJP21].

Importantly, these authors also have partial results for the cases $p=1$ and
$p=\infty$. In both cases, either initial or target spaces is not a
$\mathfrak{L}^{p}$ space. Instead, González-Pérez et al. obtain boundedness
(cf. [GPJP21, Theorem 2.18]) if we view these super operators as operators
acting between

$\displaystyle\mathrm{Op}^{A=0}\bigl{(}f_{L}\otimes
1\bigr{)}:\text{H}^{1}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\rightarrow\mathfrak{L}^{1}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)},$
(1.3) $\displaystyle\mathrm{Op}^{A=0}\bigl{(}f_{L}\otimes
1\bigr{)}:\mathcal{B}(\mathcal{H})\longrightarrow\text{BMO}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)},$
(1.4)

where $\text{H}^{1}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}$ and
$\text{BMO}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}$ are the non-commutative
versions of the Hardy space and the space of functions of bounded mean
oscillation.

This might suggest that at least in the context of pseudodifferential super
operators it is more natural to look at generalizations of (1.3) and (1.4) to
more general functions $F\neq f_{L}\otimes 1$ such as Hörmander symbols from
Definitions 2.4 and 2.5.

### Outline

Apart from the introduction, this paper consists of four sections: in Section
2 we detail our setting and give the necessary definitions and assumptions.
Then in Section 3 we characterize certain classes of (super) operators in
terms of their matrix elements. The only time we will need oscillatory
integral techniques is in Section 4, where we characterize magnetic
pseudodifferential super operators in terms of their matrix elements. Lastly,
proofs of the main result, Theorem 1.1, and its more general statements,
Theorem 5.3 as well as Corollaries 5.6 and 5.8, are given in Section 5.

### Data availability

No datasets were generated or analyzed, because this work is based on a purely
mathematical approach. Therefore, data sharing is not applicable to this
article.

### Acknowledgements

G. L. was supported by the Max Planck Institute for Mathematics, Bonn, the FWO
Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, and
the Methusalem programme of the Ghent University Special Research Fund (BOF)
(Grant number 01M01021).

M. L. acknowledges the JSPS for the support of this project through a Kiban C
grant (No. 20K03761). He would also like to thank Max Planck Institute for
Mathematics for its kind hospitality.

Both authors would like to express their gratitude towards Horia Cornean and
Radu Purice for very helpful remarks and stimulating discussions.

##  2 Setting and fundamental definitions

In this section we will fix some notation, give some basic definitions and
recount basic facts.

###  2.1 The relevant Parseval frame

Let $\mathcal{H}$ be a separable, infinite-dimensional Hilbert space endowed
with an inner product $\left\langle\,\cdot\,,\,\cdot\,\right\rangle$. A frame
generalizes the notion of a basis; it is essentially an “overcomplete basis”
of $\mathcal{H}$, which still allows for norm estimates. More precisely, a
sequence $\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ in $\mathcal{H}$
indexed by a countable set $\Gamma$ is a frame if and only if there exist two
constants $0<C_{-}\leq C_{+}<\infty$ such that

$\displaystyle
C_{-}\,\lVert\psi\rVert^{2}\leq\sum_{\alpha\in\Gamma}\bigl{\lvert}\left\langle\mathcal{G}_{\alpha},\psi\right\rangle\bigr{\rvert}^{2}\leq
C_{+}\,\lVert\psi\rVert^{2}$ (2.1)

holds for all $\psi\in\mathcal{H}$. A frame is called tight if and only if we
may choose $C_{-}=C_{+}$; a tight frame for which $C_{+}=1=C_{-}$ is called a
normalized tight or _Parseval frame_. Parseval frames act much like
orthonormal basis in that e. g. we may expand any vector

$\displaystyle\psi=\sum_{\alpha\in\Gamma}\bigl{\langle}\mathcal{G}_{\alpha},\psi\bigr{\rangle}\,\mathcal{G}_{\alpha}$

in terms of the elements of the Parseval frame. In our convention, the first
argument of the scalar product is antilinear and the second one is linear.

For Parseval frames $\\{G_{\alpha}\\}_{\alpha\in\Gamma}$ we can identify each
vector $\varphi\in\mathcal{H}$ with its collection of coefficients

$\displaystyle(\varphi_{\alpha})_{\alpha\in\Gamma}:=\bigl{(}\langle\mathcal{G}_{\alpha},\varphi\rangle\bigr{)}_{\alpha\in\Gamma}\in\ell^{2}(\Gamma).$

The Parseval properties can be rephrased as saying that the map

$\displaystyle\mathfrak{u}_{2}:\mathcal{H}\longrightarrow\ell^{2}(\Gamma),\quad\varphi\mapsto(\varphi_{\alpha})_{\alpha\in\Gamma}$

is a linear isometry between Banach spaces; when
$\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is only a Parseval frame, but
not an orthonormal basis, $\mathfrak{u}_{2}$ fails to be onto, though.
Consequently, for some arguments, we may need to restrict $\mathfrak{u}_{2}$
to its range

$\displaystyle\ell^{2}_{\mathcal{H}}(\Gamma):=\mathfrak{u}_{2}(\mathcal{H})\subseteq\ell^{2}(\Gamma).$

Frequently, frames are generated from a single vector $\chi\in\mathcal{H}$,
and we may set

$\displaystyle\mathcal{G}_{\alpha}:=\pi(\alpha)\,\chi,$

where $\Gamma\cong\mathbb{Z}^{k}$ for some $k\in\mathbb{N}$ and
$\pi:\Gamma\longrightarrow\mathcal{U}(\mathcal{H})$ is a unitary-operator-
valued map. Such frames are called _Gabor frames_. For an introduction to the
theory of frames, we refer to [Chr03].

The idea to use Gabor frames to study pseudodifferential operators goes back
to the late 1990s and early 2000s with the pioneering works of Gröchenig,
Cordero, Heil and co-workers [HG99, CG03, CG05]. More recently, it has been
applied to _magnetic_ pseudodifferential operators by Cornean, Helffer and
Purice [CHP18, CHP22], and in spirit and substance, our work borrows its main
ideas from them and applies them in new ways.

We will use a Gabor frame, which is simultaneously a Parseval frame. It starts
with a function
$\chi\in\mathcal{C}^{\infty}_{\mathrm{c}}\bigl{(}\mathbb{R}^{d},[0,\infty)\bigr{)}$
and a lattice $\Gamma\cong\mathbb{Z}^{d}$. We require that $\chi$ satisfies

  1. (a)

$\mathrm{supp}\,\chi\subseteq(-1,+1)^{d}$ and

  2. (b)

$\sum_{\gamma\in\Gamma}\chi(x-\gamma)^{2}=1$ holds for all
$x\in\mathbb{R}^{d}$.

Furthermore, we will need to introduce the _dual lattice_

$\displaystyle\Gamma^{*}:=\Bigl{\\{}\gamma^{*}\in\mathbb{R}^{d}\;\;\big{|}\;\;\gamma^{*}\cdot\gamma\in
2\pi\mathbb{Z}\;\forall\gamma\in\Gamma\Bigr{\\}},$

which is again isomorphic to $\mathbb{Z}^{d}$. Lastly, we need to introduce a
magnetic field $B=\mathrm{d}A$ and a vector potential $A$ representing it. One
may think of magnetic fields as smooth, closed two-forms
$B=\sum_{j,k=1}^{d}B_{jk}\,\mathrm{d}x_{j}\wedge\mathrm{d}x_{k}$ on
$\mathbb{R}^{d}$; we shall always tacitly identify $B$ with the matrix-valued
function $(B_{jk})_{1\leq j,k\leq d}$. A magnetic vector _potential_
$A=\sum_{j=1}^{d}A_{j}\,\mathrm{d}x_{j}$ is said to represent a magnetic field
$B$ if and only if $B=\mathrm{d}A$; whenever convenient we will think of the
one-form $A$ as the vector-valued function $(A_{1},\ldots,A_{d})$. Throughout
this article, we will make the following assumptions.

######  Assumption 2.1 (Magnetic fields and vector potentials).

  1. (a)

The components $B_{jk},1\leq j,k,\leq d$, of the magnetic field $B$ are of
class $\mathcal{C}^{\infty}_{\mathrm{b}}$, i. e. bounded, smooth and with
bounded derivatives to any order.

  2. (b)

A vector potential
$A\in\mathcal{C}^{\infty}_{\mathrm{pol}}(\mathbb{R}^{d},\mathbb{R}^{d})$
representing such magnetic fields $\mathrm{d}A=B$ has polynomially bounded,
smooth components whose derivatives are all polynomially bounded as well.

The exponential of the phase

$\displaystyle\Lambda^{A}(x,y):=\mathrm{e}^{-\mathrm{i}\int_{[x,y]}A}$

obtained by integrating the vector potential along the line segment $[x,y]$
will enter into the definition of our Gabor frame,

$\displaystyle\mathcal{G}^{A}_{\alpha,\alpha^{*}}(x):=(2\pi)^{-\nicefrac{{d}}{{2}}}\Lambda^{A}(x,\alpha)\,\mathrm{e}^{+\frac{\mathrm{i}}{2\pi}\alpha^{*}\cdot(x-\alpha)}\,\chi(x-\alpha)\in\mathcal{S}(\mathbb{R}^{d}).$
(2.2)

The presence of the factor $\nicefrac{{1}}{{2\pi}}$ stems from the embedding
$(-1,+1)^{d}\subset(-\pi,+\pi)^{d}$. Functions with compact support inside the
cube $(-1,+1)^{d}$ are then replaced by their $2\pi$-periodization (cf. the
explanations surrounding equations (2.2) and (2.3) in [CHP22]).

We will restate [CHP22, Proposition 2.2] for convenience:

######  Proposition 2.2.

  1. (1)

$\bigl{\\{}\mathcal{G}^{A}_{\alpha,\alpha^{*}}\bigr{\\}}_{(\alpha,\alpha^{*})\in\Gamma\times\Gamma^{*}}$
is a Parseval frame.

  2. (2)

We may expand any $\psi\in L^{2}(\mathbb{R}^{d})$ in terms of the Gabor frame
as

$\displaystyle\psi$
$\displaystyle=\sum_{(\alpha,\alpha^{*})\in\Gamma\times\Gamma^{*}}\bigl{\langle}\mathcal{G}^{A}_{\alpha,\alpha^{*}},\psi\bigr{\rangle}\,\mathcal{G}^{A}_{\alpha,\alpha^{*}},$

where the above series converges in the topology of $L^{2}(\mathbb{R}^{d})$.

Item (2) will allow us to identify operators $\hat{f}$ on
$L^{2}(\mathbb{R}^{d})$ with their collection of matrix elements
$\hat{f}^{A}_{\alpha,\alpha^{*},\beta,\beta^{*}}:=\bigl{\langle}\mathcal{G}^{A}_{\alpha,\alpha^{*}},\hat{f}\,\mathcal{G}^{A}_{\beta,\beta^{*}}\bigr{\rangle}$.

###  2.2 Magnetic Weyl calculus

In this subsection, we will review the construction and main properties of
magnetic Weyl calculus. For a more comprehensive account, we refer the readers
to e. g. [MP04, IMP07, Lei11].

In what follows, we denote points on the phase space
$\Xi:=T^{*}\mathbb{R}^{d}\cong\mathbb{R}^{d}\times\mathbb{R}^{d}$ by
$X=(x,\xi)$, $Y=(y,\eta)$ and $Z=(z,\zeta)$ with position variables
$x,y,z\in\mathbb{R}^{d}$ and momentum variables
$\xi,\eta,\zeta\in\mathbb{R}^{d}$.

The basic building blocks of magnetic Weyl calculus are position operators
$Q=(Q_{1},\ldots,Q_{d})$ and kinetic momentum operators
$P^{A}=(P_{1}^{A},\ldots,P_{d}^{A})$. These are self-adjoint unbounded
operators on $L^{2}(\mathbb{R}^{d})$ subject to the commutation relations

$\displaystyle\mathrm{i}\,[Q_{j},Q_{k}]=0,\qquad\mathrm{i}\,[P_{j}^{A},P_{k}^{A}]=\varepsilon\,\lambda\,B_{jk}(Q),\qquad\mathrm{i}\,[P_{j}^{A},Q_{k}]=\varepsilon\,\delta_{jk},$
$\displaystyle 1\leq j,k\leq d.$ (2.3)

Here $\varepsilon$ is a semiclassical parameter and $\lambda\leq 1$ is another
parameter quantifying the coupling of the electric charge to the magnetic
field. Both of them play an important role in the study of asymptotic
expansions and semiclassical limits [Lei10, FL13].

######  Remark 2.3.

For the sake of brevity, we will set $\lambda=1=\varepsilon$. We wish to
emphasize that all of our subsequent results hold when the small parameters
are restored.

We can put the commutation relations (2.3) into the framework of bounded
operators by using the magnetic Weyl system defined by

$\displaystyle
w^{A}(X):=\mathrm{e}^{-\mathrm{i}\sigma(X,(Q,P^{A}))}=\mathrm{e}^{+\mathrm{i}(x\cdot
P^{A}-\xi\cdot Q)},$ $\displaystyle X\in\Xi,$

where we have used the symplectic form $\sigma(X,Y):=\xi\cdot y-x\cdot\eta$ on
$\Xi$. The symplectic form also appears as the phase factor in the _symplectic
Fourier transform_

$\displaystyle(\mathcal{F}_{\sigma}f)(X):=\frac{1}{(2\pi)^{d}}\int_{\Xi}\mathrm{d}Y\,\mathrm{e}^{+\mathrm{i}\sigma(X,Y)}\,f(Y)$

that is initially defined for Schwartz functions, but its definition extends
to tempered distributions. Compared to the standard Fourier transform, it has
the added benefit of being its own inverse,
$\mathcal{F}_{\sigma}^{2}=\mathds{1}$.

The magnetic Weyl pseudodifferential operator associated with $f$ is defined
by

$\displaystyle\mathrm{op}^{A}(f):=\frac{1}{(2\pi)^{d}}\int_{\Xi}\mathrm{d}X\,(\mathcal{F}_{\sigma}f)(X)\,w^{A}(X).$
(2.4)

The magnetic Weyl quantization map $f\mapsto\mathrm{op}^{A}(f)$ yields a
continuous linear operator $\mathrm{op}^{A}$ from $\mathcal{S}(\Xi)$ to
$\mathcal{L}\bigl{(}\mathcal{S}^{\prime}(\mathbb{R}^{d})\,,\,\mathcal{S}(\mathbb{R}^{d})\bigr{)}$;
it extends to a topological vector space isomorphism from
$\mathcal{S}^{\prime}(\Xi)$ to
$\mathcal{L}\bigl{(}\mathcal{S}(\mathbb{R}^{d})\,,\,\mathcal{S}^{\prime}(\mathbb{R}^{d})\bigr{)}$.
In particular, as a Hörmander symbol space $S_{\rho,\delta}^{m}(\Xi)$ of order
$m\in\mathbb{R}$, defined below, is included in $\mathcal{S}^{\prime}(\Xi)$,
and each symbol $f\in S_{\rho,\delta}^{m}(\Xi)$ yields a continuous linear
operator from $\mathcal{S}(\Xi)$ to $\mathcal{S}^{\prime}(\Xi)$. In addition,
the magnetic Weyl quantization map $f\mapsto\mathrm{op}^{A}(f)$ also gives
rise to a unitary map from $L^{2}(\Xi)$ to the space of Hilbert-Schmidt
operators
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.

######  Definition 2.4 (Hörmander symbol classes).

Let $0\leq\delta\leq\rho\leq 1$, $\delta<1$ and $m\in\mathbb{R}$. The space
$S_{\rho,\delta}^{m}(\Xi)$ consists of smooth functions
$f:\Xi\longrightarrow\mathbb{C}$ such that, for all
$a,\alpha\in\mathbb{N}_{0}^{d}$ there exists $C_{a\alpha}>0$ such that

$\displaystyle\bigl{\lvert}\partial_{x}^{a}\partial_{\xi}^{\alpha}f(X)\bigr{\rvert}\leq
C_{a\alpha}\,\langle\xi\rangle^{m-\left\lvert\alpha\right\rvert\rho+\left\lvert
a\right\rvert\delta}$ $\displaystyle\forall X\in\Xi.$ (2.5)

The smallest constants $C_{a\alpha}$ satisfying the estimates (2.5) are the
seminorms

$\displaystyle\left\lVert
f\right\rVert_{m,a\alpha}:=\sup_{X\in\Xi}\Bigl{(}\langle\xi\rangle^{-m+\left\lvert\alpha\right\rvert\rho-\left\lvert
a\right\rvert\delta}\bigl{\lvert}\partial_{x}^{a}\partial_{\xi}^{\alpha}f(X)\bigr{\rvert}\Bigr{)},$

where $\langle\xi\rangle:=\sqrt{1+\lvert\xi|^{2}}$ is the Japanese bracket. We
endow $S_{\rho,\delta}^{m}(\Xi)$ with the Fréchet space topology generated by
the seminorms $\left\lVert\cdot\right\rVert_{m,a\alpha}$, where $a$ and
$\alpha$ range over all multi indices in $\mathbb{N}_{0}^{d}$.

The magnetic Weyl product $f\star^{B}g$ of two symbols or distributions $f$
and $g$ is defined via the composition of the pseudodifferential operators
associated with them,

$\displaystyle\mathrm{op}^{A}\bigl{(}f\star^{B}g\bigr{)}:=\mathrm{op}^{A}(f)\,\mathrm{op}^{A}(g).$

There exist multiple equivalent explicit formulas for $\star^{B}$, although we
will not need them in this work.

This product pulls back the operator product to the level of functions or
distributions on phase space. It gives rise to a continuous bilinear map
between Hörmander classes (cf. e. g. [IMP07, Theorem 2.6]),

$\displaystyle\star^{B}:S_{\rho,\delta}^{m_{1}}(\Xi)\times
S_{\rho,\delta}^{m_{2}}(\Xi)\longrightarrow
S_{\rho,\delta}^{m_{1}+m_{2}}(\Xi).$

Another standard result of magnetic Weyl calculus is the $L^{2}$-boundedness:
magnetic Weyl quantization $\mathrm{op}^{A}$ gives rise to a continuous linear
map from $S_{0,0}^{0}(\Xi)$ to
$\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$. This magnetic version of
the Calderón-Vaillancourt theorem was first proved by Iftimie, Mantoiu and
Purice in [IMP07]. An alternative approach using the magnetic Gabor frame
(2.2), the approach which we will also utilize in this article to prove the
boundedness of magnetic pseudodifferential _super_ operators, can be found in
[CHP22].

###  2.3 Magnetic pseudodifferential super operator calculus

The main motivation for introducing the calculus of magnetic Weyl
pseudodifferential super operators in [LL22] was to study generalizations of
super operators of the form

$\displaystyle\hat{g}^{A}\mapsto\mathrm{op}^{A}(f_{L})\,\hat{g}^{A}\,\mathrm{op}^{A}(f_{R}),$

where $\hat{g}^{A}\in\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$ is any
bounded operator on $L^{2}(\mathbb{R}^{d})$ and $\mathrm{op}^{A}(f_{L,R})$ are
the magnetic Weyl quantizations of symbols $f_{L,R}$. To this end we introduce
the magnetic super Weyl system

$\displaystyle
W^{A}(\mathbf{X})\,\hat{g}^{A}=w^{A}(X_{L})\,\hat{g}^{A}\,w^{A}(X_{R}).$ (2.6)

Here we have denoted a point in the doubled phase space
$\Xi^{2}:=\Xi\times\Xi$ by $\mathbf{X}=(X_{L},X_{R})$, $X_{L},X_{R}\in\Xi$.

We also introduce the symplectic Fourier transform on $\Xi^{2}$,

$\displaystyle(\mathcal{F}_{\Sigma}F)(\mathbf{X})=\frac{1}{(2\pi)^{2d}}\int_{\Xi^{2}}\mathrm{d}\mathbf{Y}\,\mathrm{e}^{+\mathrm{i}\Sigma(\mathbf{X},\mathbf{Y})}\,F(\mathbf{Y}),$

with respect to the symplectic form
$\Sigma(\mathbf{X},\mathbf{Y}):=\sigma(X_{L},Y_{L})+\sigma(X_{R},Y_{R})$,
$\mathbf{X},\mathbf{Y}\in\Xi^{2}$, on doubled phase space $\Xi^{2}$. As
before, $\mathcal{F}_{\Sigma}^{2}=\mathds{1}$ holds.

The symplectic Fourier transform $\mathcal{F}_{\Sigma}$ and the magnetic super
Weyl system (2.6) enable us to introduce the notion of magnetic _super_ Weyl
quantization

$\displaystyle\mathrm{Op}^{A}(F)\,\hat{g}^{A}:=\frac{1}{(2\pi)^{2d}}\int_{\Xi^{2}}\mathrm{d}\mathbf{X}\,(\mathcal{F}_{\Sigma}F)(\mathbf{X})\,W^{A}(\mathbf{X})\,\hat{g}^{A},$
(2.7)

whose definition is completely analogous to (2.4). For a symbol of the form
$F(\mathbf{X})=f_{L}(X_{L})\,f_{R}(X_{R})$ this prescription yields the super
operator

$\displaystyle\mathrm{Op}^{A}(F)\,\hat{g}^{A}=\mathrm{op}^{A}(f_{L})\,\hat{g}^{A}\,\mathrm{op}^{A}(f_{R}),$
$\displaystyle\hat{g}^{A}\in\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}.$

With abuse of notation we shall denote this super operator by
$\mathrm{op}^{A}(f_{L})\otimes\mathrm{op}^{A}(f_{R})=\mathrm{Op}^{A}\bigl{(}f_{L}\otimes
f_{R}\bigr{)}$.

One of the fundamental results concerning the magnetic Weyl pseudodifferential
super operators is that they map pseudodifferential operators to
pseudodifferential operators. More precisely, if $F\in\mathcal{S}(\Xi^{2})$
and $g\in\mathcal{S}(\Xi)$, there is a unique Schwartz function
$F\bullet^{B}g\in\mathcal{S}(\Xi)$ such that

$\displaystyle\mathrm{op}^{A}\bigl{(}F\bullet^{B}g\bigr{)}:=\mathrm{Op}^{A}(F)\,\mathrm{op}^{A}(g).$

The function $F\bullet^{B}g$ is called the magnetic Weyl semi-super product of
$F$ and $g$, and the explicit formula for $F\bullet^{B}g$ can be found in
[LL22, Section III]. Furthermore, the map $(F,g)\mapsto F\bullet^{B}g$ gives
rise to a continuous bilinear map from
$\mathcal{S}(\Xi^{2})\times\mathcal{S}(\Xi)$ to $\mathcal{S}(\Xi)$ (see [LL22,
Proposition IV.9]).

Given $F,G\in\mathcal{S}(\Xi^{2})$, the magnetic super Weyl product
$F\sharp^{B}G$ is the Schwartz class function satisfying

$\displaystyle\mathrm{Op}^{A}\bigl{(}F\sharp^{B}G\bigr{)}:=\mathrm{Op}^{A}(F)\,\mathrm{Op}^{A}(G).$

The super Weyl product $(F,G)\mapsto F\sharp^{B}G$ yields a continuous
bilinear map from $\mathcal{S}(\Xi^{2})\times\mathcal{S}(\Xi^{2})$ to
$\mathcal{S}(\Xi^{2})$ (cf. [LL22, Proposition IV.14]).

All the results concerning the calculus of magnetic Weyl pseudodifferential
super operators described above can be extended to Hörmander symbol classes.
In addition to the usual classes of Hörmander symbols, we introduce another
class of Hörmander symbols that allow one to study the behavior in left and
right momenta separately.

######  Definition 2.5 (Double Hörmander symbol classes).

Let $0\leq\delta\leq\rho\leq 1$, $\delta<1$ and $m_{L},m_{R}\in\mathbb{R}$.
The space $S_{\rho,\delta}^{m_{L},m_{R}}(\Xi^{2})$ consists of smooth
functions $F:\Xi^{2}\longrightarrow\mathbb{C}$ such that, for all
$a_{L},a_{R},\alpha_{L},\alpha_{R}\in\mathbb{N}_{0}^{d}$ there exists
$C_{a_{L}a_{R}\alpha_{L}\alpha_{R}}>0$ such that

$\displaystyle\bigl{\lvert}\partial_{x_{L}}^{a_{L}}\partial_{\xi_{L}}^{\alpha_{L}}\partial_{x_{R}}^{a_{R}}\partial_{\xi_{R}}^{\alpha_{R}}F(\mathbf{X})\bigr{\rvert}\leq
C_{a_{L}a_{R}\alpha_{L}\alpha_{R}}\,\langle\xi_{L}\rangle^{m_{L}-\left\lvert\alpha_{L}\right\rvert\rho+\left\lvert
a_{L}\right\rvert\delta}\,\langle\xi_{R}\rangle^{m_{R}-\left\lvert\alpha_{R}\right\rvert\rho+\left\lvert
a_{R}\right\rvert\delta}$ $\displaystyle\forall\mathbf{X}\in\Xi^{2}.$ (2.8)

We endow $S_{\rho,\delta}^{m_{L},m_{R}}(\Xi^{2})$ with the Fréchet space
topology generated by the seminorms

$\displaystyle\left\lVert
F\right\rVert_{m_{L}m_{R},a_{L}a_{R}\alpha_{L}\alpha_{R}}:=\sup_{\mathbf{X}\in\Xi^{2}}\Big{(}\langle\xi_{L}\rangle^{-m_{L}+\left\lvert\alpha_{L}\right\rvert\rho-\left\lvert
a_{L}\right\rvert\delta}\,\langle\xi_{R}\rangle^{-m_{R}+\left\lvert\alpha_{R}\right\rvert\rho-\left\lvert
a_{R}\right\rvert\delta}\big{|}\partial_{x_{L}}^{a_{L}}\partial_{\xi_{L}}^{\alpha_{L}}\partial_{x_{R}}^{a_{R}}\partial_{\xi_{R}}^{\alpha_{R}}F(\mathbf{X})\big{|}\Big{)}.$

Unfortunately, there is no simple nesting relation between the Hörmander
symbol classes $S_{\rho,\delta}^{m_{L},m_{R}}(\Xi^{2})$ and
$S_{\rho,\delta}^{m}(\Xi^{2})$, $m_{L},m_{R},m\in\mathbb{R}$. The only
exception is the inclusion

$\displaystyle S^{m_{L},m_{R}}_{\rho,0}(\Xi^{2})\subseteq S^{\lvert
m_{L}|+\lvert m_{R}|}_{0,0}(\Xi^{2})$ (2.9)

that we will use in the proof of Corollary 5.6. We refer to [LL22, Remark
VI.3] for more details on the relation between two Hörmander symbol classes.

We recall the following results on the semi-super product and super Weyl
product of Hörmander symbols:

######  Proposition 2.6 ([LL22, Proposition VI.4]).

Let $0\leq\rho\leq 1$. Then the following holds.

  1. (1)

Let $m,m^{\prime}\in\mathbb{R}$. Then the map $(F,g)\mapsto F\bullet^{B}g$
gives rise to a continuous bilinear map

$\displaystyle\bullet^{B}:S_{\rho,0}^{m}(\Xi^{2})\times
S_{\rho,0}^{m^{\prime}}(\Xi)\longrightarrow S_{\rho,0}^{m+m^{\prime}}(\Xi).$

  2. (2)

Let $m,m_{L},m_{R}\in\mathbb{R}$. Then the map $(F,g)\mapsto F\bullet^{B}g$
gives rise to a continuous bilinear map

$\displaystyle\bullet^{B}:S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})\times
S_{\rho,0}^{m}(\Xi)\longrightarrow S_{\rho,0}^{m_{L}+m_{R}+m}(\Xi).$

######  Proposition 2.7 ([LL22, Proposition VI.5]).

Let $0\leq\rho\leq 1$. Then the following holds.

  1. (1)

Let $m,m^{\prime}\in\mathbb{R}$. Then the map $(F,G)\mapsto F\sharp^{B}G$
gives rise to a continuous bilinear map

$\displaystyle\sharp^{B}:S_{\rho,0}^{m}(\Xi^{2})\times
S_{\rho,0}^{m^{\prime}}(\Xi^{2})\longrightarrow
S_{\rho,0}^{m+m^{\prime}}(\Xi^{2}).$

  2. (2)

Let $m_{L},m_{L}^{\prime},m_{R},m_{R}^{\prime}\in\mathbb{R}$. Then the map
$(F,G)\mapsto F\sharp^{B}G$ gives rise to a continuous bilinear map

$\displaystyle\sharp^{B}:S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})\times
S_{\rho,0}^{m_{L}^{\prime},m_{R}^{\prime}}(\Xi^{2})\longrightarrow
S_{\rho,0}^{m_{L}+m_{L}^{\prime},m_{R}+m_{R}^{\prime}}(\Xi^{2}).$

We can also derive the following result from [LL22]:

######  Proposition 2.8.

Let $0\leq\rho\leq 1$. Then the following holds:

  1. (1)

Let $F\in S_{\rho,0}^{m}(\Xi^{2})$, $m\in\mathbb{R}$ and
$g\in\mathcal{S}(\Xi)$. Then $F\bullet^{B}g\in\mathcal{S}(\Xi)$.

  2. (2)

Let $F\in S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})$, $m_{L},m_{R}\in\mathbb{R}$ and
$g\in\mathcal{S}(\Xi)$. Then $F\bullet^{B}g\in\mathcal{S}(\Xi)$.

######  Proof.

Suppose that $F\in S_{\rho,0}^{m}(\Xi^{2})$ and $g\in\mathcal{S}(\Xi)$. Then
also the transpose $F^{\mathrm{t}}(X_{L},X_{R}):=F(X_{R},X_{L})$,
$X_{L},X_{R}\in\Xi$, of the symbol, obtained by swapping the arguments, lies
in the same Hörmander class $S_{\rho,0}^{m}(\Xi^{2})\ni F^{\mathrm{t}}$. By
[LL22, Lemma V.5] we know that there is an inclusion from
$S_{\rho,0}^{m}(\Xi^{2})$ into the magnetic semi-super Moyal space
$\mathfrak{m}^{B}(\Xi^{2})$ defined by

$\displaystyle\mathfrak{m}^{B}(\Xi^{2}):=\Bigl{\\{}G\in\mathcal{S}^{\prime}(\Xi^{2})\;\;\big{|}\;\;\text{$\mathcal{S}(\Xi)\ni
h\mapsto G^{\mathrm{t}}\bullet^{B}h\in\mathcal{S}(\Xi)$ is continuous and
linear}\Bigr{\\}},$

where we have extended the transpose to tempered distributions by duality (cf.
the second displayed equation in [LL22, Section V.B]). Therefore, since
$F^{\mathrm{t}}\in S_{\rho,0}^{m}(\Xi^{2})\subset\mathfrak{m}^{B}(\Xi^{2})$
and $g\in\mathcal{S}(\Xi)$ we can deduce that
$(F^{\mathrm{t}})^{\mathrm{t}}\bullet^{B}g=F\bullet^{B}g\in\mathcal{S}(\Xi)$.
This proves the first assertion.

Likewise, if $F\in S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})$ is a double Hörmander
symbol, then we can combine $F^{\mathrm{t}}\in
S_{\rho,0}^{m_{R},m_{L}}(\Xi^{2})$ with [LL22, Lemma V.5],
$S_{\rho,0}^{m_{R},m_{L}}(\Xi^{2})\subset\mathfrak{m}^{B}(\Xi^{2})$. Hence,
the second assertion can be proved along similar lines as that of the proof of
the first one. This completes the proof. □

Another notable result in [LL22] is the boundedness of magnetic
pseudodifferential super operators associated with Schwartz class symbols.

######  Proposition 2.9 ([LL22, Lemma IV.6]).

Let $F\in\mathcal{S}(\Xi^{2})$. Then the following holds.

  1. (1)

$\mathrm{Op}^{A}(F)$ gives rise to a bounded linear operator from
$\mathcal{B}\big{(}L^{2}(\mathbb{R}^{d})\big{)}$ to itself.

  2. (2)

For every $p\geq 1$, $\mathrm{Op}^{A}(F)$ gives rise to a bounded linear
operator from
$\mathfrak{L}^{p}\big{(}\mathcal{B}\big{(}L^{2}(\mathbb{R}^{d})\big{)}\big{)}$
to itself.

As mentioned in the introduction the main goal of this article is to extend
the boundedness result on
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
to a suitable class of Hörmander symbols. We do this by representing linear
(super) operators as infinite matrices via tight Gabor frames described in
[CHP22].

##  3 Abstract setting: characterization of (super) operators via their matrix
elements

Before we can characterize super operators $\hat{F}^{A}$ through their matrix
elements, we need to understand what elements of the $p$-Schatten classes
$\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
look like — at least for Hilbert-Schmidt operators ($p=2$). We will also
explain our convention of matrix indices for super operators.

While the results in this section are all well-known for orthonormal bases,
their extension to various types of frames is not. During the preparation of
our manuscript, we came across a very nicely written work by Bingyang, Khoi
and Zhu [BKZ15], which characterized operators from $p$-Schatten classes in
terms of $\ell^{p}$ summability. Importantly, they showed one needs to
distinguish between the case $0<p\leq 2$ and $2\leq p<\infty$.

To clean up the presentation, we will discuss the matter in more generality:

######  Assumption 3.1 (General setting).

  1. (a)

The Hilbert space $\mathcal{H}$ is infinite-dimensional and separable.

  2. (b)

$\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is a Parseval frame indexed by
some countable set $\Gamma\cong\mathbb{Z}^{k}$ for some $k\in\mathbb{N}$.

Indeed, this allows us to omit A in (2.2) and replace pairs of multi indices
$(\alpha,\alpha^{*})$ with a single one. We also point out that the assumption
$\Gamma\cong\mathbb{Z}^{k}$ could be relaxed as any countably infinite set can
be mapped bijectively onto $\mathbb{N}$. Our assumption that $\mathcal{H}$ is
infinite-dimensional excludes the uninteresting case
$\mathcal{H}\cong\mathbb{C}^{n}$ where all $p$-Schatten classes

$\displaystyle\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}=\mathcal{B}(\mathcal{H})\cong\mathrm{Mat}_{\mathbb{C}}(n),$
$\displaystyle 1\leq p<\infty,$

coincide with the bounded operators (essentially $n\times n$ matrices). What
matters is that these assumptions are satisfied for case at hand, namely
$\mathcal{H}=L^{2}(\mathbb{R}^{d})$ and the Parseval frame from Section 2.

###  3.1 Matrix representation of operators

The idea is to write any suitable operator

$\displaystyle\hat{f}$
$\displaystyle=\sum_{\alpha,\beta\in\Gamma}\hat{f}_{\alpha,\beta}\,|\mathcal{G}_{\alpha}\rangle\langle\mathcal{G}_{\beta}|\in\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)},$
$\displaystyle 1\leq p<\infty,$ (3.1)

as a sum of rank-$1$ operators, where the sum on the right-hand side has to
converge in the relevant Banach space. Defining the matrix coefficients as

$\displaystyle\hat{f}_{\alpha,\beta}:=\bigl{\langle}\mathcal{G}_{\alpha},\hat{f}\,\mathcal{G}_{\beta}\bigr{\rangle}$
(3.2)

allows us to write operators as the sum over rank-$1$ operators,

$\displaystyle\hat{f}\varphi$
$\displaystyle=\sum_{\alpha\in\Gamma}\bigl{\langle}\mathcal{G}_{\alpha},\hat{f}\varphi\bigr{\rangle}\,\mathcal{G}_{\alpha}=\sum_{\alpha,\beta\in\Gamma}\bigl{\langle}\mathcal{G}_{\alpha},\hat{f}\mathcal{G}_{\beta}\bigr{\rangle}\,\langle\mathcal{G}_{\beta},\varphi\rangle\,\mathcal{G}_{\alpha}$
$\displaystyle=\sum_{\alpha,\beta\in\Gamma}\hat{f}_{\alpha,\beta}\,|\mathcal{G}_{\alpha}\rangle\langle\mathcal{G}_{\beta}|\varphi.$

####  3.1.1 Hilbert-Schmidt operators

In this subsection we will characterize the matrix elements of Hilbert-Schmidt
operators. They are a special case, because we can make use of the Hilbert
space structure and exploit various Hilbert space isometries. In principle,
our result is covered by combining Theorems A and B in [BKZ15]. However, we
feel the proof is so short and elegant that we give here it nonetheless.
Moreover, our proof has the advantage of giving us the equality of norms in a
straightforward fashion.

######  Proposition 3.2.

  1. (1)

There exists an isometry that maps the space of Hilbert-Schmidt operators

$\displaystyle\mathfrak{U}_{2}:\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\longrightarrow\ell^{2}(\Gamma^{2})$

into the space of infinite matrix elements whose entries are square summable.
This is accomplished via the isometry

$\displaystyle\mathfrak{U}_{2}:\hat{f}\mapsto\bigl{(}\hat{f}_{\alpha,\beta}\bigr{)}_{\alpha,\beta\in\Gamma}.$

  2. (2)

Since $\mathfrak{U}_{2}$ is an isometry, we can compute the Hilbert-Schmidt
norm from the $\ell^{2}$ norm of its matrix elements,

$\displaystyle\bigl{\lVert}\hat{f}\bigr{\rVert}_{\mathcal{L}^{2}(\mathcal{B}(\mathcal{H}))}$
$\displaystyle=\bigl{\lVert}\mathfrak{U}_{2}\hat{f}\bigr{\rVert}_{\ell^{2}(\Gamma^{2})}=\bigl{\lVert}\bigl{(}\hat{f}_{\alpha,\beta}\bigr{)}_{\alpha,\beta\in\Gamma}\bigr{\rVert}_{\ell^{2}(\Gamma^{2})}.$

  3. (3)

When $\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is in addition an
orthonormal basis, then $\mathfrak{U}_{2}$ is a Hilbert space isomorphism.

######  Proof.

First of all, the Hilbert space of Hilbert-Schmidt operators

$\displaystyle\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\cong\mathcal{H}\otimes\mathcal{H}$

acting on a Hilbert space $\mathcal{H}$ can be identified with the (Hilbert
space) tensor product of $\mathcal{H}$ with itself; this follows from [RS72,
Theorem VI.23] and using the assumed separability of $\mathcal{H}$. In fact,
this identification is an isometry, and the Hilbert-Schmidt norm of $\hat{f}$
equals the $\mathcal{H}\otimes\mathcal{H}$ norm of its image.

As tensor products of Parseval frames are again Parseval frames (cf. [KA03,
Theorem 2.3]), the map

$\displaystyle\hat{f}\mapsto\bigl{(}\hat{f}_{\alpha,\beta}\bigr{)}_{\alpha,\beta\in\Gamma}$

onto the infinite matrix is also an isometry. Therefore, the composition
$\mathfrak{U}_{2}$ of these two isometries is also an isometry; hence, we have
also obtained the claimed equality of norms.

Lastly, concerning (3), when the Parseval frame is also an orthonormal basis,
$\mathfrak{U}_{2}$ is onto as any element of $\mathcal{H}\otimes\mathcal{H}$
defines a Hilbert-Schmidt operator and does so uniquely. □

######  Corollary 3.3.

  1. (1)

We can interpret the map

$\displaystyle\mathfrak{U}_{2}:\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\longrightarrow\ell^{2}_{\mathcal{H}}(\Gamma^{2})$

as an isometric Banach space isomorphism onto

$\displaystyle\ell^{2}_{\mathcal{H}}(\Gamma^{2}):=\ell^{2}_{\mathcal{H}}(\Gamma)\otimes\ell^{2}_{\mathcal{H}}(\Gamma).$

  2. (2)

If in addition $\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is an
orthonormal basis, then
$\ell^{2}_{\mathcal{H}}(\Gamma^{2})=\ell^{2}(\Gamma^{2})$ and
$\mathfrak{U}_{2}$ is a unitary.

###  3.2 Matrix representation of super operators

The definition of matrix elements of super operators $\hat{F}$ acting on
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}$ is conceptually
trivial, but the relation between the matrix representation and the matrix
elements is not as easy as that between (3.1) and (3.2).

Our convention is such that also here the super operator is (at least
formally) the sum

$\displaystyle\hat{F}$
$\displaystyle=\sum_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}\in\Gamma}\hat{F}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}\,|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}}|\otimes|\mathcal{G}_{\alpha_{R}}\rangle\langle\mathcal{G}_{\beta_{R}}|,$
(3.3)

where product operators $\hat{F}=\hat{f}_{L}\otimes\hat{f}_{R}$,
$\hat{f}_{L,R}\in\mathcal{B}(\mathcal{H})$, by definition act as

$\displaystyle\hat{f}_{L}\otimes\hat{f}_{R}(\hat{g}):=\hat{f}_{L}\,\hat{g}\,\hat{f}_{R}.$

Our use of $\otimes$ constitutes an abuse of notation, it does _not_ denote
the tensor product of two operators, which would act on
$\mathcal{H}\otimes\mathcal{H}$.

It turns out the correct definition of matrix elements is

$\displaystyle\hat{F}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}:=\mathrm{Tr}\,_{\mathcal{H}}\Bigl{(}|\mathcal{G}_{\beta_{R}}\rangle\langle\mathcal{G}_{\alpha_{L}}|\,\hat{F}\,\bigl{(}|\mathcal{G}_{\beta_{L}}\rangle\langle\mathcal{G}_{\alpha_{R}}|\bigr{)}\Bigr{)}.$
(3.4)

As long as
$\hat{F}\,|\mathcal{G}_{\beta_{L}}\rangle\langle\mathcal{G}_{\alpha_{R}}|$
yields at least a _bounded_ operator, the super operator’s matrix element
$\hat{F}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}$ is well-defined.

Expression (3.4) can be recast as a scalar product,

$\displaystyle\hat{F}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}$
$\displaystyle=\bigl{\langle}\mathcal{G}_{\alpha_{L}},\hat{F}\,\bigl{(}|\mathcal{G}_{\beta_{L}}\rangle\langle\mathcal{G}_{\alpha_{R}}|\bigr{)}\,\mathcal{G}_{\beta_{R}}\bigr{\rangle},$
(3.5)

which is advantageous for some of the computations. We can verify all of these
equations for product super operators $\hat{F}=\hat{f}_{L}\otimes\hat{f}_{R}$:

$\displaystyle\hat{F}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}$
$\displaystyle=\mathrm{Tr}\,\Bigl{(}|\mathcal{G}_{\beta_{R}}\rangle\langle\mathcal{G}_{\alpha_{L}}|\,\hat{f}_{L}\,|\mathcal{G}_{\beta_{L}}\rangle\langle\mathcal{G}_{\alpha_{R}}|\hat{f}_{R}\Bigr{)}$
$\displaystyle=\bigl{\langle}\mathcal{G}_{\alpha_{L}},\hat{f}_{L}\,|\mathcal{G}_{\beta_{L}}\rangle\langle\mathcal{G}_{\alpha_{R}}|\,\hat{f}_{R}\,\mathcal{G}_{\beta_{R}}\bigr{\rangle}$
$\displaystyle=\bigl{\langle}\mathcal{G}_{\alpha_{L}},\hat{f}_{L}\,\mathcal{G}_{\beta_{L}}\bigr{\rangle}\,\bigl{\langle}\mathcal{G}_{\alpha_{R}},\hat{f}_{R}\,\mathcal{G}_{\beta_{R}}\bigr{\rangle}=\hat{f}_{L,\alpha_{L},\beta_{L}}\,\hat{f}_{R,\alpha_{R},\beta_{R}}$

###  3.3 Expressing products of super operators and operators in terms of
their matrix elements

When expressing products of (super) operators in terms of their matrix
elements, we can _in spirit_ just think of those operators as matrix products.
The case

$\displaystyle\hat{f}\,\hat{g}\mapsto\left(\sum_{\gamma\in\Gamma}\hat{f}_{\alpha,\gamma}\,\hat{g}_{\gamma,\beta}\right)_{\alpha,\beta\in\Gamma}$

was already covered in [CHP22, Proposition 3.4]. Consequently, we will only
deal with the two remaining cases. Here, we need to pay more attention what
indices need to be summed over.

######  Lemma 3.4.

Suppose we are in the setting of Assumption 3.1. Let $\hat{F}$ and $\hat{G}$
be elements of
$\mathcal{B}\bigl{(}\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\bigr{)}$
and $\hat{g}\in\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}$.

  1. (1)

Then we can express $\hat{F}\hat{g}$ in terms of the matrix elements as

$\displaystyle\bigl{(}\hat{F}\,\hat{g}\bigr{)}_{\alpha,\beta}=\sum_{\alpha^{\prime},\beta^{\prime}\in\Gamma}\hat{F}_{\alpha,\beta^{\prime},\alpha^{\prime},\beta}\,\hat{g}_{\beta^{\prime},\alpha^{\prime}}.$
(3.6)

  2. (2)

Likewise, the product
$\hat{F}\,\hat{G}\in\mathcal{B}\bigl{(}\mathfrak{L}^{2}\bigl{(}\mathcal{B}(\mathcal{H})\bigr{)}\bigr{)}$
has the matrix representation

$\displaystyle\bigl{(}\hat{F}\,\hat{G}\bigr{)}_{\alpha_{L},\beta_{L},\alpha_{R},\beta_{R}}$
$\displaystyle=\sum_{\alpha,\beta\in\Gamma}\hat{F}_{\alpha_{L},\alpha,\beta,\beta_{R}}\,\hat{G}_{\alpha,\beta_{L},\alpha_{R},\beta}.$
(3.7)

We will give a proof momentarily. A faster, but formal argument goes as
follows: let us assume for a moment that
$\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is not just a Parseval frame,
but an orthonormal basis. Then we can obtain an equation for
$\bigl{(}\hat{F}\,\hat{g}\bigr{)}_{\alpha,\beta}$ by considering

$\displaystyle|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}}|\otimes|\mathcal{G}_{\alpha_{R}}\rangle\langle\mathcal{G}_{\beta_{R}}|\;\bigl{(}|\mathcal{G}_{\alpha}\rangle\langle\mathcal{G}_{\beta}|\bigr{)}$
$\displaystyle=|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}}|\,|\mathcal{G}_{\alpha}\rangle\langle\mathcal{G}_{\beta}|\,|\mathcal{G}_{\alpha_{R}}\rangle\langle\mathcal{G}_{\beta_{R}}|$
$\displaystyle=\left\langle\mathcal{G}_{\beta_{L}},\mathcal{G}_{\alpha}\right\rangle\,\left\langle\mathcal{G}_{\beta},\mathcal{G}_{\alpha_{R}}\right\rangle\,|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{R}}|$
$\displaystyle=\delta_{\beta_{L},\alpha}\,\delta_{\beta,\alpha_{R}}\,|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{R}}|.$

Consequently, we arrive at equation (3.6).

######  Proof.

  1. (1)

To extend these arguments to Parseval frames, we start from first principles
and write out the matrix element

$\displaystyle\bigl{(}\hat{F}\,\hat{g}\bigr{)}_{\alpha,\beta}$
$\displaystyle=\bigl{\langle}\mathcal{G}_{\alpha},\hat{F}\,\hat{g}\,\mathcal{G}_{\beta}\bigr{\rangle}$
$\displaystyle=\sum_{\alpha^{\prime},\beta^{\prime}\in\Gamma}\hat{g}_{\beta^{\prime},\alpha^{\prime}}\;\bigl{\langle}\mathcal{G}_{\alpha},\hat{F}\,\bigl{(}|\mathcal{G}_{\beta^{\prime}}\rangle\langle\mathcal{G}_{\alpha^{\prime}}|\bigr{)}\,\mathcal{G}_{\beta}\bigr{\rangle}.$
(3.8)

We recognize that the scalar product gives
$\hat{F}_{\alpha,\beta^{\prime},\alpha^{\prime},\beta}$ (cf. equation (3.5)).
The sums converge by assumption on $\hat{F}$ and $\hat{g}$.

  2. (2)

To simplify our arguments, we will first proceed under the assumption that
$\\{\mathcal{G}_{\alpha}\\}_{\alpha\in\Gamma}$ is an orthonormal basis.
Applying the tensor product of rank-$1$ operators to some suitable bounded
operator $\hat{h}$ yields

$\displaystyle|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}}|\otimes|\mathcal{G}_{\alpha_{R}}\rangle\langle\mathcal{G}_{\beta_{R}}|\;\Bigl{(}|\mathcal{G}_{\alpha_{L}^{\prime}}\rangle\langle\mathcal{G}_{\beta_{L}^{\prime}}|\otimes|\mathcal{G}_{\alpha_{R}^{\prime}}\rangle\langle\mathcal{G}_{\beta_{R}^{\prime}}|\;(\hat{h})\Bigr{)}$
$\displaystyle\qquad=|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}}|\,|\mathcal{G}_{\alpha_{L}^{\prime}}\rangle\langle\mathcal{G}_{\beta_{L}^{\prime}}|\,\hat{h}\,|\mathcal{G}_{\alpha_{R}^{\prime}}\rangle\langle\mathcal{G}_{\beta_{R}^{\prime}}|\,|\mathcal{G}_{\alpha_{R}}\rangle\langle\mathcal{G}_{\beta_{R}}|$
$\displaystyle\qquad=\bigl{\langle}\mathcal{G}_{\beta_{L}},\mathcal{G}_{\alpha_{L}^{\prime}}\bigr{\rangle}\,\bigl{\langle}\mathcal{G}_{\beta_{R}^{\prime}},\mathcal{G}_{\alpha_{R}}\bigr{\rangle}\,\bigl{\langle}\mathcal{G}_{\beta_{L}^{\prime}},\hat{h}\,\mathcal{G}_{\alpha_{R}^{\prime}}\bigr{\rangle}\,|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{R}}|$
$\displaystyle\qquad=\delta_{\beta_{L},\alpha_{L}^{\prime}}\,\delta_{\alpha_{R},\beta_{R}^{\prime}}\,|\mathcal{G}_{\alpha_{L}}\rangle\langle\mathcal{G}_{\beta_{L}^{\prime}}|\otimes|\mathcal{G}_{\alpha_{R}^{\prime}}\rangle\langle\mathcal{G}_{\beta_{R}}|\,(\hat{h}).$

Hence, we need to contract the index pairs $(\beta_{L},\alpha_{L}^{\prime})$
and $(\alpha_{R},\beta_{R}^{\prime})$ in the sum. Relabeling indices
appropriately then gives us (3.7).

The extension to Parseval frames then follows as in (1): we use the scalar
product representation (3.5) of the matrix element for the super operator
$\hat{F}\,\hat{G}$ and successively plug in the formal series for $\hat{F}$
and $\hat{G}$. Convergence is assured by assumption in the relevant Banach
space of super operators.

□

##  4 The main technical result: characterization of symbols of super
$\Psi$DOs via their matrix elements

The previous section dealt with the abstract setting, here we will connect it
to pseudodifferential theory. For our purposes the abstract discrete group
$\Gamma$ has to be replaced with the product $\Gamma\times\Gamma^{*}$ of the
lattice and its dual lattice from Section 2. Hence, each index $\alpha$ from
Section 3 becomes a _pair_ of indices $(\alpha,\alpha^{*})$.

The matrix elements of a magnetic pseudodifferential operator
$\mathrm{op}^{A}(f)$ are defined as

$\displaystyle\mathrm{op}^{A}(f)_{\alpha,\alpha^{*},\beta,\beta^{*}}:\negmedspace$
$\displaystyle=\bigl{\langle}\mathcal{G}_{\alpha,\alpha^{*}}^{A},\mathrm{op}^{A}(f)\,\mathcal{G}_{\beta,\beta^{*}}^{A}\bigr{\rangle}$
$\displaystyle=\frac{1}{(2\pi)^{d}}\int_{\Xi}\mathrm{d}X\,(\mathcal{F}_{\sigma}f)(X)\;\bigl{\langle}\mathcal{G}_{\alpha,\alpha^{*}}^{A},w^{A}(X)\,\mathcal{G}_{\beta,\beta^{*}}^{A}\bigr{\rangle},$
(4.1)

where $\mathcal{G}^{A}_{\alpha,\alpha^{*}}$ is the Parseval frame from [CHP22]
(cf. equation (2.2)).

Likewise, magnetic pseudodifferential _super_ operators $\mathrm{Op}^{A}(F)$
have twice the number of indices,

$\displaystyle\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}:=\mathrm{Tr}\,_{L^{2}(\mathbb{R}^{d})}\Bigl{(}|\mathcal{G}^{A}_{\beta_{R},\beta_{R}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha_{L},\alpha_{L}^{*}}|\,\hat{F}\,|\mathcal{G}^{A}_{\beta_{L},\beta_{L}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha_{R},\alpha_{R}^{*}}|\Bigr{)},$
(4.2)

one set for the left and one set for the right variables. As
$\bigl{\\{}\mathcal{G}^{A}_{\alpha,\alpha^{*}}\bigr{\\}}_{(\alpha,\alpha^{*})\in\Gamma\times\Gamma^{*}}$
is a Parseval frame, despite its inherent overcompleteness, it still acts like
an orthonormal basis in that we can express

$\displaystyle\mathrm{Op}^{A}(F)=\sum_{(\alpha_{L,R},\alpha_{L,R}^{*}),(\beta_{L,R},\beta_{L,R}^{*})\in\Gamma\times\Gamma^{*}}$
$\displaystyle\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\,\cdot|\mathcal{G}^{A}_{\alpha_{L},\alpha_{L}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{L},\beta_{L}^{*}}|\otimes|\mathcal{G}^{A}_{\alpha_{R},\alpha_{R}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{R},\beta_{R}^{*}}|$
(4.3)

as an infinite linear combination of product operators.

The elegance of the approach of Cornean, Helffer and Purice is that
oscillatory integral techniques are only needed when characterizing magnetic
pseudodifferential operators in terms of their matrix elements (cf. [CHP22,
Theorem 3.1]). Proofs of other results, which would ordinarily require
oscillatory integral techniques, now translate to questions of convergence of
infinite sums over the indices that enumerate the matrix elements.

The next two results, Theorem 4.1 and Corollary 4.2, are analogs of [CHP22,
Theorem 3.1]. Their proofs are essentially identical to that given by Cornean,
Helffer and Purice, many of the expressions just need to be “doubled” to
account for the presence of left and right variables.

######  Theorem 4.1.

Suppose we are given a tempered distribution
$F\in\mathcal{S}^{\prime}(\Xi^{2})$ and some $m_{L},m_{R}\in\mathbb{R}$. Then
the following two statements are equivalent:

  1. (a)

$F\in S^{m_{L},m_{R}}_{0,0}(\Xi^{2})$ is a Hörmander symbol of order
$(m_{L},m_{R})$ and type $(0,0)$.

  2. (b)

For any $(n_{L},n_{R},n_{L}^{*},n_{R}^{*})\in\mathbb{N}_{0}^{4}$ there exists
a constant $C_{n_{L},n_{R},n_{L}^{*},n_{R}^{*}}(F,B)>0$ so that the matrix
elements satisfy the bound

$\displaystyle\sup_{(\alpha_{L},\alpha_{L}^{*}),(\alpha_{R},\alpha_{R}^{*}),(\beta_{L},\beta_{L}^{*}),(\beta_{R},\beta_{R}^{*})\in\Gamma\times\Gamma^{*}}$
$\displaystyle\langle\alpha_{L}-\beta_{L}\rangle^{n_{L}}\,\langle\alpha_{L}^{*}-\beta_{L}^{*}\rangle^{n_{L}^{*}}\,\langle\alpha_{R}-\beta_{R}\rangle^{n_{R}}\,\langle\alpha_{R}^{*}-\beta_{R}^{*}\rangle^{n_{R}^{*}}\,$
$\displaystyle\cdot\langle\alpha_{L}^{*}+\beta_{L}^{*}\rangle^{-m_{L}}\,\langle\alpha_{R}^{*}+\beta_{R}^{*}\rangle^{-m_{R}}\,\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\qquad\qquad\leq C_{n_{L},n_{R},n_{L}^{*},n_{R}^{*}}(F,B).$
(4.4)

######  Proof.

The strategy of the proof is identical to [CHP22], we will just outline the
necessary modifications.

First of all, the equations for
$\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
is essentially a “doubling” of the equation for
$\mathrm{op}^{A}(f)_{\alpha,\alpha^{*},\beta,\beta^{*}}$: indeed, for any
combination of multi indices, we can rewrite each matrix element (4.1) in
terms of the symplectic Fourier transform of $f$ and the matrix element of the
Weyl system. Writing out the action of the Weyl system and some algebra, we
recover [CHP22, equation (3.2)].

The equivalent expression for the magnetic super Weyl quantization is
essentially two copies of [CHP22, equation (3.2)], one for the left and one
for the right variables. The reason is that $\mathrm{Op}^{A}(F)$ is the
integral of the product operator $w^{A}(X_{L})\otimes w^{A}(X_{R})$. Hence,
from equation (4.2) and the definition of the magnetic super Weyl quantization
(2.7), we deduce

$\displaystyle\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\qquad=\frac{1}{(2\pi)^{2d}}\int_{\Xi}\mathrm{d}X_{L}\int_{\Xi}\mathrm{d}X_{R}\,(\mathcal{F}_{\Sigma}F)(X_{L},X_{R})\;\bigl{\langle}\mathcal{G}_{\alpha_{L},\alpha_{L}^{*}}^{A},w^{A}(X_{L})\,\mathcal{G}_{\beta_{L},\beta_{L}^{*}}^{A}\bigr{\rangle}\;\bigl{\langle}\mathcal{G}_{\alpha_{R},\alpha_{R}^{*}}^{A},w^{A}(X_{R})\,\mathcal{G}_{\beta_{R},\beta_{R}^{*}}^{A}\bigr{\rangle}$

is the double phase space integral of the “expectation values” of left and
right Weyl systems and the symplectic Fourier transform of $F$.

We will re-use the notation from the proof of [CHP22, Theorem 3.1] whenever
possible.

“(a) $\Longrightarrow$ (b):” Suppose $F\in S^{m_{L},m_{R}}_{0,0}(\Xi^{2})$ is
a Hörmander symbol of order $(m_{L},m_{R})$ and fix the four non-negative
integers $n_{L,R}\in\mathbb{N}_{0}$ and $n_{L,R}^{*}\in\mathbb{N}_{0}$.

After applying the variable transformations from [CHP22] to (4.2) twice, once
for the left and once for the right variables, one essentially obtains two
copies of [CHP22, equation (3.7)]. The only difference is that we have to
replace $\Phi(z+\nicefrac{{\mu}}{{2}},\zeta+\nicefrac{{\mu^{*}}}{{2}})$ with

$\displaystyle
F\bigl{(}z_{L}+\tfrac{1}{2}\mu_{L},\zeta_{L}+\tfrac{1}{4\pi}\mu_{L}^{*},z_{R}+\tfrac{1}{2}\mu_{R},\zeta_{R}+\tfrac{1}{4\pi}\mu_{R}^{*}\bigr{)}.$

Repeating the arguments from the proof of [CHP22, Theorem 3.1] and using
Definition 2.5 for double Hörmander symbols then gives us an estimate of the
matrix element of the form

$\displaystyle\bigl{\lvert}\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}\bigr{\rvert}$
$\displaystyle\qquad\leq
C\,\langle\nu_{L}^{*}\rangle^{-2N_{L}}\,\langle\nu_{L}\rangle^{2N_{L}+2M_{L}-2K_{L}}\,\langle\nu_{R}^{*}\rangle^{-2N_{R}}\,\langle\nu_{R}\rangle^{2N_{R}+2M_{R}-2K_{R}}$
$\displaystyle\qquad\quad\;\cdot\,\int_{\mathbb{R}^{d}}\mathrm{d}\zeta_{L}\,\langle\zeta_{L}\rangle^{-2M_{L}}\,\langle\zeta_{L}+\tfrac{1}{4\pi}\mu_{L}^{*}\rangle^{m_{L}}\,\int_{\mathbb{R}^{d}}\mathrm{d}\zeta_{R}\,\langle\zeta_{R}\rangle^{-2M_{R}}\,\langle\zeta_{R}+\tfrac{1}{4\pi}\mu_{R}^{*}\rangle^{m_{R}},$

where the indices

$\displaystyle(\mu_{L,R},\nu_{L,R})\,$
$\displaystyle:=\bigl{(}\alpha_{L,R}+\beta_{L,R},\alpha_{L,R}-\beta_{L,R}\bigr{)},$
$\displaystyle(\mu_{L,R}^{*},\nu_{L,R}^{*})\,$
$\displaystyle:=\bigl{(}\alpha_{L,R}^{*}+\beta_{L,R}^{*},\beta_{L,R}^{*}-\alpha_{L,R}^{*}\bigr{)},$

are the sums and differences of left and right multi indices.

Choosing the integers $N_{L,R}$, $M_{L,R}$ and $K_{L,R}$ large enough then
ensures the existence of the oscillatory integral (4.2); the least lower
bounds for $N_{L,R}$, $M_{L,R}$ and $K_{L,R}$ are those spelled out in [CHP22]
and involve $n_{L,R}$ and $n_{L,R}^{*}$. It turns out we may choose $N_{L,R}$
and $K_{L,R}-N_{L,R}-M_{L,R}$ as large as we like, which establishes at most
polynomial growth in the indices
$\mu_{L,R}^{*}=\alpha_{L,R}^{*}+\beta_{L,R}^{*}$ with powers $m_{L,R}$ and
rapid decay in $\nu_{L,R}$ and $\nu_{L,R}^{*}$. This shows the estimate from
(b) holds true.

“(b) $\Longrightarrow$ (a):” Suppose for any four non-negative integers
$n_{L,R}\in\mathbb{N}_{0}$ and $n_{L,R}^{*}\in\mathbb{N}_{0}$ there exists a
constant $C>0$ so that the estimate from (b) holds true. The idea of the proof
is to reconstruct the symbol of the operator as the Wigner transform of the
kernel, written as the infinite sum

$\displaystyle\mathrm{Op}^{A}(F):=\sum_{(\alpha_{L,R},\alpha_{L,R}^{*}),(\beta_{L,R},\beta_{L,R}^{*})\in\Gamma\times\Gamma^{*}}$
$\displaystyle\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\cdot\,|\mathcal{G}^{A}_{\alpha_{L},\alpha_{L}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{L},\beta_{L}^{*}}|\otimes|\mathcal{G}^{A}_{\alpha_{R},\alpha_{R}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{R},\beta_{R}^{*}}|.$

The main point of the proof is to ensure convergence of the above sum in the
appropriate sense.

Hence, we truncate the sum by fixing some $N\in\mathbb{N}_{0}$ and defining

$\displaystyle\mathrm{Op}^{A}(F_{N}):=\sum_{\lvert(\alpha_{L,R},\alpha_{L,R}^{*})|,\lvert(\beta_{L,R},\beta_{L,R}^{*})|\leq
N}$
$\displaystyle\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\cdot\,|\mathcal{G}^{A}_{\alpha_{L},\alpha_{L}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{L},\beta_{L}^{*}}|\otimes|\mathcal{G}^{A}_{\alpha_{R},\alpha_{R}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{R},\beta_{R}^{*}}|.$

For product functions $\mathrm{Op}^{A}\bigl{(}f_{L}\otimes
f_{R}\bigr{)}=\mathrm{op}^{A}(f_{L})\otimes\mathrm{op}^{A}(f_{R})$ the
magnetic _super_ Weyl quantization is the “product” of the two (regular)
magnetic Weyl quantizations. Therefore, this relationship carries over to the
inverse of the super Weyl quantization, which in this case is two copies of
the magnetic Wigner transform applied to left and right variables separately
(cf. the displayed equation below equation (3.9) in [LL22]).

The magnetic Wigner transform is a topological vector space isomorphism on
$\mathcal{S}$, and hence, for any $N\in\mathbb{N}$

$\displaystyle
F_{N}:=\sum_{\lvert(\alpha_{L,R},\alpha_{L,R}^{*})|,\lvert(\beta_{L,R},\beta_{L,R}^{*})|\leq
N}$
$\displaystyle\,\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\cdot\,(\mathrm{op}^{A})^{-1}\bigl{(}|\mathcal{G}^{A}_{\alpha_{L},\alpha_{L}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{L},\beta_{L}^{*}}|\bigr{)}\otimes(\mathrm{op}^{A})^{-1}\bigl{(}|\mathcal{G}^{A}_{\alpha_{R},\alpha_{R}^{*}}\rangle\langle\mathcal{G}^{A}_{\beta_{R},\beta_{R}^{*}}|\bigr{)}\in\mathcal{S}(\Xi^{2})$

is a Schwartz function. This defines a sequence of Schwartz functions
$F_{N}\rightarrow F\in\mathcal{S}^{\prime}(\Xi^{2})$, which converges in the
distributional sense to $F$. In fact, repeating the arguments from [CHP22]
twice, once for left and once for right variables yields the claim that

$\displaystyle F_{N}\xrightarrow{N\rightarrow\infty}F\in
S^{m_{L},m_{R}}_{0,0}(\Xi^{2})$

converges uniformly on compact subsets to a _Hörmander symbol_ of order
$(m_{L},m_{R})$. This finishes the proof. □

We get an analogous result for standard Hörmander symbols (cf. Definition
2.4):

######  Corollary 4.2.

Suppose we are given a tempered distribution
$F\in\mathcal{S}^{\prime}(\Xi^{2})$ and some $m\in\mathbb{R}$. Then the
following two statements are equivalent:

  1. (a)

$F\in S^{m}_{0,0}(\Xi^{2})$ is a Hörmander symbol of order $m$ and type
$(0,0)$.

  2. (b)

For any $n,n^{*}\in\mathbb{N}_{0}^{2}$ there exists a constant
$C_{n,n^{*}}(F,B)>0$ so that the matrix elements satisfy the bound

$\displaystyle\sup_{(\alpha_{L},\alpha_{L}^{*}),(\alpha_{R},\alpha_{R}^{*}),(\beta_{L},\beta_{L}^{*}),(\beta_{R},\beta_{R}^{*})\in\Gamma\times\Gamma^{*}}$
$\displaystyle\bigl{\langle}(\alpha_{L},\alpha_{R})-(\beta_{L},\beta_{R})\bigr{\rangle}^{n}\,\bigl{\langle}(\alpha_{L}^{*},\alpha_{R}^{*})-(\beta_{L}^{*},\beta_{R}^{*})\bigr{\rangle}^{n^{*}}$
$\displaystyle\cdot\bigl{\langle}(\alpha_{L}^{*},\alpha_{R}^{*})+(\beta_{L}^{*},\beta_{R}^{*})\bigr{\rangle}^{-m}\,\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\alpha_{R},\alpha_{R}^{*},\beta_{R},\beta_{R}^{*}}$
$\displaystyle\qquad\qquad\leq C_{n,n^{*}}(F,B).$ (4.5)

######  Proof.

The proof is essentially identical to that of Theorem 4.1, and we content
ourselves with outlining the modifications.

“(a) $\Longrightarrow$ (b):” Compared to the proof of Theorem 4.1, we will use
different $L$ operators in the proof. Namely in the notation of [CHP22,
equation (3.8)] we apply the three operators

$\displaystyle L_{(z_{L},z_{R})}\,$
$\displaystyle:=\bigl{\langle}(\nu_{L}^{*},\nu_{R}^{*})\bigr{\rangle}^{-2}\,\bigl{(}1-\Delta_{z_{L}}-\Delta_{z_{R}}\bigr{)},$
$\displaystyle L_{(\zeta_{L},\zeta_{R})}\,$
$\displaystyle:=\bigl{\langle}(\nu_{L},\nu_{R})\bigr{\rangle}^{-2}\,\bigl{(}1-\Delta_{\zeta_{L}}-\Delta_{\zeta_{R}}\bigr{)},$
$\displaystyle L_{(v_{L},v_{R})}\,$
$\displaystyle:=\bigl{\langle}(\zeta_{L},\zeta_{R})\bigr{\rangle}^{-2}\,\bigl{(}1-\Delta_{v_{L}}-\Delta_{v_{R}}\bigr{)}.$

If we introduce notation such as $z:=(z_{L},z_{R})$,
$\zeta:=(\zeta_{L},\zeta_{R})$, etc., we symbolically recover the expression
below [CHP22, equation (3.9)] and can perform all subsequent arguments almost
verbatim. This shows the first direction.

“(b) $\Longrightarrow$ (a):” Similarly to before, we can introduce the joint
left/right variables $\nu:=(\nu_{L},\nu_{R})$,
$\nu^{*}:=(\nu_{L}^{*},\nu_{R}^{*})$, etc. to make the expressions in the
proof look almost identical to that in [CHP22]. Because the matrix elements
now satisfy (4.5) instead of (4.4), we recover
$\langle\zeta\rangle^{m}=\bigl{\langle}(\zeta_{L},\zeta_{R})\bigr{\rangle}^{m}$
on the right-hand side of our estimate of
$\bigl{\lvert}\partial_{z_{L}}^{a_{L}}\partial_{z_{R}}^{a_{R}}\partial_{\zeta_{L}}^{\alpha_{L}}\partial_{\zeta_{R}}^{\alpha_{R}}F_{N}(z_{L},\zeta_{L},z_{R},\zeta_{R})\bigr{\rvert}$
with a constant that is uniform in $N$, $(z_{L},\zeta_{L})$ and
$(z_{R},\zeta_{R})$. Consequently, $F_{N}\rightarrow F\in
S^{m}_{0,0}(\Xi^{2})$ converges uniformly on compact subsets to a Hörmander
symbol of order $m$ and type $(0,0)$. □

##  5 Boundedness criteria for magnetic pseudodifferential super operators

We are now in a position to formulate and prove boundedness criteria on the
level of matrix elements: we have given characterizations of Hilbert-Schmidt
operators in Section 3 and obtained growth/decay estimates on the matrix
elements of magnetic pseudodifferential super operators (Theorem 4.1 and
Corollary 4.2). Combining both then gives very simple proofs of boundedness in
the spirit of [CHP18, CHP22].

###  5.1 Boundedness of magnetic pseudodifferential superoperators

We will need two more auxiliary statements in the proof of boundedness. The
first is a version of Schur’s test (cf. [CR20, Lemma 6.1.7]).

######  Lemma 5.1 (Schur’s Test).

Let $\Gamma$ be a countable set and
$(K_{\alpha,\beta})_{\alpha,\beta\in\Gamma}$ be a sequence with values in
$\mathbb{C}$ indexed by $\Gamma\times\Gamma$. Suppose there is $C>0$ such that

$\displaystyle\sup_{\beta\in\Gamma}\sum_{\alpha\in\Gamma}\left\lvert
K_{\alpha,\beta}\right\rvert\leq
C\quad\text{and}\quad\sup_{\alpha\in\Gamma}\sum_{\beta\in\Gamma}\left\lvert
K_{\alpha,\beta}\right\rvert\leq C.$

Then the map $K$ defined by

$\displaystyle(Kc)_{\alpha}:=\sum_{\beta\in\Gamma}K_{\alpha,\beta}\,c_{\beta},\qquad(c_{\beta})_{\beta\in\Gamma}\in\ell^{2}(\Gamma)$

gives rise to a continuous linear operator from $\ell^{2}(\Gamma)$ to itself
with $\left\lVert K\right\rVert_{\mathcal{B}(\ell^{2}(\Gamma))}\leq C$.

The second is a refinement of Lemma 3.4 (1) with more explicit control over
the convergence of the sums. As mentioned in Subsection 2.2 the magnetic Weyl
quantization $\mathrm{op}^{A}$ yields a unitary map from $L^{2}(\Xi)$ to
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.
Therefore, it follows that
$\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}:=\bigl{\\{}\mathrm{op}^{A}(f)\;|\;f\in\mathcal{S}(\Xi)\bigr{\\}}$
is dense in
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
since $\mathcal{S}(\Xi)$ is a dense subspace of $L^{2}(\Xi)$. A linear (super)
operator
$\hat{F}:\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}\longrightarrow\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
can be regarded as a densely defined unbounded operator in
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
with domain $\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}$. The scalar
product on the space of Hilbert-Schmidt operators gives rise to the adjoint
$\hat{F}^{*}:\mathscr{D}(\hat{F}^{*})\longrightarrow\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
of $\hat{F}$ by letting

$\displaystyle{\left\langle\hat{F}^{*}\,\hat{f},\hat{g}\right\rangle}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}=\left\langle\hat{f},\hat{F}\,\hat{g}\right\rangle_{\mathfrak{L}^{2}\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\qquad\forall\hat{f}\in\mathscr{D}(\hat{F}^{*})$
$\displaystyle\forall\hat{g}\in\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)},$

where as usual the domain of the adjoint is defined as

$\displaystyle\mathscr{D}(\hat{F}^{*}):=\Bigl{\\{}\hat{f}\in\mathfrak{L}^{2}\bigl{(}\mathcal{B}(L^{2}(\mathbb{R}^{d}))\bigr{)}\;\;\big{|}\;\;$
$\displaystyle\bigl{\lvert}{\left\langle\hat{f},\hat{F}\,\hat{g}\right\rangle}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\bigr{\rvert}\leq
C(\hat{F},\hat{f})\,\lVert\hat{g}\rVert_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\,\,\Bigr{.}$
$\displaystyle\Bigl{.}\forall\hat{g}\in\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\Bigr{\\}}.$

The following result is the super operator analog of [CHP22, Proposition 2.4].

######  Proposition 5.2.

Suppose that
$\hat{F}^{A}:\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}\longrightarrow\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
is a densely defined unbounded super operator as above and
$\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}\subset\mathscr{D}(\hat{F}^{*})$.
Let $g\in\mathcal{S}(\Xi)$ and $\hat{g}^{A}=\mathrm{op}^{A}(g)$ its magnetic
Weyl quantization. Then we can express the operator in terms of the matrix
elements

$\displaystyle\hat{F}^{A}\,\hat{g}^{A}=\sum_{(\alpha,\alpha^{*}),(\beta,\beta^{*})\in\Gamma\times\Gamma^{*}}\left(\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\hat{F}^{A}_{\alpha,\alpha^{*},\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*},\beta,\beta^{*}}\,\hat{g}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*}}\right)\,|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|,$

where the inner sum with respect to
$(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}$
converges in the sense of
$\ell^{1}\bigl{(}(\Gamma\times\Gamma^{*})^{2}\bigr{)}$ and the outer sum with
respect to $(\alpha,\alpha^{*}),(\beta,\beta^{*})\in\Gamma\times\Gamma^{*}$
converges in
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.

######  Proof.

Using (3.1) we can expand the application of the super operator $\hat{F}^{A}$
on the Hilbert-Schmidt operator $\hat{g}^{A}$,

$\displaystyle\hat{F}^{A}\,\hat{g}^{A}$
$\displaystyle=\sum_{(\alpha,\alpha^{*}),(\beta,\beta^{*})\in\Gamma\times\Gamma^{*}}{\bigl{\langle}|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|\,,\,\hat{F}^{A}\,\hat{g}^{A}\bigr{\rangle}}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\,|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|$
$\displaystyle=\sum_{(\alpha,\alpha^{*}),(\beta,\beta^{*})\in\Gamma\times\Gamma^{*}}{\Bigl{\langle}\hat{F}^{A\,\ast}\bigl{(}|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|\bigr{)}\,,\,\hat{g}^{A}\Bigr{\rangle}}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\,|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|,$
(5.1)

where the sum converges in
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.
Note that by using (3.1) again we also obtain

$\displaystyle\hat{g}^{A}$
$\displaystyle=\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\left\langle\mathcal{G}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*}}\,,\,\hat{g}^{A}\,\mathcal{G}^{A}_{\alpha^{\prime},{\alpha^{\prime}}^{*}}\right\rangle\,|\mathcal{G}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha^{\prime},{\alpha^{\prime}}^{*}}|$
$\displaystyle=\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\hat{g}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*}}\,|\mathcal{G}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha^{\prime},{\alpha^{\prime}}^{*}}|.$

By plugging the expansion of $\hat{g}^{A}$ back into (5.1) we get

$\displaystyle{\Bigl{\langle}\hat{F}^{A\,\ast}\bigl{(}|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|\bigr{)}\,,\,\hat{g}^{A}\Bigr{\rangle}}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}$
$\displaystyle\qquad=\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\hat{g}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*}}\,{\Bigl{\langle}\hat{F}^{A\,\ast}\bigl{(}|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|\bigr{)}\,,\,|\mathcal{G}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha^{\prime},{\alpha^{\prime}}^{*}}|\Bigr{\rangle}}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}$
$\displaystyle\qquad=\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\hat{g}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*}}\,{\Bigl{\langle}|\mathcal{G}^{A}_{\alpha,\alpha^{*}}\rangle\langle\mathcal{G}^{A}_{\beta,\beta^{*}}|\,,\,\hat{F}^{A}\,|\mathcal{G}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*}}\rangle\langle\mathcal{G}^{A}_{\alpha^{\prime},{\alpha^{\prime}}^{*}}|\Bigr{\rangle}}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}$
$\displaystyle\qquad=\sum_{(\alpha^{\prime},{\alpha^{\prime}}^{*}),(\beta^{\prime},{\beta^{\prime}}^{*})\in\Gamma\times\Gamma^{*}}\hat{F}^{A}_{\alpha,\alpha^{*},\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*},\beta,\beta^{*}}\,\hat{g}^{A}_{\beta^{\prime},{\beta^{\prime}}^{*},\alpha^{\prime},{\alpha^{\prime}}^{*}}.$

This completes the proof. □

Now we can furnish the proof of our main theorem.

######  Theorem 5.3.

Assume the function $F\in S^{m}_{\rho,0}(\Xi^{2})$ is a Hörmander symbol of
order $m\leq 0$ with $0\leq\rho\leq 1$ and. Then $\mathrm{Op}^{A}(F)$ defines
a bounded super operator
$\mathrm{Op}^{A}(F)\in\mathcal{B}\bigl{(}\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\bigr{)}$
on the Hilbert space of Hilbert-Schmidt operators.

######  Proof.

First of all, by the nesting property of Hörmander classes,

$\displaystyle S^{m}_{\rho,0}(\Xi^{2})\subseteq S^{m}_{0,0}(\Xi^{2})$
$\displaystyle\forall\rho\in[0,1],$ (5.2)

it suffices to verify the statement for $\rho=0$.

Let $F\in S_{0,0}^{m}(\Xi^{2})$ and $g\in\mathcal{S}(\Xi)$. Then it follows
from Proposition 2.8 that the semi-super product $F\bullet^{B}g$ belongs to
$\mathcal{S}(\Xi)$. Since $\mathrm{op}^{A}$ yields a unitary map from
$L^{2}(\Xi)$ to
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
(cf. Subsection 2.2) and $\mathcal{S}(\Xi)\subset L^{2}(\Xi)$ we can deduce
that
$\mathrm{Op}^{A}(F)\,\hat{g}^{A}:=\mathrm{op}^{A}\bigl{(}F\bullet^{B}g\bigr{)}$
belongs to
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$.

By using the Parseval identity and Proposition 5.2 we obatin

$\displaystyle\bigl{\lVert}\mathrm{Op}^{A}(F)\,\hat{g}^{A}\bigr{\rVert}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}^{2}$
$\displaystyle=\sum_{(\alpha_{L,R},\alpha_{L,R}^{*})\in\Gamma\times\Gamma^{*}}\left\lvert\bigl{(}\mathrm{Op}^{A}(F)\,\hat{g}^{A}\bigr{)}_{\alpha_{L},\alpha_{L}^{*},\alpha_{R},\alpha_{R}^{*}}\right\rvert^{2}$
$\displaystyle=\sum_{(\alpha_{L,R},\alpha_{L,R}^{*})\in\Gamma\times\Gamma^{*}}\biggl{|}\sum_{(\beta_{L,R},\beta_{L,R}^{*})\in\Gamma\times\Gamma^{*}}\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\beta_{R},\beta_{R}^{*},\alpha_{R},\alpha_{R}^{*}}\,\hat{g}^{A}_{\beta_{L},\beta_{L}^{*},\beta_{R},\beta_{R}^{*}}\biggr{|}^{2}.$
(5.3)

It follows from Schur’s Test (Lemma 5.1) and Proposition 3.2 that we have

$\displaystyle\sum_{(\alpha_{L,R},\alpha_{L,R}^{*})\in\Gamma\times\Gamma^{*}}$
$\displaystyle\left\lvert\sum_{(\beta_{L,R},\beta_{L,R}^{*})\in\Gamma\times\Gamma^{*}}\mathrm{Op}^{A}(F)_{\alpha_{L},\alpha_{L}^{*},\beta_{L},\beta_{L}^{*},\beta_{R},\beta_{R}^{*},\alpha_{R},\alpha_{R}^{*}}\,\hat{g}^{A}_{\beta_{L},\beta_{L}^{*},\beta_{R},\beta_{R}^{*}}\right\rvert^{2}$
$\displaystyle\leq
C(F,B)^{2}\sum_{(\beta_{L,R},\beta_{L,R}^{*})\in\Gamma\times\Gamma^{*}}\left\lvert\hat{g}^{A}_{\beta_{L},\beta_{L}^{*},\beta_{R},\beta_{R}^{*}}\right\rvert^{2}=C(F,B)^{2}\left\lVert\hat{g}^{A}\right\rVert_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}^{2}.$

Combining this with (5.3) we get

$\displaystyle\bigl{\lVert}\mathrm{Op}^{A}(F)\,\hat{g}^{A}\bigr{\rVert}_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\leq
C(F,B)\left\lVert\hat{g}^{A}\right\rVert_{\mathfrak{L}^{2}(\mathcal{B}(L^{2}(\mathbb{R}^{d})))}\qquad\forall\hat{g}^{A}\in\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}.$

The magnetic Weyl quantization map $\mathrm{op}^{A}$ yields a unitary map from
$L^{2}(\Xi)$ to
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
(cf. Subsection 2.2) and $\mathcal{S}(\Xi)$ is dense in $L^{2}(\Xi)$. This
shows that
$\mathrm{op}^{A}\bigl{(}\mathcal{S}(\Xi)\bigr{)}:=\bigl{\\{}\mathrm{op}^{A}(g)\;\;|\;\;g\in\mathcal{S}(\Xi)\bigr{\\}}$
is a dense subspace of
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$,
and hence $\mathrm{Op}^{A}(F)$ uniquely extends to a bounded linear operator
from
$\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
to itself. The proof is complete. □

######  Remark 5.4.

If we _could_ prove that

$\displaystyle\mathrm{Op}^{A}(F)\in\mathcal{B}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}=\mathcal{B}\Bigl{(}\mathfrak{L}^{\infty}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\Bigr{)}$

was bounded on $\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$ under the
_same_ conditions as in Theorem 5.3, the non-commutative version of the Riesz-
Thorin Interpolation Theorem (cf. e. g. Theorems 2.9 and 2.10 as well as
Remark 1 on p. 23 in [Sim05] or [GK69, Theorem 13.1]) would imply that

$\displaystyle\mathrm{Op}^{A}(F):\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\longrightarrow\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$

defined a bounded operator for _any_ $2\leq p\leq\infty$.

Likewise, a proof of boundedness on the trace-class operators
$\mathfrak{L}^{1}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}$
would imply boundedness for any $1\leq p\leq 2$.

######  Remark 5.5 (Extension to operator-valued symbols).

In many applications, one encounters pseudodifferential operators defined from
_operator-valued_ functions (see e. g. [Cor83, PST03, PST07, FL13]); a
systematic construction of the operator-valued calculus from first principles
can be found in [DLS22]. Combining the ideas from [DLS22] and [LL22] should
give a pseudodifferential _super_ calculus defined for operator-valued and
equivariant operator-valued Hörmander symbols.

Specifically, the ideas from [DLS22, Section 3.4.3] should apply verbatim and
would allow for an extension of our boundedness result, Theorem 5.3, to the
operator-valued context. The abstract results from Section 3 apply verbatim to
e. g. $L^{2}(\mathbb{R}^{d},\mathcal{H})$, where $\mathcal{H}$ is some
separable Hilbert space. The main technical result, Theorem 4.1, should also
extend, following the strategy and arguments in [DLS22].

The inclusion $S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})\subseteq
S_{0,0}^{0}(\Xi^{2})$ for $m_{L},m_{R}\leq 0$ allows us to apply Theorem 5.3
also to double Hörmander classes from Definition 2.5 and obtain an boundedness
result for them.

######  Corollary 5.6.

Suppose that $F\in S_{\rho,0}^{m_{L},m_{R}}(\Xi^{2})$, $m_{L},m_{R}\leq 0$,
with $0\leq\rho\leq 1$. Then $\mathrm{Op}^{A}(F)$ defines a bounded super
operator
$\mathrm{Op}^{A}(F)\in\mathcal{B}\bigl{(}\mathfrak{L}^{2}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\bigr{)}$
on the Hilbert space of Hilbert-Schmidt operators.

###  5.2 More advanced boundedness results

Let us give a brief outlook on how to obtain more advanced boundedness results
for super operators from Theorem 5.3 and Corollary 5.6.

One that should be useful in applications is an extension from
$\mathfrak{L}^{2}$ to non-commutative magnetic Sobolev spaces. We first
revisit the case of magnetic pseudodifferential operators: if $f\in
S^{m}_{\rho,0}(\Xi)$ is a Hörmander symbol of order $m$, then the associated
magnetic seudodifferential operator $\mathrm{op}^{A}(f)$ defines a bounded
operator

$\displaystyle\mathrm{op}^{A}(f):H^{s}_{A}(\mathbb{R}^{d})\longrightarrow
H^{s-m}_{A}(\mathbb{R}^{d})$

between magnetic Sobolev spaces for any $s\in\mathbb{R}$ (cf. [IMP07,
Proposition 4.12]). Rather than use the standard magnetic Sobolev norm, we
instead use an equivalent norm defined from

$\displaystyle w_{m}(x,\xi):=\langle\xi\rangle^{m}+\lambda(m)$

for $m\geq 0$. To extend this definition to negative orders, we choose the
constant $\lambda(m)>0$ large enough so that the inverse

$\displaystyle
w_{-m}=\bigl{(}\langle\xi\rangle^{m}+\lambda(m)\bigr{)}^{(-1)_{\star^{B}}}$

with respect to the magnetic Weyl product $\star^{B}$ exists (cf. [IMP10,
Proposition 6.31] or [MPR07, Theorem 1.8]). Importantly, $w_{m}\in
S^{m}_{1,0}(\Xi)$ is a Hörmander symbol of order $m\in\mathbb{R}$ and
$\bigl{\lVert}\mathrm{op}^{A}(w_{m})\psi\bigr{\rVert}_{L^{2}(\mathbb{R}^{d})}$
is an equivalent norm for the Banach space $H^{m}_{A}(\mathbb{R}^{d})$.
Consequently, we have proven the following little

######  Lemma 5.7.

$\hat{f}^{A}\in\mathcal{B}\bigl{(}H^{m}_{A}(\mathbb{R}^{d})\,,\,H^{s}_{A}(\mathbb{R}^{d})\bigr{)}$
is a bounded operator between magnetic Sobolev spaces of orders
$m\in\mathbb{R}$ and $s\in\mathbb{R}$ if and only if

$\displaystyle\mathrm{Op}^{A}\bigl{(}w_{-s}\otimes
w_{-m}\bigr{)}\,\hat{f}^{A}$
$\displaystyle=\mathrm{op}^{A}(w_{-s})\,\hat{f}^{A}\,\mathrm{op}^{A}(w_{-m})\in\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}$

defines a bounded operator on $L^{2}(\mathbb{R}^{d})$.

The other piece we will need is a notion of $p$-Schatten classes
$\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})\bigr{)}$
for operators that map between two different separable Hilbert spaces. While
the ideas are all contained in [Tre67], a short, guided introduction to the
topic can be found in [DLS22, Appendix B]. For the present purposes it
suffices to note that we may set

$\displaystyle\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})\bigr{)}:=\Bigl{\\{}\hat{f}\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})\;\;\big{|}\;\;\bigl{\lvert}\hat{f}\bigr{\rvert}\in\mathfrak{L}^{p}\bigl{(}\mathcal{B}(\mathcal{H}_{1})\bigr{)}\Bigr{\\}}$

where we have defined the absolute value as
$\bigl{\lvert}\hat{f}\bigr{\rvert}:=\sqrt{\hat{f}^{*}\,\hat{f}}$.
Equivalently, we could have put the adjoint operator to the right and base our
definition on the absolute value
$\bigl{\lvert}\hat{f}\bigr{\rvert}:=\sqrt{\hat{f}\,\hat{f}^{*}}$.

That being said, we can now exploit Lemma 5.7 to derive boundedness results
for magnetic pseudodifferential super operators.

######  Corollary 5.8.

Suppose $F\in S^{m_{L},m_{R}}_{\rho,0}(\Xi^{2})$ is a double Hörmander symbol
of orders $m_{L},m_{R}\in\mathbb{R}$ with $0\leq\rho\leq 1$. Then
$\mathrm{Op}^{A}(F)$ defines a bounded operator

$\displaystyle\mathrm{Op}^{A}(F)\in\mathcal{B}\Bigl{(}\mathfrak{L}^{2}\bigl{(}H^{m_{\mathrm{i},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{i},L}}_{A}(\mathbb{R}^{d})\bigr{)}\,,\,\mathfrak{L}^{2}\bigl{(}H^{m_{\mathrm{f},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{f},L}}_{A}(\mathbb{R}^{d})\bigr{)}\Bigr{)}$
(5.4)

whenever the coefficients satisfy

$\displaystyle m_{L}$ $\displaystyle\leq m_{\mathrm{i},L}+m_{\mathrm{f},L},$
$\displaystyle m_{R}$ $\displaystyle\leq m_{\mathrm{i},R}+m_{\mathrm{f},R}.$

######  Proof.

First of all, with the help of Lemma 5.7 we can reformulate equation (5.4) as

$\displaystyle\mathrm{Op}^{A}\bigl{(}w_{-m_{\mathrm{f},L}}\otimes
w_{-m_{\mathrm{f},R}}\bigr{)}\,\mathrm{Op}^{A}(F)\,\mathrm{Op}^{A}\bigl{(}w_{-m_{\mathrm{i},L}}\otimes
w_{-m_{\mathrm{i},R}}\bigr{)}=$
$\displaystyle\qquad\qquad=\mathrm{Op}^{A}\Bigl{(}\bigl{(}w_{-m_{\mathrm{f},L}}\otimes
w_{-m_{\mathrm{f},R}}\bigr{)}\sharp^{B}F\sharp^{B}\bigl{(}w_{-m_{\mathrm{i},L}}\otimes
w_{-m_{\mathrm{i},R}}\bigr{)}\Bigr{)}\in\mathcal{B}\Bigl{(}\mathfrak{L}^{p}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\Bigr{)}$

being a bounded operator on the standard $p$-Schatten class for the Hilbert
space $L^{2}(\mathbb{R}^{d})$.

We know from [LL22, Proposition VI.5 (2)] that the right-hand side is a
magnetic pseudodifferential operator defined by a symbol from the double
symbol class of orders
$\bigl{(}m_{L}-m_{\mathrm{i},L}-m_{\mathrm{f},L}\,,\,m_{R}-m_{\mathrm{i},R}-m_{\mathrm{f},R}\bigr{)}$.
Combining this with Corollary 5.6 yields the conditions

$\displaystyle m_{L}-m_{\mathrm{i},L}-m_{\mathrm{f},L}$ $\displaystyle\leq 0,$
$\displaystyle m_{R}-m_{\mathrm{i},R}-m_{\mathrm{f},R}$ $\displaystyle\leq 0,$

which are equivalent to the ones stated in the proof. □

By the same token, we get a similar statement for regular Hörmander classes on
$\Xi^{2}$ via Theorem 5.3:

######  Corollary 5.9.

Suppose $F\in S^{m}_{\rho,0}(\Xi^{2})$ is a double Hörmander symbol of orders
$m\in\mathbb{R}$ with $0\leq\rho\leq 1$. Then $\mathrm{Op}^{A}(F)$ defines a
bounded operator

$\displaystyle\mathrm{Op}^{A}(F)\in\mathcal{B}\Bigl{(}\mathfrak{L}^{2}\bigl{(}H^{m_{\mathrm{i},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{i},L}}_{A}(\mathbb{R}^{d})\bigr{)}\,,\,\mathfrak{L}^{2}\bigl{(}H^{m_{\mathrm{f},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{f},L}}_{A}(\mathbb{R}^{d})\bigr{)}\Bigr{)}$
(5.5)

whenever the coefficients satisfy

$\displaystyle m$ $\displaystyle\leq m_{\mathrm{i},L}+m_{\mathrm{f},L},$
$\displaystyle m$ $\displaystyle\leq m_{\mathrm{i},R}+m_{\mathrm{f},R}.$

######  Remark 5.10.

The limitation to the case $p=2$ is entirely due to the absence of boundedness
results akin to Theorem 5.3 and Corollary 5.6 for $p\neq 2$. However, _if_ one
can prove that

$\displaystyle\mathrm{Op}^{A}(F)\in\mathcal{B}\bigl{(}\mathfrak{L}^{p}\bigl{(}\mathcal{B}\bigl{(}L^{2}(\mathbb{R}^{d})\bigr{)}\bigr{)}\bigr{)}$

holds true for some $F\in S^{0,0}_{0,0}(\Xi^{2})$ or $F\in
S^{0}_{0,0}(\Xi^{2})$ for $p\neq 2$, then the above arguments immediately
imply boundedness in the sense that

$\displaystyle\mathrm{Op}^{A}(F)\in\mathcal{B}\Bigl{(}\mathfrak{L}^{p}\bigl{(}H^{m_{\mathrm{i},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{i},L}}_{A}(\mathbb{R}^{d})\bigr{)}\,,\,\mathfrak{L}^{p}\bigl{(}H^{m_{\mathrm{f},R}}_{A}(\mathbb{R}^{d})\,,\,H^{m_{\mathrm{f},L}}_{A}(\mathbb{R}^{d})\bigr{)}\Bigr{)}$

where the $m_{\mathrm{i},L/R}$ and $m_{\mathrm{f},L/R}$ satisfy the same
conditions as in Corollaries 5.8 or 5.9 above.

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