# Landau levels on a compact manifold

L. Charles

###### Abstract

We consider a magnetic Laplacian on a compact manifold, with a constant non-
degenerate magnetic field. In the large field limit, it is known that the
eigenvalues are grouped in clusters, the corresponding sums of eigenspaces
being called the Landau levels. The first level has been studied in-depth as a
natural generalization of the Kähler quantization. The current paper is
devoted to the higher levels: we compute their dimensions as Riemann-Roch
numbers, study the associated Toeplitz algebras and prove that each level is
isomorphic with a quantization twisted by a convenient auxiliary bundle.

##  1 Introduction

The main theme of this paper and its companion [10] is the Landau levels of
compact manifolds. For a physicist, the Landau quantization refers to a
charged particle confined to two dimensions and exposed to a magnetic field.
It has discrete energy levels connected by ladder operators. Besides the
planar geometry considered by Landau [18], the case of Riemann surfaces has
been investigated in the context of the quantum Hall effect, see [15] and
references therein.

On a mathematical point of view, a natural generalisation is the Bochner
Laplacian acting on the sections of a Hermitian line bundle $L$ on a compact
manifold. This Laplacian is defined from two data: a Riemannian metric of the
base and a connection of the line bundle. The idea underlying this work is
that when the curvature of the connection is non-degenerate and large with
respect to the metric, the spectrum of the Laplacian exhibits a structure
similar to the Landau quantization.

More specifically, let us assume that the curvature is related to the
Riemannian metric by a complex structure, and consider the spectrum of the
Laplacian of $L^{k}$ in the large $k$ limit. In this setting, Faure-Tsujii
[13] have shown that the eigenvalues are grouped in clusters, each of them
representing a generalised Landau level. The first level was previously
identified by Guillemin-Uribe [14] and studied further by Borthwick-Uribe [2]
as a generalization of Kähler quantization. In particular, its dimension is
given by a Riemann-Roch number and it comes with an algebra of Toeplitz
operators quantizing the classical Poisson algebra.

Our goal in this paper is to extend these results to the higher Landau levels.
Our main results are:

  1. 1.

the dimension of the $m$-th Landau level is the Riemann-Roch number of
$L^{k}\otimes F_{m}$ when $k$ is sufficiently large, where $F_{m}$ is the
symmetric $m$-th power of the complex tangent bundle of the base.

  2. 2.

there is an algebra of Berezin-Toeplitz operators associated to the $m$-th
Landau level, the symbol of these operators being sections of the endomorphism
bundle $\operatorname{End}F_{m}$.

  3. 3.

the $m$-th Landau level is isomorphic with the first Landau level twisted by
$F_{m}$ through a ladder operator, these isomorphisms are compatible with the
Berezin-Toeplitz operators.

The main ingredient to establish these results is an asymptotic expansion of
the Schwartz kernel of the spectral projector of each level. For the first
level, when the complex structure is integrable, the Kähler case, this kernel
is the Szegö kernel. Its asymptotic is well-understood since the seminal work
by Boutet de Monvel and Sjöstrand [5] and has been used in numerous papers
starting from [4], [1], [27]. In the non-Kähler case, the asymptotics of the
first level projector kernel has been obtained by Borthwick-Uribe [3] and Ma-
Marinescu [21]. For the higher Landau levels, this asymptotic expansion will
be the main result of our second paper [10].

In the current paper, we will rely on this asymptotic expansion or more
generally we will show that the previous results hold for higher Landau levels
defined as the image of any projector whose Schwartz kernel has the convenient
asymptotics. Here the inspiration is the generalised Toeplitz structure of
Boutet de Monvel-Guillemin [4] and our previous work [9], the idea being that
the only important feature of the Landau levels is this asymptotic expansion.

The main tool we will use for the proofs is a particular class of operators
containing the Landau level projectors, the associated Toeplitz operators and
also the generalised ladder operators. The operators in this class are
controlled at first order by their symbols, which are defined as sections of a
bundle of non-commutative algebras. Each of these algebras is generated by the
spectral projectors and ladder operators of a Landau Hamiltonian. By this
mechanism, the basic properties of the Landau quantization are transferred to
the Bochner Laplacian.

To finish this general introduction, let us mention the two contemporaneous
papers [17], [16] by Yuri Kordyukov on the same subject. Let us mention as
well that in a related but different context, belonging to homogeneous
microlocal analysis instead of semi-classical analysis, Boutet de Monvel-
Guillemin [4, Chapter 15] and Epstein-Melrose [12, Chapter 6] have considered
generalised Szegö projections at higher level with associated Toeplitz
algebras, which are similar to our constructions.

###  1.1 Magnetic Laplacian

#### Constant magnetic intensity

Consider a Riemannian manifold $(M,g)$ with a Hermitian line bundle $L$
equipped with a connection $\nabla$. Associated to these data is a Laplacian
$\frac{1}{2}\nabla^{*}\nabla$ acting on ${\mathcal{C}}^{\infty}(M,L)$, which
from the physical point of view is a Schrödinger operator with a magnetic
field $\Omega=i\operatorname{curv}(\nabla)\in\Omega^{2}(M,{\mathbb{R}})$.

We will assume that $\Omega$ is non-degenerate at each point and has a
constant magnetic intensity with respect to $g$ in the following sense. In the
case where $M$ is a surface, the magnetic intensity is the positive function
defined by $|\Omega|=B\operatorname{vol}_{g}$, where $\operatorname{vol}_{g}$
is the Riemannian volume, and we merely assume that $B$ is constant. In higher
dimension, $\Omega$ being non-degenerate, the dimension of $M$ is even, say
$2n$. At any $p\in M$, there exists a skew-symmetric endomorphism $j_{B}(p)$
of $(T_{p}M,g_{p})$ such that $\Omega_{p}(X,Y)=g_{p}(j_{B}(p)X,Y)$. The
eigenvalues of $j_{B}(p)$ are $\pm iB_{\ell}(p)$ with
$0<B_{1}(p)\leqslant\ldots\leqslant B_{n}(p)$. We assume that these
eigenvalues are all equal, $B_{1}=\ldots=B_{n}$, and do not depend on $p$.
Equivalently $j_{B}(p)=Bj(p)$ with $B$ a positive constant and $j$ an almost
complex structure of $M$ compatible with $g$, cf. Proposition 2.5.6. of [22].

So we have that $\Omega=B{\omega}$ where $B>0$ is constant and ${\omega}$ is a
symplectic form of $M$ defined by ${\omega}(X,Y)=g(jX,Y)$. We will consider
the large $B$ limit. To do this, we will replace $L$ by $L^{k}$,
$k\in{\mathbb{N}}$, so that the curvature of $\nabla^{L^{k}}$ is $kB{\omega}$,
and let $k$ tend to infinity. We will also normalise the metric so that $B=1$,
and our magnetic intensity is simply $k$.

Alternatively, we can introduce our data as follows. Consider a compact
symplectic manifold $(M^{2n},{\omega})$ with a compatible almost complex
structure $j$ and a Hermitian line bundle $L\rightarrow M$ with a connection
$\nabla$ having curvature $\frac{1}{i}{\omega}$. Such a bundle is called a
prequantum bundle in the Kostant-Souriau theory, where it is used to define
the geometric quantization of $M$. For any positive integer $k$, we consider
the Laplacian

$\displaystyle\Delta_{k}=\tfrac{1}{2}(\nabla^{L^{k}})^{*}\nabla^{L^{k}}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k})$
(1)

with
$\nabla^{L^{k}}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow\Omega^{1}(M,L^{k})$
the covariant derivative induced by $\nabla$, and the Riemannian metric
$g(X,Y)={\omega}(jX,Y)$ independent of $k$.

#### Earlier results

It is known that the spectrum ${\sigma}(\Delta_{k})$ of $\Delta_{k}$ is
partitioned into clusters around each point of $k(\frac{n}{2}+{\mathbb{N}})$
in the large $k$ limit. More precisely, for any $m\in{\mathbb{N}}$, define the
interval $I_{m}$

$I_{0}=[0,\tfrac{n}{2}+\tfrac{1}{2}],\qquad
I_{m}=(\tfrac{n}{2}+m)+[-\tfrac{1}{2},\tfrac{1}{2}[\text{ if }m\geqslant 1,$

so that we have a partition $[0,\infty[=\bigcup_{m\in{\mathbb{N}}}I_{m}$. Then
we set

$\displaystyle{\Sigma}_{m,k}:=\bigl{(}k^{-1}{\sigma}(\Delta_{k})\bigr{)}\cap
I_{m},\qquad\mathcal{H}_{m,k}:=\bigoplus_{{\lambda}\in{\Sigma}_{m,k}}\ker(k^{-1}\Delta_{k}-{\lambda}).$
(2)

It was proved by Faure-Tsuji [13] that

$\displaystyle{\Sigma}_{m,k}\subset\Bigl{(}\tfrac{n}{2}+m+C_{m}k^{-\frac{1}{4}}[-1,1]\Bigr{)}$
(3)

and by Demailly [11] that

$\displaystyle\operatorname{dim}\mathcal{H}_{m,k}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}{{m+n-1}\choose{n-1}}\operatorname{vol}(M)+\operatorname{o}(k^{n})$
(4)

For a surface $(n=1)$ with a constant Gauss curvature $S$, more precise
results have been obtained by Iengo-Li [15]: if $k+mS>0$, then

$\displaystyle\begin{split}{\Sigma}_{m,k}=\bigl{\\{}\tfrac{1}{2}+m+k^{-1}S\tfrac{m(m+1)}{2}\bigr{\\}},\\\
\operatorname{dim}\mathcal{H}_{m,k}=\tfrac{k}{2\pi}\operatorname{vol}(M)+(\tfrac{1}{2}+m)\chi(M).\end{split}$
(5)

So in this case, when $k$ is sufficiently large, the $m$-th eigenvalue is
degenerate with multiplicity equal to $\operatorname{dim}\mathcal{H}_{m,k}$.

The first cluster has been further studied. In the Kähler case, that is when
the complex structure $j$ is integrable, $L$ has itself a natural holomorphic
structure such that $\overline{\partial}_{L}=\nabla^{0,1}$ and by Kodaira
identities, we have when $k$ is sufficiently large that

$\displaystyle{\Sigma}_{0,k}=\\{\tfrac{n}{2}\\},\qquad\mathcal{H}_{0,k}=H^{0}(M,L^{k}).$
(6)

and the dimension of $\mathcal{H}_{0,k}$ is given by the Riemann-Roch-
Hirzebruch Theorem

$\displaystyle\dim\mathcal{H}_{0,k}=\int_{M}\exp\Bigl{(}\frac{k{\omega}}{2\pi}\Bigr{)}\;\operatorname{Todd}M$
(7)

Here $\operatorname{Todd}M$ is the Todd class of $(M,j)$.

More generally, when $j$ is not necessarily integrable, it was proved by
Guillemin-Uribe [14] that

$\displaystyle{\Sigma}_{0,k}\subset\Bigl{(}\tfrac{n}{2}+C_{0}k^{-1}[-1,1]\Bigr{)}$
(8)

and by Borthwick-Uribe [2] that the dimension of $\mathcal{H}_{0,k}$ is given
by (7) when $k$ is sufficiently large.

###  1.2 Main results

In the sequel $m\in{\mathbb{N}}$ is a fixed non negative integer and all the
results holds in the large $k$ limit, with estimates, bounds depending on $m$.

#### Dimension

Our first result is the computation of the dimension of $\mathcal{H}_{m,k}$,
as the Riemann-Roch number of $L^{k}\otimes\mathcal{D}_{m}(TM)$ where
$\mathcal{D}_{m}(TM)$ is the $m$-th symmetric power of $(T^{0,1}M)^{*}$. Here,
$T^{0,1}M=\ker(j+i)$ and $j$ is the almost complex structure introduced
previously. The reason why we prefer to work with $(T^{0,1}M)^{*}$ instead of
the isomorphic bundle $T^{1,0}M$ should be clear later.

######  Theorem 1.1.

If $k$ is sufficiently large, then

$\operatorname{dim}\mathcal{H}_{m,k}=\int_{M}\exp\Bigl{(}\frac{k{\omega}}{2\pi}\Bigr{)}\;\operatorname{ch}(\mathcal{D}_{m}(TM)))\operatorname{Todd}M$

with $\operatorname{ch}$ the Chern character and $\operatorname{Todd}M$ the
Todd class of $(M,j)$.

As far as we know, Theorem 1.1 is a new result, except in the cases already
mentionned ($n=1$ with constant curvature or $m=0$).

######  Remark 1.2.

  * -

When $n=1$, $\mathcal{D}_{m}(TM)$ is isomorphic with $K^{-m}$, $K$ being the
canonical bundle, and it is easy to see that we recover the second equation
(5). However, even for $n=1$, Theorem 1.1 goes further since we don’t assume
that the Gauss curvature is constant. In this generality, it is not likely
that ${\Sigma}_{m,k}$ consists of a single degenerate eigenvalue, but the
dimension of the $m$-th cluster is given by the same formula.

  * -

For a general dimension $n$, $\mathcal{D}_{m}(TM)$ has rank
${{m+n-1}\choose{n-1}}$ and we recover the asymptotic (4).

#### Symbol spaces

In the sequel, we will use $\mathcal{D}_{m}(TM)$ as a bosonic space, with
associated creation an annihilation operators defined as follows. For any
$x\in M$, let us view $\mathcal{D}_{m}(T_{x}M)$ as the space of homogeneous
polynomials maps $T_{x}^{0,1}M\rightarrow{\mathbb{C}}$ with degree $m$. Set

$\displaystyle\mathcal{D}(T_{x}M):=\bigoplus_{m\in{\mathbb{N}}}\mathcal{D}_{m}(T_{x}M)$
(9)

and let $\pi_{m}(x)$ be the corresponding projector of $\mathcal{D}(T_{x}M)$
onto $\mathcal{D}_{m}(T_{x}M)$. For any $Y\in T_{x}M\otimes{\mathbb{C}}$, let
$\rho(Y)$ be the endomorphism of $\mathcal{D}(T_{x}M)$ defined as follows.
Write $Y=U+\overline{V}$ with $U,V\in T_{x}^{1,0}M$. Then

$\displaystyle\rho(Y)=\rho(U)+\rho(\overline{V})\;\text{ with
}\;\begin{cases}\rho(U)=\text{ multiplication by }i{\omega}(U,\cdot)\\\
\rho(\overline{V})=\text{ derivation with respect to }\overline{V}\end{cases}$
(10)

More concretely, let $(U_{i})$ be a basis of $T^{1,0}_{x}M$ such that
$\frac{1}{i}{\omega}(U_{i},\overline{U}_{j})=\delta_{ij}$. Let $(z_{i})$ be
the basis of $(T^{1,0}_{x}M)^{*}$ dual to $(U_{i})$, so
$z_{i}=\frac{1}{i}{\omega}(\cdot,\overline{U}_{i})$ and
$\mathcal{D}(T_{x}M)={\mathbb{C}}[\overline{z}_{1},\ldots,\overline{z}_{n}]$.
Then for any polynomial $P$ of the variable $\overline{z}_{1}$, $\ldots$,
$\overline{z}_{n}$

$\displaystyle\rho(U_{i})P=-\overline{z}_{i}P,\qquad\rho(\overline{U}_{i})P=\frac{\partial
P}{\partial\overline{z}_{i}}.$ (11)

So $-\rho(U_{i})$ and $\rho(\overline{U}_{i})$ are respectively the creation
and annihilation operators.

#### Berezin-Toeplitz operators

Our second result is about Berezin-Toeplitz operators. By [2], the spaces
$\mathcal{H}_{0,k}$ can be considered as quantizations of $M$, replacing the
standard Kähler quantization $H^{0}(M,L^{k})$, for symplectic manifold not
necessarily having an integrable complex structure. An important feature is
that there is a natural way to pass from classical to quantum Hamiltonians,
provided by the Berezin-Toeplitz quantization. In the semi-classical limit,
defined here as the large $k$ limit, the product and commutator of quantum
observables correspond to the product and Poisson bracket of classical
observables, up to some error terms. More precisely, let $\Pi_{m,k}$ be the
orthogonal projector of ${\mathcal{C}}^{\infty}(M,L^{k})$ onto
$\mathcal{H}_{m,k}$ and for any $f\in{\mathcal{C}}^{\infty}(M)$, let
$T_{m,k}(f)$ be the endomorphism of $\mathcal{H}_{m,k}$ defined by

$\displaystyle
T_{m,k}(f)\psi=\Pi_{m,k}(f\psi)\qquad\forall\;\psi\in\mathcal{H}_{m,k}.$ (12)

For the first Landau level, it is known [1], [7], [20] that for any $N$

$\displaystyle
T_{0,k}(f)T_{0,k}(g)=\sum_{\ell=0}^{N}k^{-\ell}T_{0,k}(B_{\ell}(f,g))+\mathcal{O}(k^{-(N+1)})$
(13)

for some bidifferential operators
$B_{\ell}:{\mathcal{C}}^{\infty}(M)\times{\mathcal{C}}^{\infty}(M)\rightarrow{\mathcal{C}}^{\infty}(M)$,
where

$\displaystyle B_{0}(f,g)=fg,\qquad
B_{1}(f,g)=-\tfrac{1}{2}g(X,Y)+\tfrac{1}{2i}{\omega}(X,Y),$ (14)

$X$ and $Y$ being the Hamiltonian vector fields of $f$ and $g$ respectively.

For the generalisation to higher Landau levels, we will use in addition to the
$T_{m,k}(f)$’s the following operators: let $p\in{\mathbb{N}}$ and $X_{1}$, …,
$X_{2p}$ be vector fields of $M$. Define
$T_{m,k}(X_{1},\ldots,X_{2p}):\mathcal{H}_{m,k}\rightarrow\mathcal{H}_{m,k}$
by

$\displaystyle
T_{m,k}(X_{1},\ldots,X_{2p})(\Psi)=k^{-p}\Pi_{m,k}(\nabla^{L^{k}}_{X_{1}}\ldots\nabla_{X_{2p}}^{L^{k}}\Psi)$
(15)

Let $\mathcal{T}_{m}$ be the vector space of families
$(P_{k}:\mathcal{H}_{m,k}\rightarrow\mathcal{H}_{m,k},\;k\in{\mathbb{N}})$
spanned by the $(T_{m,k}(f),k\in{\mathbb{N}})$’s and
$(T_{m,k}(X_{1},\ldots,X_{2p}),\;k\in{\mathbb{N}})^{\prime}s$. Here the
functions $f$ or vector fields $X_{1},\ldots X_{2p}$ do not depend on $k$.

As we will see, for any $(P_{k})$ in $\mathcal{T}_{m}$, the operator norm
$\|P_{k}\|$ is bounded independently of $k$. Define the semiclassical
completion $\mathcal{T}_{m}^{\operatorname{sc}}$ as the vector space of
families
$(P_{k}:\mathcal{H}_{m,k}\rightarrow\mathcal{H}_{m,k},\;k\in{\mathbb{N}})$
such that for any $N$,

$P_{k}=\sum_{\ell=0}^{N}k^{-\ell}P_{\ell,k}+\mathcal{O}(k^{-N-1})$

where the coefficients $(P_{\ell,k})_{k}$, $\ell\in{\mathbb{N}}$ all belong to
$\mathcal{T}_{m}$, and the $\mathcal{O}$ is for the operator norm.

######  Theorem 1.3.

For any $m\in{\mathbb{N}}$, we have:

  1. 1.

$\mathcal{T}_{m}^{\operatorname{sc}}$ is closed under product.

  2. 2.

There exists a linear map
$\mathcal{\tau}:\mathcal{T}_{m}^{\operatorname{sc}}\rightarrow{\mathcal{C}}^{\infty}(M,\operatorname{End}(\mathcal{D}_{m}(TM)))$
defined on the generators (12) and (15) by

$\displaystyle\begin{split}\tau(T_{m,k}(f))_{x}&=f(x)\operatorname{id}_{\mathcal{D}_{m}(T_{x}M)},\\\
\tau(T_{m,k}(X_{1},\ldots,X_{2p}))_{x}&=\pi_{m}(x)\rho(X_{1}(x))\ldots\rho(X_{2p}(x))\end{split}$
(16)

$\tau$ is onto, its kernel consists of
$k^{-1}\mathcal{T}_{m}^{\operatorname{sc}}$.

  3. 3.

For any $P,Q\in\mathcal{T}_{m}^{\operatorname{sc}}$

$\displaystyle\tau(PQ)=\tau(P)\tau(Q),$ (17)
$\displaystyle\|P_{k}\|=\sup\\{\|\tau(P)_{x}\|,\;x\in
M\\}+\mathcal{O}(k^{-1}),$ (18) $\displaystyle
P_{k}(x,x)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}(\operatorname{tr}(\tau(P)_{x})+\mathcal{O}(k^{-1})).$
(19)

  4. 4.

For any $f$, $g$ in ${\mathcal{C}}^{\infty}(M)$, we have

$\displaystyle
T_{m,k}(f)T_{m,k}(g)=T_{m,k}(fg)+k^{-1}T_{m,k}(X,Y)+\mathcal{O}(k^{-2})$ (20)

where $X$, $Y$ are the Hamiltonian vector fields of $f$ and $g$. In
particular,

$\displaystyle
ik[T_{m,k}(f),T_{m,k}(g)]=T_{m,k}(\\{f,g\\})+\mathcal{O}(k^{-1})$ (21)

with $\\{\cdot,\cdot\\}$ the Poisson Bracket of $(M,{\omega})$.

We call $\tau$ the symbol map. The symbol of the generators (16) is defined in
terms of the endormorphisms (10). The product of symbols in the left-hand side
of (17) is the pointwise composition. In the norm estimate (18), the norm of
$\tau(P)_{x}$ is defined in terms of the hermitian structure of
$\mathcal{D}_{m}(T_{x}M)$. In (19), $P_{k}(x,x)$ is the value of the Schwartz
kernel of $P_{k}$ at $(x,x)$. Integrating (19), we obtain the following
estimate of the trace of $P_{k}$,

$\displaystyle\operatorname{tr}P_{k}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int_{M}\operatorname{tr}(\tau(P)_{x})\mu_{M}(x)+\mathcal{O}(k^{n-1})$
(22)

where $\mu_{M}={\omega}^{n}/n!$. Since $P_{k}=T_{m,k}(1)$ is the identity of
$\mathcal{H}_{m,k}$, we recover the estimate (4).

The main estimates for the Toeplitz operators associated to functions, that is
$T_{m,k}(f)T_{m,k}(g)=T_{m,k}(fg)+\mathcal{O}(k^{-1})$ and
$ik[T_{m,k}(f),T_{m,k}(g)]=T_{m,k}(\\{f,g\\})+\mathcal{O}(k^{-1})$ have been
proved independently in [16]. An important difference between the first and
the higher Landau levels, which does not appear in [16], is that we use the
Toeplitz operators $T_{m,k}(X_{1},\ldots,X_{2p})$’s in addition to the
$T_{m,k}(f)$’s and the related fact that the symbol is a section of
$\operatorname{End}(\mathcal{D}_{m}(TM))$ instead of a scalar valued function.

In the surface case, $n=1$, $\mathcal{D}_{m}(TM)$ is a line bundle, so any
endomorphism of $\mathcal{D}_{m}(T_{x}M)$ is scalar and as a consequence of
Theorem 1.3, $\mathcal{T}_{m}^{\operatorname{sc}}$ consists of the families
$(P_{k}=T_{m,k}(f(\cdot,k))+\mathcal{O}(k^{-\infty}),\;k\in{\mathbb{N}})$
where the multiplicator $f(\cdot,k)$ depends on $k$ in such a way that it
admits an expansion $f(\cdot,k)=f_{0}+k^{-1}f_{1}+\ldots$. We could as well
prove that the $T_{m,k}(f)$ satisfy (13) for some bidifferential operators
$B_{\ell}^{m}$ depending on $m$.

In the general case, we may ask if there exists a subalgebra of
$\mathcal{T}_{m}^{\operatorname{sc}}$ consisting of Toeplitz operators
associated to scalar multiplicators. This is not the case because, as shows a
short computation, if $X$ and $Y$ are any real vector fields of $M$, by (16),
the symbol of $(T_{m,k}(X,Y))$ at $x\in M$ is the endomorphism of
$\mathcal{D}_{m}(T_{x}M)$ given in terms of a basis $(U_{i})$ of
$T_{x}^{0,1}M$ as above by

$\displaystyle\tau(T_{m}(X,Y))(x)=-\sum_{i,j=1}^{n}\pi_{m}(x)({\alpha}_{i}\overline{{\beta}}_{j}\overline{z}_{i}\partial_{\overline{z}_{j}}+\overline{{\alpha}}_{i}{\beta}_{j}\partial_{\overline{z}_{i}}\overline{z}_{j})\pi_{m}(x)$
(23)

where $X(x)=\sum{\alpha}_{i}U_{i}+\overline{{\alpha}}_{i}\overline{U}_{i}$ and
$Y(x)=\sum{\beta}_{i}U_{i}+\overline{{\beta}}_{i}\overline{U}_{i}$.

Assume that $n\geqslant 2$ and $m\geqslant 1$. Then it is not difficult to see
from (23) that $\tau(T_{m}(X,X))(x)$ is a scalar multiple of $\pi_{m}(x)$ only
when $X(x)=0$. So by the second assertion of Theorem 1.3 and (20), for any
$f\in{\mathcal{C}}^{\infty}(M,{\mathbb{R}})$ not locally constant, we do not
have that
$T_{m,k}(f)T_{m,k}(f)=T_{m,k}(f^{2})+k^{-1}T_{m,k}(g)+\mathcal{O}(k^{-2})$ for
a function $g$.

For $m=0$, we recover from (23) and (20) the formula for $B_{1}$ in (14). For
$n=1$, we have similarly that
$T_{m,k}(f)T_{m,k}(g)=T_{m,k}(fg)+k^{-1}T_{m,k}(B_{1}(f,g))+\mathcal{O}(k^{-2})$
with

$B_{1}(f,g)=-(\tfrac{1}{2}+m)g(X,Y)+\tfrac{1}{2i}{\omega}(X,Y)$

where $X,Y$ are the Hamiltonian vector fields of $f$ and $g$.

#### Ladder operators

The last result we would like to emphasize in this introduction is the
construction of some ladder operators for the spaces $\mathcal{H}_{m,k}$. In
the surface case with constant Gauss curvature, $\mathcal{H}_{m,k}$ is
naturally isomorphic with the space of holomorphic sections of $L^{k}\otimes
K^{-m}$ where $K$ is the canonical bundle [26], the isomorphism being the
ladder operator $\overline{\partial}_{L^{k}\otimes
K^{-m+1}}\circ\ldots\circ\overline{\partial}_{L^{k}}$, cf. the appendix 7.
Here we will show that the family $(\mathcal{H}_{m,k},\;k\in{\mathbb{N}})$ is
isomorphic to a quantization of $M$ twisted by the vector bundle
$\mathcal{D}_{m}(TM)$.

Recall that for any Hermitian vector bundle $F\rightarrow M$, we can define a
family of finite dimensional subspaces
$\mathcal{H}_{F,k}\subset{\mathcal{C}}^{\infty}(M,L^{k}\otimes F)$,
$k\in{\mathbb{N}}$, having the following properties:

  1. 1.

$\operatorname{dim}\mathcal{H}_{F,k}=\int_{M}\operatorname{ch}(L^{k}\otimes
F)\;\operatorname{Todd}M$, when $k$ is sufficiently large.

  2. 2.

the space $\mathcal{T}_{F}^{\operatorname{sc}}$, consisting of families of
$(T_{k}\in\operatorname{End}(\mathcal{H}_{F,k}),\;k\in{\mathbb{N}})$ having an
expansion of the form

$T_{k}=\sum_{\ell=0}^{N}k^{-\ell}T_{F,k}(f_{\ell})+\mathcal{O}(k^{-N+1}),\qquad\forall
N\in{\mathbb{N}}$

for a sequence $(f_{\ell})$ of
${\mathcal{C}}^{\infty}(M,\operatorname{End}F)$, is closed under product.
Here, $T_{F,k}(f_{\ell})(\psi)=\Pi_{F,k}(f_{\ell}\psi)$ for any
$\psi\in\mathcal{H}_{F,k}$ where $\Pi_{F,k}$ is the orthogonal projector of
${\mathcal{C}}^{\infty}(M,L^{k}\otimes F)$ onto $\mathcal{H}_{F,k}$.

  3. 3.

At first order, the product is given by the pointwise product, that is
$T_{F,k}(f)T_{F,k}(g)=T_{F,k}(fg)+\mathcal{O}(k^{-1})$ for any
$f,g\in{\mathcal{C}}^{\infty}(M,\operatorname{End}F)$.

The algebra $\mathcal{T}_{F}^{\operatorname{sc}}$ is called the Toeplitz
algebra. In the Kähler case, that is when $(M,{\omega})$ is Kähler, $L$
holomorphic with $\nabla$ the Chern connection, and $F$ holomorphic as well,
the space $\mathcal{H}_{F,k}$ can be defined as the space $H^{0}(L^{k}\otimes
F)$ of holomorphic sections. In the non-Kähler case, various constructions
have been developed [2], [20]: Spin-c quantization, first Landau level of a
Laplacian acting on ${\mathcal{C}}^{\infty}(M,L^{k}\otimes F)$, or more
generally any image of a projector of ${\mathcal{C}}^{\infty}(M,L^{k}\otimes
F)$ having a specific Schwartz kernel [9]. Let us call such a family
$(\mathcal{H}_{F,k},\;k\in{\mathbb{N}})$ a quantization of $(M,L)$ twisted by
$F$. These twisted quantizations have sometimes better properties than the non
twisted one (corresponding to $F={\mathbb{C}}$), typically when $F$ is a half-
form bundle [8]. The general case where the rank of $F$ is $\geqslant 2$ may
be viewed as a free generalization without any application, but interestingly,
this is exactly what we need.

Assume $F$ is equipped with a connection
$\nabla^{F}:{\mathcal{C}}^{\infty}(M,F)\rightarrow\Omega^{1}(M,F)$. Let
$G=(T^{0,1}M)^{*}$ and $D_{F,k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
F)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes F\otimes G)$ be the
$(0,1)$-part of the connection $\nabla^{F\otimes L^{k}}$ induced by
$\nabla^{F}$ and $\nabla^{L^{k}}$. Endow $G$ with a connection and define the
differential operators

$\displaystyle\begin{split}W_{k}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes\mathcal{D}_{m}(TM))\\\
W_{k}=R_{m}D_{G^{\otimes(m-1)},k}\circ D_{G^{\otimes(m-2)},k}\circ\ldots\circ
D_{G,k}\circ D_{{\mathbb{C}},k}\end{split}$ (24)

where $R_{m}$ is the projection from $G^{\otimes m}$ onto
$D_{m}(TM)=\operatorname{Sym}^{m}G$.

######  Theorem 1.4.

For any quantization $(\mathcal{H}_{F,k},\;k\in{\mathbb{N}})$ of $(M,L)$
twisted by $F=\mathcal{D}_{m}(TM)$, the linear maps

$V_{k}=\tfrac{1}{m!}k^{-\frac{m}{2}}\Pi_{F,k}W_{k}:\mathcal{H}_{m,k}\rightarrow\mathcal{H}_{F,k},\qquad
k\in{\mathbb{N}}$

satisfy:

  1. 1.

$V_{k}V_{k}^{*}=\operatorname{id}_{\mathcal{H}_{F,k}}+\mathcal{O}(k^{-1})$ and
$V_{k}^{*}V_{k}=\operatorname{id}_{\mathcal{H}_{m,k}}+\mathcal{O}(k^{-1})$. In
particular, $V_{k}$ is an isomorphism when $k$ is sufficiently large.

  2. 2.

the conjugation by $V=(V_{k})$ is an isomorphism between the Toeplitz algebra
$\mathcal{T}^{\operatorname{sc}}_{m}$ and
$\mathcal{T}_{F}^{\operatorname{sc}}$ modulo $\mathcal{O}(k^{-\infty})$. In
particular, for any $(P_{k})\in\mathcal{T}^{\operatorname{sc}}_{m}$,
$(V_{k}P_{k}V_{k}^{*})_{k}$ belongs to $\mathcal{T}_{F}^{\operatorname{sc}}$
and if $f\in{\mathcal{C}}^{\infty}(M,\operatorname{End}F)$ is the symbol
$\tau(P_{k})$, then $V_{k}P_{k}V_{k}^{*}=T_{F,k}(f)+\mathcal{O}(k^{-1})$.

The first assertion of Theorem 1.4 tells us that $V_{k}$ is almost unitary.
This can be improved by setting $U_{k}:=A_{k}V_{k}$ with $A_{k}$ the
endomorphism of $\mathcal{H}_{F,k}$ equal to
$(V_{k}V_{k}^{*})^{-1/2}|_{\mathcal{H}_{m,k}}$ when $k$ is sufficiently large
and to $0$ for the first values of $k$. Then
$U_{k}U_{k}^{*}=\operatorname{id}_{\mathcal{H}_{F,k}}$ and
$U_{k}^{*}U_{k}=\operatorname{id}_{\mathcal{H}_{m,k}}$ when $k$ is
sufficiently large. Furthermore the second assertion of 1.4 holds with
$(U_{k})$ instead of $(V_{k})$.

###  1.3 Generalised Landau level and Schwartz kernel expansion

#### Generalised Landau level

In the previous results, the $m$-th Landau level $\mathcal{H}_{m,k}$,
$k\in{\mathbb{N}}$ can be replaced by any family
$(\mathcal{H}_{m,k}\subset{\mathcal{C}}^{\infty}(M,L^{k}),\;k\in{\mathbb{N}})$
of finite dimensional subspaces, such that the Schwartz kernel of the
orthogonal projector $\Pi_{m,k}$ of ${\mathcal{C}}^{\infty}(M,L^{k})$ onto
$\mathcal{H}_{m,k}$ has a specific form in the large $k$ limit. We require
first that

$\displaystyle\Pi_{m,k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)Q^{(n-1)}_{m}(k\delta(x,y))+\mathcal{O}(k^{n-1})$
(25)

where

  * -

$E$ is a section of $L\boxtimes\overline{L}$ such that $|E(x,y)|<1$ when
$x\neq y$, and its second order Taylor expansion along the diagonal has a
specific form, cf. Equation (54). In particular,
$\ln|E(x+\xi,x)|=-\frac{1}{4}|\xi|^{2}+\mathcal{O}(|\xi|^{3})$.

  * -

$\delta\in{\mathcal{C}}^{\infty}(M^{2})$ is any function vanishing to second
order along the diagonal and satisfying
$\delta(x+\xi,x)=|\xi|^{2}+\mathcal{O}(|\xi|^{3})$, $\xi\in T_{x}M$.
$Q_{m}^{(p)}$ is the generalised Laguerre polynomial
$Q^{(p)}_{m}(x)=\frac{x^{-p}}{m!}\bigl{(}\frac{d}{dx}-1\bigr{)}^{m}x^{m+p}$.

In addition to (25), we require a full expansion of the form

$\displaystyle\Pi_{m,k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)\sum_{\ell\in{\mathbb{Z}}}k^{-\ell}a_{\ell}(x,y)+\mathcal{O}(k^{-\infty})$
(26)

with coefficients $a_{\ell}\in{\mathcal{C}}^{\infty}(M^{2})$ such that for
$\ell<0$, $a_{\ell}$ vanishes to order $m(\ell)\geqslant-2\ell$ along the
diagonal and $m(\ell)+2\ell\rightarrow\infty$ as $\ell\rightarrow-\infty$. The
meaning of this expansion is not obvious because the negative $\ell$’s give
positive powers of $k$. Actually, the condition satisfied by $|E|$ implies
that $|E^{k}(x,y)b(x,y)|=\mathcal{O}(k^{-m/2})$ when $b$ vanishes to order $m$
along the diagonal. So the $\ell$-th summand in (26) is in
$\mathcal{O}(k^{n-\frac{1}{2}(m(\ell)+2\ell)})$, and the expansion is
meaningful because of the conditions satisfied by $m(\ell)$.

We will prove that for any family $(\mathcal{H}_{m,k})$ whose associated
projector $\Pi_{m,k}$ satisfies (25) and (26), Theorems 1.1, 1.3 and 1.4 hold.
On the other hand, in the second part of this work [10], cf. also [17], it is
proved that the Schwartz kernel of the orthogonal projector $\Pi_{m,k}$ onto
the Landau levels $\mathcal{H}_{m,k}$ defined in (2) from the Laplacian
$\Delta_{k}$, satisfies (25) and (26). The assumption that the magnetic field
is constant with respect to the metric can be relaxed. It is actually possible
to define some Landau levels and describe the asymptotic expansion of the
associated projector as soon as a particular gap condition is satisfied, cf
[10].

#### The class $\mathcal{L}(A,B)$

To establish our results, we will introduce a specific class of operators,
containing the projector $\Pi_{m,k}$, the Berezin Toeplitz operators
$T_{m,k}(f)$ and $T_{m,k}(X_{1},\ldots,X_{2p})$ and also the projector
$\Pi_{F,k}$ of any twisted quantization, the corresponding Toeplitz operators
$T_{F,k}(g)$, the isomorphisms $V_{k}$ of Theorem 1.4 and their unitarizations
$(U_{k})$. This operator class has a natural filtration, with associated
symbol spaces, which allows to prove most of the results by successive
approximations as often in microlocal analysis. Interestingly, in the symbolic
calculus appear the eigenprojectors of the Laplacian of ${\mathbb{C}}^{n}$,
providing another link between the Landau Laplacian and our geometric Landau
levels.

Introduce two auxiliary Hermitian vector bundles $A$, $B$ over $M$. Then
$\mathcal{L}(A,B)$ consists of families
$(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes B),\;k\in{\mathbb{N}})$ of
operators having a smooth Schwartz kernel satisfying

$\displaystyle
P_{k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)\sum_{\ell\in{\mathbb{Z}}}k^{-\frac{\ell}{2}}b_{\ell}(x,y)+\mathcal{O}(k^{-\infty})$
(27)

where $E$ is defined as in (25); the coefficients $b_{\ell}$ are in
${\mathcal{C}}^{\infty}(M^{2},B\boxtimes\overline{A})$; for $\ell<0$,
$b_{\ell}$ vanishes to order $m(\ell)\geqslant-\ell$ along the diagonal;
$m(\ell)+\ell\rightarrow\infty$ as $\ell\rightarrow-\infty$, and the meaning
of this expansion is the same as in (26). We have a decomposition into
even/odd elements: $(P_{k})\in\mathcal{L}^{+}(A,B)$ (resp.
$\mathcal{L}^{-}(A,B)$) if the expansion (27) holds with a sum over the
$\ell$’s even (resp. odd).

The main property is that this class of operators is closed under composition:

$\displaystyle\mathcal{L}^{{\beta}}(B,C)\cdot\mathcal{L}^{{\alpha}}(A,B)\subset\mathcal{L}^{{\alpha}{\beta}}(A,C),\qquad{\alpha},{\beta}\in\\{\pm
1\\}.$ (28)

In particular, $\mathcal{L}^{+}(A):=\mathcal{L}^{+}(A,A)$ is an algebra. We
also have a filtration
$\mathcal{L}_{q}(A,B):=\mathcal{L}(A,B)\cap\mathcal{O}(k^{-q/2})$,
$q\in{\mathbb{N}}$ and the corresponding graduation is described by symbol
maps ${\sigma}_{q}$

$0\rightarrow\mathcal{L}_{q+1}(A,B)\rightarrow\mathcal{L}_{q}(A,B)\xrightarrow{{\sigma}_{q}}{\mathcal{C}}^{\infty}(M,\mathcal{S}(M)\otimes\operatorname{Hom}(A,B))\rightarrow
0$

Here, $\mathcal{S}(M)$ is an infinite rank vector bundle over $M$, each fiber
$\mathcal{S}_{x}(M)$ is a subalgebra of the algebra of endomorphisms of the
space $\mathcal{D}(T_{x}M)$ defined in (9). This is compatible with the
composition (28) in the sense that
$\mathcal{L}_{p}(B,C)\cdot\mathcal{L}_{q}(A,B)\subset\mathcal{L}_{p+q}(A,C)$
and the corresponding product of symbols is the pointwise product of
$\mathcal{S}(M)$ tensored by
$\operatorname{Hom}(B,C)\otimes\operatorname{Hom}(A,B)\rightarrow\operatorname{Hom}(A,C)$.

The projector $(\Pi_{m,k})_{k}$ is an idempotent of
$\mathcal{L}^{+}({\mathbb{C}})$ with symbol ${\sigma}_{0}(\Pi_{m})$ equal at
$x$ to the projector $\pi_{m}(x)$ onto the $m$-th summand in (9). The Toeplitz
algebra introduced previously is

$\displaystyle\mathcal{T}_{m}^{\operatorname{sc}}=\\{P\in\mathcal{L}^{+}({\mathbb{C}})/\;\Pi_{m}P\Pi_{m}=P\\}.$
(29)

The isomorphism $V$ of Theorem 1.4 belongs to $\mathcal{L}({\mathbb{C}},F)$
and has the same parity as $m$.

Interestingly, $\mathcal{S}_{x}(M)$ has a representation as operators of
$L^{2}({\mathbb{C}}^{n})$, and in this representation,
$\pi_{m}(x)={\sigma}_{0}(\Pi_{m})(x)$ is the projector onto the $m$-th Landau
level of a magnetic Laplacian of ${\mathbb{C}}^{n}$.

#### Outline of the paper

In Section 2, we introduce the class $\mathcal{L}(A,B)$, and state its main
properties. In section 3, we prove variations of the theorems stated before,
where the $\mathcal{H}_{m,k}$ are subspaces of
${\mathcal{C}}^{\infty}(M,L^{k}\otimes A)$ such that the corresponding family
$(\Pi_{m,k})$ of orthogonal projectors belongs to $\mathcal{L}^{+}(A)$ with a
convenient symbol. Sections 4, 5 and 6 are devoted to the proof of the
properties of $\mathcal{L}(A,B)$. The proofs of Theorems 1.1, 1.3 and 1.4 is
given in the last subsection 6.3. In the appendix 7, we prove formulas (5) on
constant curvature surface.

#### Acknowledgment

I would like to thank Yuri Kordyukov for his collaboration at an early stage
of this work.

##  2 The class $\mathcal{L}(A,B)$

We start the discussion with the algebra in which the symbol of operators of
$\mathcal{L}(A,B)$ takes their values. The class $\mathcal{L}(A,B)$ is defined
in Subsection 2.2.

###  2.1 Symbol spaces

#### The algebra $\mathcal{S}({\mathbb{C}}^{n})$

Let $n$ be a positive integer and denote by $z_{1},\ldots,z_{n}$ the linear
coordinates of ${\mathbb{C}}^{n}$. Let
$\mathcal{D}({\mathbb{C}}^{n})={\mathbb{C}}[\overline{z}_{1},\ldots,\overline{z}_{n}]$
be the space of antiholomorphic polynomial maps from ${\mathbb{C}}^{n}$ to
${\mathbb{C}}$. Introduce the scalar product

$\displaystyle\langle
f,g\rangle=(2\pi)^{-n}\int_{{\mathbb{C}}^{n}}e^{-|z|^{2}}f(z)\,\overline{g(z)}\;d\mu_{n}(z),\qquad
f,\;g\in\mathcal{D}({\mathbb{C}}^{n})$ (30)

where $|z|^{2}=\sum_{i=1}^{n}|z_{i}|^{2}$ and $\mu_{n}$ is the measure
$\prod_{i=1}^{n}dz_{i}d\overline{z}_{i}$. The family
$(({\alpha}!)^{-\frac{1}{2}}\overline{z}^{\alpha},\,{\alpha}\in{\mathbb{N}}^{n})$
is an orthonormal basis of $\mathcal{D}({\mathbb{C}}^{n})$. We will also need
the decomposition into even and odd functions

$\displaystyle\mathcal{D}({\mathbb{C}}^{n})=\mathcal{D}^{+}({\mathbb{C}}^{n})\oplus\mathcal{D}^{-}({\mathbb{C}}^{n})$
(31)

where $\mathcal{D}^{+}({\mathbb{C}}^{n})$ is spanned by the
$\overline{z}^{\alpha}$ with $|{\alpha}|=\sum{\alpha}(i)$ even and
$\mathcal{D}^{-}({\mathbb{C}}^{n})$ by the $\overline{z}^{\alpha}$ with
$|{\alpha}|$ odd.

Let $\mathcal{S}({\mathbb{C}}^{n})$ be the space of endomorphisms $s$ of
$\mathcal{D}({\mathbb{C}}^{n})$ such that $s(\overline{z}^{\alpha})=0$ except
for a finite number of ${\alpha}\in{\mathbb{N}}^{n}$. We claim that
$\mathcal{S}({\mathbb{C}}^{n})$ is closed under product and taking adjoint. To
see that, simply observe that $\mathcal{S}({\mathbb{C}}^{n})$ is the space of
endomorphisms having a matrix in the basis $(\overline{z}^{\alpha})$ whose
almost all entries are equal to zero. Notice as well that the family
$(\rho_{{\alpha},{\beta}},\;{\alpha},{\beta}\in{\mathbb{N}}^{n})$ of
$\mathcal{S}({\mathbb{C}}^{n})$ defined by

$\displaystyle\begin{split}\rho_{{\alpha}{\beta}}\bigl{(}({\beta}!)^{-\frac{1}{2}}\overline{z}^{\beta}\bigr{)}=({\alpha}!)^{-\frac{1}{2}}\overline{z}^{{\alpha}},\\\
\rho_{{\alpha}{\beta}}(\overline{z}^{{\gamma}})=0,\quad\forall{\gamma}\in{\mathbb{N}}^{n}\setminus\\{{\beta}\\}\end{split}$
(32)

is a vector space basis of $\mathcal{S}({\mathbb{C}}^{n})$. And we have

$\displaystyle\rho_{{\alpha}{\beta}}\circ\rho_{\tilde{\alpha}\tilde{\beta}}=\delta_{{\beta}\tilde{{\alpha}}}\rho_{{\alpha}\tilde{\beta}},\qquad\rho_{{\alpha}{\beta}}^{*}=\rho_{{\beta}{\alpha}}$
(33)

for all ${\alpha}$, ${\beta}$, $\tilde{\alpha}$, $\tilde{\beta}$ in
${\mathbb{N}}^{n}$.

Each element $s\in\mathcal{S}({\mathbb{C}}^{n})$ can be written in a block
matrix $s=\begin{pmatrix}s_{++}&s_{-+}\\\ s_{+-}&s_{--}\end{pmatrix}$ in the
decomposition (31), which leads to a decomposition into even/odd endomorphisms

$\displaystyle\mathcal{S}({\mathbb{C}}^{n})=\mathcal{S}^{+}({\mathbb{C}}^{n})\oplus\mathcal{S}^{-}({\mathbb{C}}^{n})$
(34)

where $s\in\mathcal{S}^{+}({\mathbb{C}}^{n})$ iff $s_{-+}=s_{+-}=0$, and
$s\in\mathcal{S}^{-}({\mathbb{C}}^{n})$ iff $s_{++}=s_{--}=0$. Observe that
$\rho_{{\alpha}{\beta}}$ has the same parity as $|{\alpha}|+|{\beta}|$.
Furthermore

$\displaystyle\mathcal{S}^{\epsilon}({\mathbb{C}}^{n})\,\cdot\,\mathcal{S}^{\epsilon^{\prime}}({\mathbb{C}}^{n})\subset\mathcal{S}^{\epsilon\epsilon^{\prime}}({\mathbb{C}}^{n})$
(35)

for any $\epsilon,\epsilon^{\prime}\in\\{1,-1\\}$.

#### Extension to vector bundles

In the previous definitions, we can replace ${\mathbb{C}}^{n}$ by any
$n$-dimensional Hermitian vector space $\mathbf{E}$. We denote by
$\mathcal{D}(\mathbf{E})$ the space of antiholomorphic polynomial maps
$\mathbf{E}\rightarrow{\mathbb{C}}$. Choosing an orthonormal basis $(e_{i})$
of $\mathbf{E}$, we can identify $\mathbf{E}$ with ${\mathbb{C}}^{n}$ and then
define the scalar product of $\mathcal{D}(\mathbf{E})$ by the formula (30).
Since the weight $|z|^{2}$ and the measure $d\mu_{n}$ are invariant by unitary
change of coordinates, the resulting scalar product of
$\mathcal{D}(\mathbf{E})$ is independent of $(e_{i})$. Similarly, we define
the subspace $\mathcal{S}(\mathbf{E})$ of the space of endomorphisms of
$\mathcal{D}(\mathbf{E})$ and associated to the basis $(e_{i})$ of
$\mathbf{E}$, we have a basis
$(\rho_{{\alpha},{\beta}},\;{\alpha},{\beta}\in{\mathbb{N}}^{n})$ of
$\mathcal{S}(\mathbf{E})$. The decompositions into even/odd elements are
defined and denoted as for ${\mathbb{C}}^{n}$ by

$\displaystyle\mathcal{D}(\mathbf{E})=\mathcal{D}^{+}(\mathbf{E})\oplus\mathcal{D}^{-}(\mathbf{E}),\qquad\mathcal{S}(\mathbf{E})=\mathcal{S}^{+}(\mathbf{E})\oplus\mathcal{S}^{-}(\mathbf{E}).$
(36)

We can extend all these constructions to vector bundles. Let
$\mathbf{E}\rightarrow M$ be a Hermitian vector bundle with rank $n$. Define
the infinite-dimensional vector bundles $\mathcal{D}(\mathbf{E})$ and
$\mathcal{S}(\mathbf{E})$ over $M$ with fibers
$\mathcal{D}(\mathbf{E})_{x}=D(\mathbf{E}_{x})$ and
$\mathcal{S}(\mathbf{E})_{x}=\mathcal{S}(\mathbf{E}_{x})$. Later, we will
choose for $\mathbf{E}$ the complex tangent bundle of an almost-complex
manifold, and we will construct operator whose symbols are smooth sections of
$\mathcal{S}(\mathbf{E})\otimes A$, where $A$ is an auxiliary vector bundle.
Since the bundle $\mathcal{S}(\mathbf{E})$ has infinite rank, let us make
precise the definition of its smooth sections: a section
$s\in{\mathcal{C}}^{\infty}(M,\mathcal{S}(\mathbf{E})\otimes A)$ is a family
$(s(x)\in\mathcal{S}(\mathbf{E}_{x})\otimes A_{x},\;x\in M)$ such that for any
orthonormal frame $(e_{i})$ of $\mathbf{E}$ and $(a_{j})$ of $A$ over the same
open set $U$ of $M$, if $(\rho_{{\alpha},{\beta}}(x))$ is the basis of
$\mathcal{S}(\mathbf{E}_{x})$ associated to the basis $(e_{i}(x))$, then

$s(x)=\sum{\lambda}_{{\alpha},{\beta},j}(x)\,\rho_{{\alpha},{\beta}}(x)\otimes
a_{j}(x),\qquad x\in U$

where the ${\lambda}_{{\alpha},{\beta},j}$ are smooth functions on $U$, almost
all equal to zero.

###  2.2 Operators

#### Schwartz kernel

Consider a compact symplectic manifold $(M,{\omega})$ with a compatible
almost-complex structure $j$ and a prequantum bundle $L\rightarrow M$. Assume
we have two auxiliary Hermitian vector bundles $A$ and $B$ over $M$. The
dimension of $M$ is $2n$ with $n\in{\mathbb{N}}$.

We will define a space $\mathcal{L}(A,B)$ consisting of families of operators

$\displaystyle\bigl{(}P_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes
B),\;k\in{\mathbb{N}}\bigr{)}$ (37)

having smooth Schwartz kernels satisfying some conditions. Let us first recall
some standard definitions and notations.

The Schwartz kernel of $P_{k}$ is the section $K_{k}$ of $(L^{k}\otimes
B)\boxtimes(\overline{L}^{k}\otimes\overline{A})$ such that

$(P_{k}f)(x)=\int_{M}K_{k}(x,y)\cdot
f(y)\;\mu_{M}(y),\qquad\forall\;f\in{\mathcal{C}}^{\infty}(M,L^{k}\otimes A)$

where the $\cdot$ stands for the scalar product
$(\overline{L}_{y}^{k}\otimes\overline{A}_{y})\times(L_{y}^{k}\otimes
A_{y})\rightarrow{\mathbb{C}}$, and $\mu_{M}={\omega}^{n}/n!$. We will denote
the operator and its Schwartz kernel by the same letter, hoping it is not too
confusing.

Since $L$, $A$ and $B$ are Hermitian bundles, the bundle $(L^{k}\otimes
B)\boxtimes(\overline{L}^{k}\otimes\overline{A})$ has a natural metric, so the
pointwise norm $|P_{k}(x,y)|$ is well-defined. For any $N\in{\mathbb{N}}$, we
will say that $(P_{k})$ is in $\mathcal{O}(k^{-N})$ on an open set $U$ of
$M^{2}$ if $|P_{k}(x,y)|=\mathcal{O}(k^{-N})$ for $(x,y)\in U$ with a
$\mathcal{O}$ uniform on any compact subsets of $U$. We say that $(P_{k})$ is
in $\mathcal{O}(k^{-\infty})$ on $U$ if $(P_{k})$ is in $\mathcal{O}(k^{-N})$
on $U$ for any $N$.

We will also use the uniform norm $\|P_{k}\|=\sup\|P_{k}(f)\|/\|f\|$ with
respect to the usual $L^{2}$ norms of section:
$\|f\|^{2}=\int_{M}|f(x)|^{2}\;d\mu_{M}(x)$.

#### Definition of $\mathcal{L}(A,B)$

By definition, a family $(P_{k})$ as in (37) belongs to $\mathcal{L}(A,B)$ if
each $P_{k}$ has smooth Schwartz kernel, the Schwartz kernel family is in
$\mathcal{O}(k^{-\infty})$ on $M^{2}\setminus\operatorname{diag}M$ and we have
the following expansion on a neighborhood of the diagonal. Assume first that
$A$ and $B$ are the trivial line bundle
${\mathbb{C}}_{M}:=M\times{\mathbb{C}}$. Choose a coordinate chart $U\subset
M$ and a unitary frame $t:U\rightarrow L$ so that we can identify $U$ with an
open set of ${\mathbb{R}}^{2n}$, the sections of $L^{k}$ over $U$ with
functions of $U$ and our Schwartz kernels on $U^{2}$ become functions as well.
Then for any $N\in{\mathbb{N}}$, we require that over $U^{2}$

$\displaystyle
P_{k}(x+\xi,x)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}e^{-k\varphi(x,\xi)}\sum_{p=0}^{N}k^{-\frac{p}{2}}a_{p}\bigl{(}x,k^{\frac{1}{2}}\xi\bigr{)}+\mathcal{O}\bigl{(}k^{n-\frac{N+1}{2}}\bigr{)}$
(38)

where

  * -

$\varphi(x,\xi)=-i\bigl{(}\sum_{i=1}^{2n}{\alpha}_{i}(x)\xi_{i}+\frac{1}{2}\sum_{i,j=1}^{2n}(\partial_{x_{i}}{\alpha}_{j})(x)\xi_{i}\xi_{j}\bigr{)}+\frac{1}{4}|\xi|_{x}^{2}$,
with ${\alpha}=\sum{\alpha}_{i}dx_{i}\in\Omega^{1}(U,{\mathbb{R}})$ the one-
connection form defined by $\nabla t=\frac{1}{i}{\alpha}\otimes t$.

  * -

$a_{p}(x,\xi)\in{\mathbb{C}}$ depends polynomially on $\xi$, meaning that for
some $d(p)\in{\mathbb{N}}$, $a_{p}(x,\xi)=\sum_{|{\alpha}|\leqslant
d(p)}a_{p,{\alpha}}(x)\xi^{\alpha}$ with smooth coefficients $a_{p,{\alpha}}$.

Since the real part of $\varphi(x,\xi)$ is $\frac{1}{4}|\xi|_{x}^{2}$, we have
for any $p$

$e^{-k\varphi(x,\xi)}a_{p}\bigl{(}x,k^{\frac{1}{2}}\xi\bigr{)}=\mathcal{O}(1).$

So the $p$-th summand in (38) is in $\mathcal{O}(k^{n-\frac{p}{2}}\bigr{)}$
and the expansion is meaningful. We require an expansion (38) for any
coordinate chart $U$ and unitary frame $t$ on $U$. In the case where $A$ and
$B$ are general vector bundles, we introduce frames of $A$ and $B$ on $U$, so
that the Schwartz kernel $P_{k}$ on $U^{2}$ becomes a
${\mathbb{C}}^{r}$-valued functions with
$r=(\operatorname{rank}A)(\operatorname{rank}B)$, and we can adopt the same
definition with ${\mathbb{C}}^{r}$-valued coefficients $a_{p}$.

It is not obvious that this definition is equivalent with the one given in the
introduction, cf. (27). This will be proved in Proposition 5.5. The advantage
of the expansion (38) is that its analytical meaning is more transparent, the
drawback is that it depends on local choices (coordinates, frames, rescaling
$k^{\frac{1}{2}}\xi$) whereas the expansion (27) is global.

#### Properties of $\mathcal{L}(A,B)$

$\mathcal{L}(A,B)$ has a natural filtration defined as follows. For any
$q\in{\mathbb{N}}$, $\mathcal{L}_{q}(A,B)$ is the subspace of
$\mathcal{L}(A,B)$ consisting of the operators such that the local expansions
(38) hold with a sum starting at $p=q$, that is the coefficients $a_{0}$,
…$a_{q-1}$ are zero.

######  Proposition 2.1.

  1. 1.

$\mathcal{L}_{q}(A,B)=k^{-\frac{q}{2}}\mathcal{L}(A,B)$ and if $q\geqslant
q^{\prime}$, then $\mathcal{L}_{q}(A,B)\subset\mathcal{L}_{q^{\prime}}(A,B)$.

  2. 2.

For any $(P_{k})$ in $\mathcal{L}(A,B)$,

$\displaystyle(P_{k})\in\mathcal{L}_{q}(A,B)\Leftrightarrow\|P_{k}\|=\mathcal{O}(k^{-\frac{q}{2}})$
$\displaystyle\Leftrightarrow\text{the Schwartz kernel family of $(P_{k})$
belongs to }\mathcal{O}(k^{n-\frac{q}{2}})$

  3. 3.

$\mathcal{L}_{\infty}(A,B):=\bigcap_{q}\mathcal{L}_{q}(A,B)$ consists of the
families (37) with a smooth Schwartz kernel in $\mathcal{O}(k^{-\infty})$.

  4. 4.

for any sequence $(P_{q})_{q\in{\mathbb{N}}}$ of $\mathcal{L}(A,B)$ such that
$P_{q}\in\mathcal{L}_{q}(A,B)$ for any $q$, there exists
$P\in\mathcal{L}(A,B)$ satisfying $P=\sum_{p=0}^{q}P_{p}$ modulo
$\mathcal{L}_{q+1}(A,B)$ for any $q$.

We will now describe the quotients
$\mathcal{L}_{q}(A,B)/\mathcal{L}_{q+1}(A,B)$ by using the material introduced
in Section 2.1. Since $M$ has an almost complex structure $j$ compatible with
${\omega}$, the tangent bundle $TM$ is a complex vector bundle with a
Hermitian metric, which defines our bundle $\mathcal{S}(M):=\mathcal{S}(TM)$.

######  Theorem 2.2.

For any $q\in{\mathbb{N}}$, there exists a linear map

${\sigma}_{q}:\mathcal{L}_{q}(A,B)\rightarrow{\mathcal{C}}^{\infty}(M,\mathcal{S}(M)\otimes\operatorname{Hom}(A,B))$

which is onto and has kernel $\mathcal{L}_{q+1}(A,B)$. Furthermore, the
following holds for any $P\in\mathcal{L}_{q}(A,B)$:

  1. 1.

${\sigma}_{q}(P)={\sigma}_{0}(k^{\frac{q}{2}}P)$.

  2. 2.

For any $f\in{\mathcal{C}}^{\infty}(M,\operatorname{Hom}(B,C))$, $(f\circ
P_{k})$ belongs to $\mathcal{L}_{q}(A,C)$ and ${\sigma}_{q}(f\circ
P)=f\circ{\sigma}_{q}(P)$. For any
$g\in{\mathcal{C}}^{\infty}(M,\operatorname{Hom}(C,A))$, $(P_{k}\circ g)$
belongs to $\mathcal{L}_{q}(C,B)$ and ${\sigma}_{q}(P\circ
g)={\sigma}_{q}(P)\circ g$.

  3. 3.

$P^{*}$ belongs to $\mathcal{L}_{q}(B,A)$ and
${\sigma}(P^{*})={\sigma}(P)^{*}$.

  4. 4.

For any $P^{\prime}\in\mathcal{L}_{q^{\prime}}(B,C)$, $P^{\prime}\circ P$
belongs to $\mathcal{L}_{q^{\prime}+q}(A,C)$ and

${\sigma}_{q^{\prime}+q}(P^{\prime}\circ
P)={\sigma}_{q^{\prime}}(P^{\prime})\circ{\sigma}_{q}(P).$

  5. 5.

The Schwartz kernel of $P_{k}$ on the diagonal satisfies

$P_{k}(x,x)=\frac{k^{n-\frac{q}{2}}}{(2\pi)^{n}}\Bigl{[}\operatorname{tr}({\sigma}_{q}(P)(x))+\mathcal{O}(k^{-1/2})\Bigr{]}$

where $\operatorname{tr}$ is the map
$\mathcal{S}(T_{x}M)\otimes\operatorname{Hom}(A_{x},B_{x})\rightarrow(L_{x}\otimes\overline{L}_{x})^{k}\otimes
B_{x}\otimes\overline{A}_{x}\simeq\operatorname{Hom}(A_{x},B_{x})$ sending
$s\otimes f$ to $(\operatorname{tr}s)f$.

Let us explain how is defined the symbol map ${\sigma}_{0}$ for
$A=B={\mathbb{C}}_{M}$. Consider $P\in\mathcal{L}(A,B)$ and the local
expansion (38). We view $(x,\xi)$ as a tangent vector of $M$, that is $\xi\in
T_{x}M$, so we consider $a_{0}(x,\cdot)$ as a polynomial of $T_{x}M$. Then it
is not obvious but nevertheless true that this polynomial does not depend on
the choice of the coordinate chart $U$ and the unitary frame $t$. To compare,
the coefficients $a_{p}$ in (38) with $p\geqslant 1$ do depend on the choice
of the coordinates and the frame of $L$.

To pass from $a_{0}(x,\cdot)$ to the symbol of $P$ at $x$, we first choose a
unitary frame $(e_{i})$ of $T_{x}M$. So $T_{x}M\simeq{\mathbb{C}}^{n}$ by
sending $\xi=\sum z_{i}e_{i}(x)$ to $z(\xi)=(z_{i})$. We also have a basis
$\rho_{{\alpha},{\beta}}(x)$ of $\mathcal{S}(T_{x}M)$ defined in (32). Then

${\sigma}_{0}(P)(x)=\sum
f_{{\alpha},{\beta}}(x)\rho_{{\alpha},{\beta}}(x)\Leftrightarrow
a_{0}(x,\xi)=\sum f_{{\alpha},{\beta}}(x)p_{{\alpha},{\beta}}(z(\xi))$

where we use the polynomials
$p_{{\alpha},{\beta}}(z)=\bigl{(}\frac{1}{{\alpha}!{\beta}!}\bigr{)}^{1/2}\bigl{(}\partial_{z}-\overline{z})^{{\alpha}}z^{\beta}.$
These polynomials form a basis of ${\mathbb{C}}[z,\overline{z}]$, cf. proof of
Proposition 4.4.

######  Example 2.3.

Choose a connection of $A$ and let $\Delta_{k}$ be the Laplacian

$\displaystyle\Delta_{k}=\tfrac{1}{2}(\nabla^{L^{k}\otimes
A})^{*}\nabla^{L^{k}\otimes A}:{\mathcal{C}}^{\infty}(L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(L^{k}\otimes A).$ (39)

For any $m\in{\mathbb{N}}$, let $\Pi_{m,k}$ be the spectral projector

$\displaystyle\Pi_{m,k}:=1_{[m-\frac{1}{2},m+\frac{1}{2}]}(k^{-1}\Delta_{k}).$
(40)

By [10], the family $(\Pi_{m,k})$ belongs to $\mathcal{L}(A,A)$, its
${\sigma}_{0}$-symbol at $x$ is $\pi_{m}(x)\otimes\operatorname{id}_{A_{x}}$
where $\pi_{m}(x)$ is the projector of $\mathcal{D}(T_{x}M)$ onto the subspace
$\mathcal{D}_{m}(T_{x}M)$ of homogeneous degree $m$ polynomials.

Since $\pi_{m}(x)=\sum_{|{\alpha}|=m}\rho_{{\alpha},{\alpha}}(x)$, if the
auxiliary bundle $A$ is trivial, the corresponding function $a_{0}$ is

$\displaystyle
a_{0}(x,\xi)=\sum_{|{\alpha}|=m}p_{{\alpha},{\alpha}}(z(\xi))=Q_{m}^{(n-1)}(|z(\xi)|^{2})$
(41)

where $Q_{m}^{(p)}$ is the Laguerre polynomial
$Q^{(p)}_{m}(x)=\frac{x^{-p}}{m!}\bigl{(}\frac{d}{dx}-1\bigr{)}^{m}x^{m+p}$.
The second equality in (41) follows from $p_{m,m}(z)=Q_{m}^{(0)}(|z|^{2})$ and
the identity

$Q_{m}^{(n-1)}(x_{1}+\ldots+x_{n})=\sum_{|{\alpha}|=m}Q_{{\alpha}(1)}^{(0)}(x_{1})\ldots
Q_{{\alpha}(n)}^{(0)}(x_{n}).$

Actually we won’t use the expression in terms of Laguerre polynomials, what
really matters is the fact that ${\sigma}_{0}(\Pi_{m})(x)$ is the orthogonal
projector onto $\mathcal{D}_{m}(T_{x}M)$. ∎

The definition of the symbol map ${\sigma}_{0}$ is motivated by Theorem 2.2
and its efficiency in the proofs of Section 3. But this definition does not
explain why it is natural to associate to $P\in\mathcal{L}(A,B)$ an
endomorphism of $\mathcal{D}(T_{x}M)$. A first explanation is provided by the
following construction of peaked sections. A deeper reason will be provided
later in Section 4.

We still assume that $A=B={\mathbb{C}}_{M}$ to simplify the exposition. Let
$x\in M$ be a base point, with a coordinate chart $U$ at $x$ and a unitary
frame $t:U\rightarrow L$. Let $\psi\in{\mathcal{C}}^{\infty}_{0}(U)$ be equal
to $1$ on a neighborhood of $x$. To any $f\in\mathcal{D}(T_{x}M)$, we
associate a family $\Phi_{k}^{f}\in{\mathcal{C}}^{\infty}(M,L^{k})$ defined by

$\Phi_{k}^{f}(x+\xi)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{\frac{n}{2}}e^{-k\varphi(x,\xi)}\,f(k^{\frac{1}{2}}\xi)\,\psi(x+\xi)\,t^{k}(x+\xi),\qquad
k\in{\mathbb{N}}$

where $\varphi$ is the same function as in (38).

######  Proposition 2.4.

For any $f\in\mathcal{D}(T_{x}M)$,
$\|\Phi^{f}_{k}\|=\|f\|+\mathcal{O}(k^{-1/2})$ and for any
$P\in\mathcal{L}({\mathbb{C}}_{M},{\mathbb{C}}_{M})$,

$P_{k}\Phi^{f}_{k}=\Phi^{g}_{k}+\mathcal{O}(k^{-1/2})$

where $g={\sigma}_{0}(P)(x)\cdot f$.

For a more general result with auxiliary bundles $A$, $B$ and the estimates of
the scalar product of peaked sections, cf. Proposition 5.9.

We say that an element $P$ of $\mathcal{L}(A,B)$ is even (resp. odd) if in the
local expansions (38), every polynomial $a_{p}(x,\cdot)$ has the same (resp.
the opposite) parity as $p$. Denote by $\mathcal{L}^{+}(A,B)$ and
$\mathcal{L}^{-}(A,B)$ the subspaces of even and odd elements respectively.

######  Theorem 2.5.

We have

  1. 1.

$\mathcal{L}(A,B)=\mathcal{L}^{+}(A,B)+\mathcal{L}^{-}(A,B)$,
$\mathcal{L}^{+}(A,B)\cap\mathcal{L}^{-}(A,B)=\mathcal{L}_{\infty}(A,B)$.

  2. 2.

$\mathcal{L}^{\epsilon}(A,B)\cdot\mathcal{L}^{\epsilon^{\prime}}(B,C)\subset\mathcal{L}^{\epsilon\epsilon^{\prime}}(A,C)$
for any choice of signs $\epsilon$, $\epsilon^{\prime}$.

  3. 3.

${\sigma}_{q}(\mathcal{L}_{q}(A,B)\cap\mathcal{L}^{\epsilon}(A,B))={\mathcal{C}}^{\infty}(M,\mathcal{S}^{\epsilon(-1)^{q}}(M)\otimes\operatorname{Hom}(A,B))$.

The proofs of Proposition 2.1, Theorem 2.2 and Theorem 2.5 are postponed to
Section 5.4.

###  2.3 Comparison with earlier works

The expansions (27), (38) or similar versions appeared in the literature [24],
[7], [20] to describe Bergman kernels of ample line bundles and their
symplectic generalizations as well as the associated Toeplitz operators. In a
more general context, the Boutet de Monvel-Guillemin theory [4] is built on
two classes of operators: Hermite operators and Fourier integral operators
respectively. The spaces $\mathcal{L}^{+}(A,B)$ may be viewed as an
intermediate choice in the semi-classical setting.

In [9], we considered a subalgebra of $\mathcal{L}^{+}(A,A)$, denoted by
$\mathcal{A}(A)$, consisting of operators having an expansions (38) in which
each $a_{p}(x,\cdot)$ has degree $\leqslant\frac{3}{2}p$. For our applications
in this paper, it is necessary to consider the larger spaces
$\mathcal{L}(A,B)$, because our generalized projectors $\Pi_{m}$ and unitary
equivalences do not belong to $\mathcal{A}(A)$. More precisely, only the
projector corresponding to the first Landau level belongs to $\mathcal{A}(A)$.

Theorem 2.2 is a generalization of similar results for $\mathcal{A}(A)$
established in [9], and surprisingly the proofs are somehow easier in this new
generality. However, a crucial difference with [9] relies in the symbols.
Roughly, the symbols of the elements of $\mathcal{A}(A)$ were defined directly
as the polynomials $a_{0}(x,\cdot)$. This had the advantage that it is easier
to pass from the Schwartz kernel of the operator to the symbol. The drawback
of this definition is that the product for these symbols is given by the
mysterious formula

$\displaystyle(u\star
v)(x,z,\overline{z})=\bigl{[}\exp(\Box)(u(x,-\zeta,\overline{z}-\overline{\zeta})v(x,z+\zeta,\overline{\zeta}))\bigr{]}_{\zeta=\overline{\zeta}=0}$
(42)

where $\Box=\sum\partial^{2}/\partial\zeta^{i}\partial\overline{\zeta}^{i}$.
This was actually tractable for what we did in [9] because our main interest
was the projector $\Pi$ for the first Landau level, with symbol
${\sigma}_{0}(\Pi)=\rho_{00}$, so its old symbol was the function
$u(x,\xi)=1$. But for our new projectors whose symbols are typically a sum of
the $\rho_{{\alpha}{\alpha}}$’s, it is essential to work with the symbol in
$\mathcal{S}(M)$. For instance, it is even not obvious how to recover the
relations (33) from the product (42).

##  3 Projectors of $\mathcal{L}(A)$ and Toeplitz operators

In this section, we consider an auxiliary Hermitian vector bundle $A$ with
arbitrary rank. We denote by $\mathcal{L}(A):=\mathcal{L}(A,A)$ the associated
algebra and by $\mathcal{L}^{+}(A)$ the subalgebra consisting of even
elements. The symbols of operators of $\mathcal{L}(A)$ are sections of
$\mathcal{S}(M)\otimes\operatorname{End}A$. We will view
$\mathcal{S}(M)_{x}\otimes\operatorname{End}A_{x}$ as a subspace of
$\operatorname{End}(\mathcal{D}(T_{x}M)\otimes A_{x})$.

Let $F$ be a subbundle of $\mathcal{D}_{\leqslant m}(TM)\otimes A$ for some
$m\in{\mathbb{N}}$, where $\mathcal{D}_{\leqslant m}(T_{x}M)$ is the subspace
of $\mathcal{D}(T_{x}M)$ of polynomial with degree $\leqslant m$. We assume
that $F$ has a definite parity, that is
$F\subset\mathcal{D}^{\epsilon}(TM)\otimes A$ with $\epsilon\in\\{\pm 1\\}$.
Associated to $F$ is the section $\pi$ of
$\mathcal{S}(M)\otimes\operatorname{End}A$ such that $\pi(x)$ is the
orthogonal projector of $\mathcal{D}(T_{x}M)\otimes A_{x}$ onto $F_{x}$ at
each point $x\in M$. The content of the following subsections is:

  * -

3.1: we construct a selfadjoint projector $\Pi\in\mathcal{L}^{+}(A)$ with
symbol $\pi$.

  * -

3.2: we study the Toeplitz algebra $\mathcal{T}=\\{\Pi
P\Pi,\;P\in\mathcal{L}^{+}(A)\\}$

  * -

3.3: we prove $(\operatorname{Im}\Pi_{k})$ is isomorphic with any quantization
of $(L,M)$ twisted by $F$, and deduce that the dimension of
$\operatorname{Im}\Pi_{k}$ is the Riemann-Roch number of $L^{k}\otimes F$ when
$k$ is sufficiently large.

A possible choice for $F$ is $F=\mathcal{D}_{m}(TM)\otimes A$ where
$m\in{\mathbb{N}}$ and $\mathcal{D}_{m}(T_{x}M)$ is the subspace of
$\mathcal{D}(T_{x}M)$ consisting of homogeneous polynomials with degree $m$.
As explained in example 2.3, the projector $\Pi_{m,k}$ onto the $m$-th Landau
level $\mathcal{H}_{m,k}$ belongs to $\mathcal{L}^{+}(A)$ and has symbol the
projector onto $F$, so it can be used as the projector $\Pi$. Theorems 1.1,
1.3 and 1.4 will mainly follow from the results in Sections 3.2 and 3.3.

By [10], the spectral projectors of Laplacians with a magnetic field, not
necessarily constant but still satisfying some convenient assumptions, give
other instances of projectors in $\mathcal{L}^{+}(A)$.

Another choice for $F$ is $F=\mathcal{D}_{0}(TM)\otimes A$ where
$\mathcal{D}_{0}(T_{x}M)={\mathbb{C}}$ is the subspace of
$\mathcal{D}(T_{x}M)$ of constant polynomials. The corresponding quantum space
and Toeplitz algebra is the quantization of $(M,L)$ twisted by $A$.

A last example is the Spin-c Dirac quantization twisted by an auxiliary bundle
$B$. In this case, $A=S\otimes B$ where $S$ is the spinor bundle
$\bigoplus\bigwedge^{k}(T^{*}M)^{0,1}$ and
$F=\mathcal{D}_{0}(TM)\otimes\bigwedge^{0}(T^{*}M)^{0,1}\otimes B$. This
example will be used to compute the dimension of our quantum spaces from the
Atiyah-Singer Theorem.

###  3.1 Construction of the projector

Let $\chi:{\mathbb{R}}\rightarrow{\mathbb{R}}$ be defined by $\chi(x)=1$ if
$x\geqslant\frac{1}{2}$ and $\chi(x)=0$ otherwise. If $P$ is a bounded self-
adjoint operator of a Hilbert space $\mathcal{H}$, then using the functional
calculus for Borel bounded functions, we define a new bounded operator
$\chi(P)$ of $\mathcal{H}$, cf. as instance [23, Theorem VII.2]. Since $\chi$
is real valued and $\chi^{2}=\chi$, $\chi(P)$ is a self-adjoint projector.

######  Theorem 3.1.

Let $P\in\mathcal{L}(A)$ be self-adjoint and having symbol
${\sigma}_{0}(P)=\pi$. Then $\chi(P)$ belongs to $\mathcal{L}(A)$ and
${\sigma}_{0}(\chi(P))=\pi$. If furthermore $P\in\mathcal{L}^{+}(A)$, then
$\chi(P)$ is in $\mathcal{L}^{+}(A)$.

Theorem 3.1 holds without the assumption that $F$ has a definite parity. When
$F$ does have a definite parity, $\pi$ is even, so we can choose
$P\in\mathcal{L}^{+}(A)$ with symbol $\pi$.

###### Proof.

To prove that the $\chi(P_{k})$’s have smooth kernels, we will use the
following basic fact: let $Q$, $Q^{\prime}$ be two operators with smooth
kernels acting on ${\mathcal{C}}^{\infty}(M,A)$ and $Q^{\prime\prime}$ be a
bounded operator of $L^{2}(M,A)$. Then $QQ^{\prime\prime}Q^{\prime}$ has a
smooth kernel. This follows from the Schwartz theorem saying that the
operators with smooth kernel are the operators which can be continuously
extended $\mathcal{C}^{-\infty}\rightarrow{\mathcal{C}}^{\infty}$. We will
also need the following pointwise norm estimates: consider families of
operator $Q_{k},Q^{\prime}_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes A)$ and
$Q^{\prime\prime}_{k}:L^{2}(M,L^{k}\otimes A)\rightarrow L^{2}(M,L^{k}\otimes
A)$. Then by [9, Section 4.3], if the Schwartz kernel families of $(Q_{k})$
and $(Q^{\prime}_{k})$ are respectively in $\mathcal{O}(k^{-N})$ and
$\mathcal{O}(k^{-N^{\prime}})$, and the operator norms of
$Q_{k}^{{}^{\prime\prime}}$ are in $\mathcal{O}(1)$, then the Schwartz kernel
family of $Q_{k}Q^{\prime\prime}_{k}Q_{k}^{\prime}$ is in
$\mathcal{O}(k^{-(N+N)^{\prime}})$.

Back to our problem, we can write $\chi(P_{k})=P_{k}\tilde{\chi}(P_{k})P_{k}$
with $\tilde{\chi}(x)=\chi(x)/x^{2}$. Since $\tilde{\chi}(P_{k})$ is bounded,
this shows that $\chi(P_{k})$ has a smooth kernel. This also shows that the
Schwartz kernel family of $\chi(P)$ is in $\mathcal{O}(k^{2n})$. To improve
this, observe that $Q=P^{2}-P$ is in $\mathcal{L}(A)$ and
${\sigma}_{0}(Q)=\pi^{2}-\pi=0$, so $\|Q_{k}\|=\mathcal{O}(k^{-1/2})$, which
implies easily that $\frac{1}{2}$ is not in the spectrum of $P_{k}$ when $k$
is sufficiently large, cf. [9, Proposition 4.2].

Now for $x\in{\mathbb{R}}\setminus\\{\frac{1}{2}\\}$, $y=x^{2}-x>-1/4$ and we
have

$\chi(x)=x+(1-2x)f(x^{2}-x)\quad\text{with }\quad
f(y)=\tfrac{1}{2}(1-(1+4y)^{-1/2})$

Write the Taylor expansion of $f$ at $0$ as
$f(y)=\sum_{\ell=0}^{m}a_{\ell}y^{\ell}+y^{m+1}f_{m}(y)$ with
$f_{m}\in\mathcal{C}^{0}(]-\frac{1}{4},\infty[,{\mathbb{R}})$. Then

$\displaystyle\chi(P)=P+\sum_{\ell=0}^{m}a_{\ell}(1-2P)Q^{\ell}+(1-2P)Q^{m+1}f_{m}(Q).$
(43)

Now ${\sigma}_{0}(Q)=0$ implies that $Q^{\ell}$ and $PQ^{\ell}$ belong both to
$\mathcal{L}_{\ell}(A)$. Furthermore, $\|f_{m}(Q_{k})\|=\mathcal{O}(1)$. Since
$Q^{m+1}f_{m}(Q)=Q^{m}f(Q)Q$ and similarly $PQ^{m+1}f_{m}(Q)=PQ^{m}f(Q)Q$, it
follows from the preliminary observation that the Schwartz kernel family of
$(1-2P)Q^{m+1}f_{m}(Q)$ is in $\mathcal{O}_{\infty}(k^{2n-m})$. We can now
conclude easily the proof from (43) by choosing at each step sufficiently
large value of $m$: first the Schwartz kernel family of $\chi(P)$ is in
$\mathcal{O}(k^{-\infty})$ outside the diagonal and second the local
expansions (38) hold. ∎

###  3.2 Toeplitz algebra

Choose a self-adjoint projector $\Pi\in\mathcal{L}^{+}(A)$ with symbol $\pi$,
which exists by Theorem 3.1. For any $k\in{\mathbb{N}}$, let
$\mathcal{H}_{k}=\operatorname{Im}\Pi_{k}\subset{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)$. Computing the trace of $\Pi_{k}$ by integrating its Schwartz kernel over
the diagonal, we deduce from the last assertion of Theorem 2.2 that
$\mathcal{H}_{k}$ is finite dimensional and

$\displaystyle\operatorname{dim}\mathcal{H}_{k}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}(\operatorname{rank}F)\;\operatorname{vol}(M,{\omega})\;\bigl{(}1+\mathcal{O}(k^{-1})\bigr{)}.$
(44)

We will now work with families of operators
$(T_{k}\in\operatorname{End}\mathcal{H}_{k}$, $k\in{\mathbb{N}})$.
Equivalently, we can consider that each $T_{k}$ acts on the larger space
${\mathcal{C}}^{\infty}(M,L^{k}\otimes A)$ and satisfies
$\Pi_{k}T_{k}\Pi_{k}=T_{k}$. Define the space

$\displaystyle\mathcal{T}=\\{T\in\mathcal{L}^{+}(A)/\,\Pi T\Pi=T\\}.$ (45)

For any $q\in{\mathbb{N}}$, set
$\mathcal{T}_{q}:=k^{-q}\mathcal{T}=\mathcal{L}_{2q}(A)\cap\mathcal{T}$ and
$\mathcal{T}_{\infty}=\bigcap_{q}\mathcal{T}_{q}$. Clearly,

$\mathcal{T}_{\infty}\subset\mathcal{T}_{q}\subset\mathcal{T}_{p}\subset\mathcal{T}$

when $q\geqslant p$.

######  Theorem 3.2.

  1. 1.

For any $T\in\mathcal{T}$, $T$ belongs to $\mathcal{T}_{q}$ iff
$\|T_{k}\|=\mathcal{O}(k^{-q})$. Furthermore, $\mathcal{T}_{\infty}$ consists
of the families
$(T_{k}\in\operatorname{End}(\mathcal{H}_{k}),\,k\in{\mathbb{N}}^{*})$ such
that $\|T_{k}\|=\mathcal{O}(k^{-N})$ for any $N$.

  2. 2.

$\mathcal{T}$ is closed under composition and taking adjoint:
$\bigl{(}\mathcal{T}_{q}\bigr{)}^{*}=\mathcal{T}_{q}$ and
$\mathcal{T}_{q}\cdot\mathcal{T}_{p}\subset\mathcal{T}_{q+p}$ for any $q$ and
$p$.

  3. 3.

For any $q$, there exists a linear map
$\tau_{q}:\mathcal{T}_{q}\rightarrow{\mathcal{C}}^{\infty}(M,\operatorname{End}F)$,
which is onto, has kernel $\mathcal{T}_{q+1}$ and is determined by
${\sigma}_{2q}(T)=\tau_{q}(T)\pi$. Furthermore, if $P\in\mathcal{T}_{q}$ and
$Q\in\mathcal{T}_{p}$, then

$\displaystyle\begin{split}&\tau_{q}(P)=\tau_{0}(k^{q}P),\qquad\tau_{q}(P^{*})=\tau_{q}(P)^{*},\\\
&\tau_{q}(P)\tau_{p}(Q)=\tau_{q+p}(PQ)\end{split}$ (46)

and the restriction to the diagonal of the Schwartz kernel of $P_{k}$
satisfies

$P_{k}(x,x)=\frac{k^{n-q}}{(2\pi)^{n}}\Bigl{[}\operatorname{tr}(\tau_{q}(P)(x))+\mathcal{O}(k^{-1})\Bigr{]}.$

Let us give more details on the equation ${\sigma}_{2q}(T)=\tau_{q}(T)\pi$
defining the symbol map $\tau_{q}$. Recall that $\pi(x)$ is the orthogonal
projector of $\mathcal{D}(T_{x}M)\otimes A_{x}$ onto $F_{x}$.

Then for an endomorphism $s$ of $F_{x}$, we define
$s\pi(x)\in\mathcal{S}(T_{x}M)\otimes\operatorname{End}A_{x}$ as the
endomorphism of $\mathcal{D}(T_{x}M)\otimes A_{x}$ sending $\psi$ into
$s(\pi(x)\psi)$.

###### Proof.

1\. The first assertion follows from Part 2 of Proposition 2.1. To establish
the second assertion, we deduce from the first part of the proof of Theorem
3.1 that if a family $(P_{k}\in\operatorname{End}(\mathcal{H}_{k}))$ satisfies
$\|P_{k}\|=\mathcal{O}(k^{-\infty})$, then its Schwartz kernel is in
$\mathcal{O}(k^{-\infty})$ because $\Pi_{k}P_{k}\Pi_{k}=P_{k}$ and the
Schwartz kernel of $\Pi_{k}$ is in $\mathcal{O}(k^{n})$.

Parts 2 and 3 follow from Theorem 2.2 and the fact that $\mathcal{L}^{+}(A)$
is a subalgebra of $\mathcal{L}(A)$ by theorem 2.5. To define $\tau_{q}(P)$,
simply observe that $\Pi P\Pi=P$ implies that
$\pi{\sigma}_{2q}(P)\pi={\sigma}_{2q}(P)$, so we can write
${\sigma}_{2q}(P)=\tau_{q}(P)\pi$ with $\tau_{q}(P)$ a section of
$\operatorname{End}F$. The map $\tau_{q}$ is onto because for any section $s$
of $\operatorname{End}F$, there exists $P$ in $\mathcal{L}_{2q}(A)$ with
${\sigma}_{2q}(P)=s\pi$. Since $\pi(s\pi)\pi=s\pi$, we have ${\sigma}_{2q}(\Pi
P\Pi)=s\pi$ and clearly $\Pi P\Pi\in\mathcal{T}_{q}$.

The kernel of $\tau_{q}$ is $\mathcal{T}_{q+1}$ because $\tau_{q}(P)=0$
implies that ${\sigma}_{2q}(P)=0$ so $P\in\mathcal{L}^{+}_{2q+1}(A)$ so
${\sigma}_{2q+1}(P)$ is odd by Theorem 2.5. But $\Pi P\Pi=P$ implies that
${\sigma}_{2q+1}(P)=\pi{\sigma}_{2q+1}(P)\pi$. This implies that
${\sigma}_{2q+1}(P)$ is even because $F$ has a definite parity. Indeed, if for
instance $F\subset\mathcal{D}^{+}(TM)\otimes A$, and $f=\pi g\pi$ with
$g\in\mathcal{S}(T_{x}M)\otimes\operatorname{End}A_{x}$, then in the
decomposition

$\mathcal{D}(T_{x}M)\otimes A_{x}=(\mathcal{D}^{+}(T_{x}M)\otimes
A_{x})\oplus(\mathcal{D}^{-}(T_{x}M)\otimes A_{x}),$

$f$ has the form $\begin{pmatrix}f_{++}&0\\\ 0&0\end{pmatrix}$, so $f$ is
even. Consequently, ${\sigma}_{2q+1}(P)=0$ so $P\in\mathcal{L}_{2q+2}(A)$. The
formulas giving the symbol of products, adjoints and the Schwartz kernel on
the diagonal follow directly from Theorem 2.2. ∎

For any $f\in{\mathcal{C}}^{\infty}(M)$ and $k\in{\mathbb{N}}$, define the
endomorphism $T_{k}(f)$ of $\mathcal{H}_{k}$ such that

$\langle T_{k}(f)\psi,\psi^{\prime}\rangle=\langle
f\psi,\psi^{\prime}\rangle,\qquad\forall\psi,\psi^{\prime}\in\mathcal{H}_{k}.$

Viewed as an operator of ${\mathcal{C}}^{\infty}(M,L^{k}\otimes A)$,
$T_{k}(f)$ is merely $\Pi_{k}f\Pi_{k}$. It follows from part 2 of Theorem 2.2
that the family $(T_{k}(f))$ belongs to $\mathcal{T}$ and has symbol
$\tau_{0}(T_{k}(f))=f\operatorname{id}_{F}$. By part 4 of Theorem 3.2, we
deduce that

$T_{k}(f)T_{k}(g)=T_{k}(fg)+\mathcal{O}(k^{-1}).$

A consequence of Theorem 3.4 will be that

$[T_{k}(f),T_{k}(g)]=ik^{-1}T_{k}(\\{f,g\\})+\mathcal{O}(k^{-2})$

with $\\{f,g\\}$ the Poisson bracket of $f$ and $g$ with respect to
${\omega}$. This equality does not follow from Theorem 3.2. However, the
following characterization of the Toeplitz operators having a scalar symbol
follows from Theorem 3.2: for any $P\in\mathcal{T}$,

$P=T(f)+\mathcal{O}(k^{-1})\text{ for some
}f\in{\mathcal{C}}^{\infty}(M)\,\Leftrightarrow\;\forall
Q\in\mathcal{T},\,[P,Q]\in\mathcal{T}_{1}.$

By Jacobi identity, this proves that
$[T_{k}(f),T_{k}(g)]=k^{-1}T_{k}(h)+\mathcal{O}(k^{-2})$ for some function
$h\in{\mathcal{C}}^{\infty}(M)$.

###  3.3 A unitary equivalence

Consider an auxiliary vector bundle $B$ with an arbitrary rank. Set
$F^{\prime}=\mathcal{D}_{0}(TM)\otimes B$ with
$\mathcal{D}_{0}(T_{x}M)\subset\mathcal{D}(T_{x}M)$ the subspace of constant
polynomials. Then by Theorem 3.1, there exists a projector $\Pi^{\prime}$ in
$\mathcal{L}^{+}(B)$ having symbol

${\sigma}_{0}(\Pi^{\prime})=\rho_{00}\otimes\operatorname{id}_{B}\in{\mathcal{C}}^{\infty}(M,\mathcal{S}(M)\otimes\operatorname{End}B),$

where $\rho_{00}(x)\in\mathcal{S}(T_{x}M)$ is the orthogonal projector of
$\mathcal{D}(T_{x}M)$ onto $\mathcal{D}_{0}(T_{x}M)$, the notation being the
same as in (32).

Starting from $\Pi^{\prime}$, we define
$\mathcal{H}_{k}^{\prime}:=\operatorname{Im}\Pi^{\prime}_{k}$ and the
corresponding Toeplitz space
$\mathcal{T}^{\prime}:=\\{P\in\mathcal{L}^{+}(B)/\;\Pi^{\prime}P\Pi^{\prime}=P\\}$.
Since $\mathcal{D}_{0}(TM)={\mathbb{C}}$, $F\simeq B$, so the symbols of the
Toeplitz operators of $\mathcal{T}^{\prime}$ are sections of
$\operatorname{End}B$:

$0\rightarrow\mathcal{T}^{\prime}_{q+1}\rightarrow\mathcal{T}^{\prime}_{q}\xrightarrow{\tau^{\prime}_{q}}{\mathcal{C}}^{\infty}(M,\operatorname{End}B)\rightarrow
0.$

Our goal now is to establish an equivalence between the
$(\mathcal{H}_{k},\mathcal{T})$ and
$(\mathcal{H}_{k}^{\prime},\mathcal{T}^{\prime})$ when the bundle $B$ is $F$.
The critical point is the existence of a convenient symbol. Recall our
assumption that $F\subset\mathcal{D}^{\epsilon}(TM)\otimes A$ with
$\epsilon\in\\{\pm 1\\}$.

######  Lemma 3.3.

If $B=F$, then there is a canonical symbol
$\rho\in{\mathcal{C}}^{\infty}(M,\mathcal{S}^{\epsilon}(M)\otimes\operatorname{Hom}(A,B))$
such that $\rho^{*}\rho=\pi$ and
$\rho\rho^{*}=\rho_{00}\otimes\operatorname{id}_{B}.$

###### Proof.

On the one hand, $\pi(x)$ is the orthogonal projector of
$\mathcal{D}(T_{x}M)\otimes A_{x}$ onto $F_{x}$. On the other hand,
$\pi^{\prime}(x):=\rho_{00}(x)\otimes\operatorname{id}_{B_{x}}$ is the
orthogonal projector of $\mathcal{D}(T_{x}M)\otimes B_{x}$ onto
${\mathbb{C}}\otimes B_{x}$. Since $B=F$, the images of $\pi(x)$ and
$\pi^{\prime}(x)$ are isomorphic by the map
$\xi(x):F_{x}\rightarrow\operatorname{Im}\pi^{\prime}(x)$ sending $f$ into
$1\otimes f$. We define $\rho(x)$ as the extension of $\xi(x)$

$\displaystyle\rho(x):F_{x}\oplus
F_{x}^{\perp}\rightarrow(\operatorname{Im}\pi^{\prime}(x))\oplus(\operatorname{Im}\pi^{\prime}(x))^{\perp}$
(47)

having the block decomposition $\begin{pmatrix}\xi(x)&0\\\ 0&0\end{pmatrix}$.
So $\rho(x)$ is canonically defined. The equalities
$\rho(x)^{*}\rho(x)=\pi(x)$ and $\rho(x)\rho(x)^{*}=\pi^{\prime}(x)$ are
easily verified by using that $\xi(x)$ is unitary. Writing $\rho$ in terms of
a local frame of $F$, we see that $\rho(x)$ depends smoothly on $x$. Finally,
$F_{x}\subset\mathcal{D}^{\epsilon}(T_{x}M)\otimes A_{x}$ and
$\operatorname{Im}\pi^{\prime}(x)\subset\mathcal{D}^{+}(T_{x}M)\otimes B_{x}$,
so
$\rho(x)\in\mathcal{S}^{\epsilon}(M)_{x}\otimes\operatorname{Hom}(A_{x},B_{x})$.
∎

######  Theorem 3.4.

Assume that $B=F$ and $\rho$ is the symbol defined above. Then there exists
$U\in\mathcal{L}^{\epsilon}(A,B)$ with symbol ${\sigma}_{0}(U)=\rho$ and such
that

$\displaystyle U^{*}_{k}U_{k}=\Pi_{k},\qquad U_{k}U_{k}^{*}=\Pi^{\prime}_{k}$
(48)

when $k$ is sufficiently large. Modifying $\Pi_{k}^{\prime}$ for a finite
number of $k$, we can choose $U$ so that (48) holds for any $k$. In this case,
the Toeplitz algebras $\mathcal{T}$ and $\mathcal{T}^{\prime}$ are isomorphic
by the map sending $P$ into $UPU^{*}$. Furthermore, $P\in\mathcal{T}_{q}$ if
and only if $UPU^{*}\in\mathcal{T}^{\prime}_{q}$ and when this is satisfied

$\displaystyle\tau^{\prime}_{q}(UPU^{*})=\tau_{q}(P).$ (49)

###### Proof.

Choose $W\in\mathcal{L}^{\epsilon}(A,B)$ with symbol $\rho$ and set
$V:=\Pi^{\prime}W\Pi$. Then $V\in\mathcal{L}^{\epsilon}(A,B)$ with
${\sigma}_{0}(V)=\pi\rho\pi^{\prime}=\rho$ and since $\rho^{*}\rho=\pi$ and
$\rho\rho^{*}=\pi^{\prime}$, we have

$V^{*}_{k}V_{k}=\Pi_{k}+\mathcal{O}(k^{-\frac{1}{2}}),\qquad
V_{k}V^{*}_{k}=\Pi_{k}^{\prime}+\mathcal{O}(k^{-\frac{1}{2}})$

So $V_{k}$, viewed as an operator from $\mathcal{H}_{k}$ to
$\mathcal{H}_{k}^{\prime}$, is invertible when $k$ is sufficiently large.

Observe also that $V^{*}V$ is a Toeplitz operator of $\mathcal{T}$ with symbol
$\operatorname{id}_{F}$. So $V^{*}V=\Pi+Q$ with $Q\in\mathcal{T}_{1}$. Since
$\|Q_{k}\|=\mathcal{O}(k^{-1})$, the spectrum of $Q_{k}$ is contained in
$[-\frac{1}{2},\frac{1}{2}]$ when $k$ is sufficiently large. Modifying $Q_{k}$
for a finite number of $k$, we can assume this holds for any $k$, and when $k$
is sufficiently large, we still have $V_{k}^{*}V_{k}=\Pi_{k}+Q_{k}$. Let
$P_{k}$ be the endomorphism of ${\mathcal{C}}^{\infty}(M,L^{k}\otimes A$)
which is zero on $\mathcal{H}_{k}^{\perp}$ and equal to
$(\operatorname{Id}_{\mathcal{H}_{k}}+Q_{k})^{-1/2}$ on $\mathcal{H}_{k}$. We
claim that $(P_{k})$ belongs to $\mathcal{T}$ and has symbol
$\operatorname{id}_{F}$.

Assuming this temporarily, it follows that $U_{k}:=V_{k}P_{k}$ belongs to
$\mathcal{L}^{\epsilon}(A,B)$, has symbol $\rho$ and satisfies when $k$ is
sufficiently large $U_{k}^{*}U_{k}=P_{k}V_{k}^{*}V_{k}P_{k}=\Pi_{k}$. Since
$U_{k}$, viewed as an operator from $\mathcal{H}_{k}$ to
$\mathcal{H}^{\prime}_{k}$ is invertible, this also implies that
$U_{k}U_{k}^{*}=\Pi^{\prime}_{k}$.

To prove the claim above, we write the Taylor expansion
$(1+x)^{-1/2}=1+\sum_{\ell=0}^{m}a_{\ell}x^{\ell}+x^{m+1}f_{m}(x)$ with
$f_{m}$ a continuous function
$[-\frac{1}{2},\frac{1}{2}]\rightarrow{\mathbb{R}}$. Then

$\displaystyle
P_{k}=\Pi_{k}+\sum_{\ell=0}^{m}a_{\ell}Q_{k}^{\ell}+Q_{k}^{m+1}f_{m}(Q_{k}).$
(50)

Then we show that $P$ belongs to $\mathcal{L}^{+}(A)$ by arguing as in the
proof of Theorem 3.1: $Q^{\ell}\in\mathcal{T}_{\ell}$ and
$\|f_{m}(Q_{k})\|=\mathcal{O}(1)$, so the Schwartz kernel family of
$Q_{k}^{m+1}f_{m}(Q_{k})=Q_{k}^{m}f_{m}(Q_{k})Q_{m}$ is in
$\mathcal{O}(k^{2n-m-1})$. Choosing $m$ sufficiently large at each step, we
then deduce from (50) that the Schwartz kernel family of $P_{k}$ is
$\mathcal{O}(k^{-\infty})$ outside the diagonal and that the local expansions
(38) hold.

So we have proved the existence of $U\in\mathcal{L}^{\epsilon}(A,B)$ with
${\sigma}_{0}(U)=\rho$ and satisfying (48) for any $k$ except a finite set.
For the missing $k$’s, we modify $\Pi_{k}^{\prime}$ by choosing any subspace
$\mathcal{H}^{\prime}_{k}$ of ${\mathcal{C}}^{\infty}(M,L^{k}\otimes B)$
having the same dimension as $\mathcal{H}_{k}$, define $\Pi_{k}^{\prime}$ as
the orthogonal projector onto $\mathcal{H}_{k}^{\prime}$ and $U_{k}$ as any
isometry $\mathcal{H}_{k}\rightarrow\mathcal{H}^{\prime}_{k}$ extended to zero
on $\mathcal{H}_{k}^{\perp}$. Then $\Pi_{k}^{\prime}$ and $U_{k}$ have a
smooth Schwartz kernel, so the new families $\Pi^{\prime}$ and $U$ are still
in $\mathcal{L}^{+}(B)$ and $\mathcal{L}^{\epsilon}(A,B)$ respectively.

It is now easy to prove the last assertion: if $P\in\mathcal{L}^{+}(A)$, then
$UPU^{*}\in\mathcal{L}^{+}(B)$ because $U\in\mathcal{L}^{\epsilon}(A,B)$ and
$U^{*}\in\mathcal{L}^{\epsilon}(B,A)$. If $\Pi P\Pi=P$, then
$\Pi^{\prime}(UPU^{*})\Pi^{\prime}=UPU^{*}$ by (48). So $P\in\mathcal{T}$
implies that $UPU^{*}\in\mathcal{T}^{\prime}$, which defines an isomorphism
from $\mathcal{T}$ into $\mathcal{T}^{\prime}$ because we can invert it by
sending $Q$ into $U^{*}QU$. Furthermore,
${\sigma}_{0}(UPU^{*})=\rho^{*}{\sigma}_{0}(P)\rho$ which leads to (49). ∎

A first corollary is the computation of the symbols of commutators in terms of
Poisson bracket. Recall the Toeplitz operators
$T_{k}(f):\mathcal{H}_{k}\rightarrow\mathcal{H}_{k}$ associated to
$f\in{\mathcal{C}}^{\infty}(M)$. Define similarly
$T^{\prime}_{k}(f):\mathcal{H}^{\prime}_{k}\rightarrow\mathcal{H}_{k}^{\prime}$.

######  Corollary 3.5.

$[T_{k}(f),T_{k}(g)]=ik^{-1}T_{k}(\\{f,g\\})+\mathcal{O}(k^{-2})$ for any
$f,g\in{\mathcal{C}}^{\infty}(M)$.

###### Proof.

This amounts to show that for any two Toeplitz operators $T$, $S$ of
$\mathcal{T}$ with symbol $\tau_{0}(T)=f\operatorname{id}_{F}$,
$\tau_{0}(S)=g\operatorname{id}_{F}$, we have
$\tau_{1}([T,S])=i\\{f,g\\}\operatorname{id}_{F}$. By Theorem 3.4, this holds
for $\mathcal{T}$ if and only if this holds for $\mathcal{T}^{\prime}$. The
results for $\mathcal{T}^{\prime}$ has been proved in [9], when the projector
is chosen in a specific subalgebra of $\mathcal{L}^{+}(B)$, which is always
possible. ∎

The operators $T^{\prime}_{k}(f)$ are defined not only for
$f\in{\mathcal{C}}^{\infty}(M)$ but also for
$f\in{\mathcal{C}}^{\infty}(M,\operatorname{End}B)$. Since
$\tau_{q}^{\prime}(k^{-q}T_{k}^{\prime}(f))=f$, it follows that we can define
the Toeplitz operators of $\mathcal{T}^{\prime}$ as the families $(T_{k})$
such that for any $N$,

$T_{k}=\sum_{\ell=0}^{N}k^{-\ell}T_{k}(f_{\ell})+\mathcal{O}(k^{-(N+1)})$

for a sequence $(f_{\ell})$ of
${\mathcal{C}}^{\infty}(M,\operatorname{End}B)$. This provides a definition of
$\mathcal{T}^{\prime}$ without any reference to the algebra
$\mathcal{L}^{+}(B)$. Observe also that the coefficients $f_{\ell}$ are
uniquely determined by $T$ and the map
$\mathcal{T}^{\prime}\rightarrow{\mathcal{C}}^{\infty}(M,\operatorname{End}B)[[\hbar]]$
sending $T$ into $\sum\hbar^{\ell}f_{\ell}$ is a full symbol map, meaning that
it is onto and its kernel is
$\mathcal{T}^{\prime}\cap\mathcal{O}(k^{-\infty})$. This full symbol map can
also be used to get uniform control of the product of Toeplitz operators, cf.
[9]. But unfortunately, this does not hold for $\mathcal{T}$, except in the
particular case where $F$ has rank one, so that
$\operatorname{End}F\simeq{\mathbb{C}}$. This happens in particular for higher
Landau level in dimension $n=1$.

A second consequence of Theorem 3.4 is the computation of the dimension of our
quantum spaces.

######  Theorem 3.6.

If $\Pi\in\mathcal{L}^{+}(A)$ is any self-adjoint projector with symbol $\pi$,
then the dimension of $\mathcal{H}_{k}=\operatorname{Im}(\Pi_{k})$ is

$\operatorname{dim}\mathcal{H}_{k}=\int_{M}\operatorname{ch}(L^{k}\otimes
F)\;\operatorname{Td}(M)$

when $k$ is sufficiently large.

###### Proof.

We introduce a new family
$(\mathcal{H}^{\prime\prime}_{k}:=\operatorname{Ker}D_{k})$ where $D_{k}$ is
the spin-c Dirac operator acting on ${\mathcal{C}}^{\infty}(M,L^{k}\otimes
B\otimes S)$ with $S:=\bigwedge(T^{*}M)^{0,1}$ the spinor bundle. By the
Atiyah-Singer Theorem and a vanishing theorem [2], [19], the dimension of
$\mathcal{H}^{\prime\prime}_{k}$ is given by the Riemann-Roch number of
$L^{k}\otimes B$ when $k$ is sufficiently large. We claim that the projector
$\Pi^{\prime\prime}_{k}$ of ${\mathcal{C}}^{\infty}(M,L^{k}\otimes B\otimes
S)$ onto $\mathcal{H}_{k}^{\prime\prime}$ belongs to $\mathcal{L}^{+}(B\otimes
S)$ and has symbol $\rho_{00}\otimes p_{B}$ where $p_{B}$ is the section of
$\operatorname{End}(B\otimes S)$ equal at each $x\in M$ to the projector of
$B_{x}\otimes S_{x}$ onto $B_{x}\otimes{\mathbb{C}}$. This is actually a
reformulation of results by Ma and Marinescu [20], as is explained in [9,
Appendice A]. Alternatively this follows from the companion paper [10].

Now the image of the symbol $\rho_{00}\otimes p_{B}$ is isomorphic with $B$,
so by Theorem 3.4, when $k$ is sufficiently large,
$\mathcal{H}^{\prime\prime}_{k}$ has the same definition as
$\mathcal{H}_{k}^{\prime}=\operatorname{Im}\Pi^{\prime}_{k}$, where $\Pi_{k}$
is any self-adjoint projector of $\mathcal{L}^{+}(B)$ with symbol
$\rho_{00}\otimes\operatorname{id}_{B}$. And by another application of theorem
3.4, for $B=F$, $\mathcal{H}^{\prime}_{k}$ and $\mathcal{H}_{k}$ have the same
dimension when $k$ is sufficiently large. ∎

##  4 Landau Hamiltonian algebra

In this section, we come back to the algebra $\mathcal{S}({\mathbb{C}}^{n})$
introduced in Section 2.1. We extend the action of the elements of
$\mathcal{S}({\mathbb{C}}^{n})$ on $\mathcal{D}({\mathbb{C}}^{n})$ to the
complete polynomial space and we compute the corresponding Schwartz kernel.
This will be used in the sequel to give an intrinsic definition of the symbol
maps ${\sigma}_{q}$, cf. Definition 5.8, and to understand the composition
properties of the class $\mathcal{L}(A,B)$.

Let $\mathcal{P}({\mathbb{C}}^{n})$ be the space of polynomials map from
${\mathbb{C}}^{n}$ to ${\mathbb{C}}$, so any
$f\in\mathcal{P}({\mathbb{C}}^{n})$ has the form $f=\sum
a_{{\alpha}{\beta}}z^{\alpha}\overline{z}^{\beta}$ where the sum is finite and
the $a_{{\alpha}{\beta}}$ are complex numbers. The space
$\mathcal{D}({\mathbb{C}}^{n})$ introduced in Section 2.1 is the subspace of
$\mathcal{P}({\mathbb{C}}^{n})$ of antiholomorphic maps. We endow
$\mathcal{P}({\mathbb{C}}^{n})$ with the same scalar product

$\displaystyle\langle
f,g\rangle=(2\pi)^{-n}\int_{{\mathbb{C}}^{n}}e^{-|z|^{2}}f(z)\overline{g(z)}\;d\mu_{n}(z)$
(51)

as in (30) for $\mathcal{D}({\mathbb{C}}^{n})$. The family
$((({\alpha}+{\beta})!)^{-\frac{1}{2}}z^{{\alpha}}\overline{z}^{\beta},\;{\alpha},{\beta}\in{\mathbb{N}}^{n})$
is an orthonormal basis of $\mathcal{P}({\mathbb{C}}^{n})$.

For any $i=1,\ldots,n$, introduce the endomorphism
$a_{i}=\partial_{\overline{z}_{i}}$ and its adjoint
$a_{i}^{*}=\overline{z}_{i}-\partial_{z_{i}}$. They satisfy the bosonic
commutation relations

$[a_{i},a_{j}]=[a_{i}^{*},a_{j}^{*}]=0,\qquad[a_{i},a_{j}^{*}]=\delta_{ij}.$

So the $a_{i}^{*}a_{i}$’s are mutually commuting Hermitian endomorphisms.
Their eigenspaces are the Landau levels of ${\mathbb{C}}^{n}$. In the sequel,
we use the notation $a^{\alpha}:=a_{1}^{{\alpha}(1)}\ldots
a_{n}^{{\alpha}(n)}$ and
$(a^{*})^{{\alpha}}:=(a_{1}^{*})^{{\alpha}(1)}\ldots(a_{n}^{*})^{{\alpha}(n)}$.

######  Proposition 4.1.

  1. 1.

For $i=1,\ldots,n$, $a_{i}^{*}a_{i}$ is diagonalisable with spectrum
${\mathbb{N}}$. So we have a decomposition into mutually orthogonal joint
eigenspaces
$\mathcal{P}({\mathbb{C}}^{n})=\bigoplus_{{\alpha}\in{\mathbb{N}}^{n}}\mathcal{L}_{\alpha}$
with
$\mathcal{L}_{{\alpha}}=\bigcap_{i=1}^{n}\operatorname{ker}(a_{i}^{*}a_{i}-{\alpha}(i)).$

  2. 2.

$\mathcal{L}_{0}={\mathbb{C}}[z_{1},\ldots,z_{n}]$ and for any
${\alpha}\in{\mathbb{N}}^{n}$,
$\mathcal{L}_{{\alpha}}=(a^{*})^{\alpha}\mathcal{L}_{0}$.

  3. 3.

For any ${\alpha},{\beta}\in{\mathbb{N}}^{n}$, let
$\tilde{\rho}_{{\alpha}{\beta}}:=({\alpha}!{\beta}!)^{-\frac{1}{2}}(a^{*})^{{\alpha}}\tilde{\rho}_{00}a^{{\beta}}$
with $\tilde{\rho}_{00}$ the orthogonal projector of
$\mathcal{P}({\mathbb{C}}^{n})$ onto $\mathcal{L}_{0}$. Then

    1. (a)

$\tilde{\rho}_{{\alpha}{\beta}}$ is zero on the $\mathcal{L}_{{\gamma}}$’s
with ${\gamma}\neq{\beta}$ and restricts to a unitary isomorphism from
$\mathcal{L}_{{\beta}}$ to $\mathcal{L}_{{\alpha}}$

    2. (b)

$\tilde{\rho}_{{\alpha}{\alpha}}$ is the orthogonal projector onto
$\mathcal{L}_{\alpha}$.

    3. (c)

$\tilde{\rho}_{{\alpha}{\beta}}\circ\tilde{\rho}_{\tilde{{\alpha}}\tilde{{\beta}}}=\delta_{{\beta}\tilde{{\alpha}}}\tilde{\rho}_{{\alpha}\tilde{\beta}}$
and $\tilde{\rho}_{{\alpha}{\beta}}^{*}=\tilde{\rho}_{{\beta}{\alpha}}$

###### Proof.

The result is certainly standard in condensed matter theory. For the
convenience of the reader, we explain briefly the proof for $n=1$. The
extension in higher dimension is straightforward. We write $a:=a_{1}$, recall
the commutation relation $[a,a^{*}]=1$ and set
$\mathcal{L}_{m}:=(a^{*})^{m}({\mathbb{C}}[z])$ for any $m\in{\mathbb{N}}$.

We check by induction that $\mathcal{L}_{m}=\ker(a^{*}a-m)$. First, writing
$\langle a^{*}af,f\rangle=\|af\|^{2}$, it comes that $\ker a^{*}a=\ker
a=\mathcal{L}_{0}$. Assume now that $\mathcal{L}_{m}=\ker(a^{*}a-m)$. By the
commutation relation, $f\in\mathcal{L}_{m}$ implies that
$a^{*}aa^{*}f=(m+1)a^{*}f$, so $\mathcal{L}_{m+1}\subset\ker(a^{*}a-(m+1))$.
Conversely, by the commutation relation again, $a^{*}af=(m+1)f$ implies that
$(a^{*}a)af=maf$ so $af\in\mathcal{L}_{m}$ and
$f=(m+1)^{-1}a^{*}(af)\in\mathcal{L}_{m+1}$.

To conclude that $a^{*}a$ is diagonalizable with eigenvalues in
${\mathbb{N}}$, it suffices to prove that $\mathcal{P}({\mathbb{C}})$ is
spanned by the $\mathcal{L}_{m}$. Introduce the filtration
$\mathcal{F}_{m}:=\oplus_{\ell=0}^{m}\overline{z}^{\ell}{\mathbb{C}}[z]$,
$m\in{\mathbb{N}}$. If $f\in{\mathbb{C}}[z]$, then
$\overline{z}^{m}f=(a^{*})^{m}f\mod\mathcal{F}_{m-1}$. So
$\mathcal{F}_{m}=\mathcal{L}_{m}+\mathcal{F}_{m-1}=\ldots=\mathcal{L}_{m}+\mathcal{L}_{m-1}+\ldots+\mathcal{L}_{0}$
by reiterating.

So we have proved that $\mathcal{P}({\mathbb{C}})=\bigoplus\mathcal{L}_{m}$
with $\mathcal{L}_{m}=\ker(a^{*}a-m)$, which shows the first and second
assertions of the proposition. By the commutation relation, $aa^{*}=(m+1)$ on
$\mathcal{L}_{m}$. So $a^{*}:\mathcal{L}_{m}\rightarrow\mathcal{L}_{m+1}$ is
invertible with inverse
$(m+1)^{-1}a:\mathcal{L}_{m+1}\rightarrow\mathcal{L}_{m}$. We conclude that

  * •

if $p\leqslant m$, then $a^{p}$ restricts to an isomorphism from
$\mathcal{L}_{m}$ to $\mathcal{L}_{m-p}$, whose inverse is the restriction of
$\frac{(m-p)!}{m!}(a^{*})^{m}$ to $\mathcal{L}_{m-p}$.

  * •

if $p>m$, then $a^{p}(\mathcal{L}_{m})=\\{0\\}$.

With these two facts, we easily check the third assertion. ∎

By the last assertion of Proposition 4.1, the space
$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$ of endomorphisms of
$\mathcal{P}({\mathbb{C}}^{n})$ generated by the
$\tilde{\rho}_{{\alpha}{\beta}}$’s is closed under composition, so it is an
algebra. By the following proposition,
$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$ is isomorphic with the algebra
$\mathcal{S}({\mathbb{C}}^{n})$ introduced in Section 2.1, through the map
sending $\tilde{\rho}_{{\alpha}{\beta}}$ into $\rho_{{\alpha}{\beta}}$.

######  Proposition 4.2.

The elements of $\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$ preserve the
subspace $\mathcal{D}({\mathbb{C}}^{n})$ of $\mathcal{P}({\mathbb{C}}^{n})$.
Furthermore, the restriction map
$\operatorname{res}:\widetilde{\mathcal{S}}({\mathbb{C}}^{n})\rightarrow\operatorname{End}(\mathcal{D}({\mathbb{C}}^{n}))$
is injective, with image $\mathcal{S}({\mathbb{C}}^{n})$ and
$\operatorname{res}(\tilde{\rho}_{{\alpha}{\beta}})=\rho_{{\alpha}{\beta}}$.

Recall the decomposition (34) of $\mathcal{S}({\mathbb{C}}^{n})$ into the
subspaces of even and odd elements. Since
$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})\simeq\mathcal{S}({\mathbb{C}}^{n})$,
this gives us a new decomposition

$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})=\widetilde{\mathcal{S}}^{+}({\mathbb{C}}^{n})\oplus\widetilde{\mathcal{S}}^{-}({\mathbb{C}}^{n}).$

###### Proof.

Observe first that the operators $a_{i}$, $a_{i}^{*}$ and the projector
$\tilde{\rho}_{00}$ preserves $\mathcal{D}({\mathbb{C}}^{n})$. Furthermore,
for any $f\in\mathcal{D}({\mathbb{C}}^{n})$,
$a_{i}f=\partial_{\overline{z}_{i}}f$, $a_{i}^{*}f=\overline{z}_{i}f$ and
$\tilde{\rho}_{00}f=f(0)$. Consequently, the operators
$\tilde{\rho}_{{\alpha}{\beta}}$ preserve $\mathcal{D}({\mathbb{C}}^{n})$ and
an easy computation shows that

$\tilde{\rho}_{{\alpha}{\beta}}\bigl{(}({\beta}!)^{-\frac{1}{2}}\overline{z}^{\beta}\bigr{)}=({\alpha}!)^{-\frac{1}{2}}\overline{z}^{{\alpha}},\qquad\tilde{\rho}_{{\alpha}{\beta}}(\overline{z}^{{\gamma}})=0,\quad\forall{\gamma}\in{\mathbb{N}}^{n}\setminus\\{{\beta}\\}.$

This means that the restriction of $\tilde{\rho}_{{\alpha}{\beta}}$ to
$\mathcal{D}({\mathbb{C}}^{n})$ is exactly the endomorphism
$\rho_{{\alpha}{\beta}}$ introduced in Section 2.1, cf. Equation (32). So the
restriction map $\operatorname{res}$ is well-defined, its image is
$\mathcal{S}({\mathbb{C}}^{n})$, and the $\rho_{{\alpha}{\beta}}$’s being
linearly independent, it is injective. ∎

Let us compute the Schwartz kernel of each $\tilde{\rho}_{{\alpha}{\beta}}$.

######  Lemma 4.3.

For any $f\in\mathcal{P}({\mathbb{C}}^{n})$, we have

$(\tilde{\rho}_{{\alpha}{\beta}}f)(u)=(2\pi)^{-n}\int_{{\mathbb{C}}^{n}}e^{u\cdot\overline{v}-|v|^{2}}p_{{\alpha}{\beta}}(u-v)\,f(v)\;d\mu_{n}(v)$

where $u\cdot\overline{v}=\sum u_{i}\overline{v}_{i}$,
$p_{{\alpha},{\beta}}(z)=\bigl{(}{\alpha}!{\beta}!\bigr{)}^{-\frac{1}{2}}\bigl{(}\partial_{z}-\overline{z})^{{\alpha}}z^{\beta}$.

###### Proof.

For ${\alpha}={\beta}=0$, this is the well-known formula for the Schwartz
kernel $K(u,v)=(2\pi)^{-n}e^{u\cdot\overline{v}-|v|^{2}}$ of the projector
onto the Bargmann space, which in our setting is the $L^{2}$-completion of
$\mathcal{L}_{0}$. So the Schwartz kernel of
$(a^{*})^{{\alpha}}\tilde{\rho}_{00}a^{{\beta}}$ is
$K_{{\alpha},{\beta}}(u,v)=(\overline{u}-\partial_{u})^{{\alpha}}(-\partial_{\overline{v}})^{{\beta}}K(u,v)$.
To compute this, we use that for a polynomial $g(z,\overline{z})$

$\displaystyle(\overline{u}_{i}-\partial_{u_{i}})(K(u,v)g(u-v))$
$\displaystyle=K(u,v)(a_{i}^{*}g)(u-v),$
$\displaystyle(\partial_{\overline{v}_{i}})(K(u,v)g(u-v))$
$\displaystyle=K(u,v)(b_{i}^{*}g)(u-v)$

where $b_{i}^{*}:=z_{i}-\partial_{\overline{z}_{i}}$. So
$K_{{\alpha},{\beta}}(u,v)=K(u,v)p(u-v)$ with
$p(z,\overline{z})=(a^{*})^{{\alpha}}(b^{*})^{\beta}1=\bigl{(}\partial_{z}-\overline{z})^{{\alpha}}z^{\beta}$,
which ends the proof. ∎

So on one hand, the elements of $\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$
act on $\mathcal{P}({\mathbb{C}}^{n})$, on the other hand, the Schwartz kernel
of $\tilde{\rho}_{{\alpha}{\beta}}$ is given by a polynomial
$p_{{\alpha}{\beta}}\in\mathcal{P}({\mathbb{C}}^{n})$.

######  Proposition 4.4.

$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$ consists of the endomorphisms $V$
having the form

$\displaystyle(Vf)(u)=(2\pi)^{-n}\int_{{\mathbb{C}}^{n}}e^{u\cdot\overline{v}-|v|^{2}}q(u-v)f(v)\;d\mu_{n}(v)$
(52)

with $q\in\mathcal{P}({\mathbb{C}}^{n})$. Furthermore, the map
$\operatorname{Op}:\mathcal{P}({\mathbb{C}}^{n})\rightarrow\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$,
sending $q$ into $V$, is an isomorphism which preserves the parity and

$\displaystyle\begin{split}&\operatorname{tr}\bigl{(}\operatorname{Op}(f)|_{\mathcal{D}({\mathbb{C}}^{n})}\bigr{)}=f(0)\\\
&\operatorname{Op}(f)\circ\operatorname{Op}(g)=\operatorname{Op}(\operatorname{Op}(f)g)\end{split}$
(53)

for any $f,g\in\mathcal{P}({\mathbb{C}}^{n})$.

###### Proof.

Lemma 4.3 says that
$\operatorname{Op}(p_{{\alpha}{\beta}})=\tilde{\rho}_{{\alpha}{\beta}}$. The
family $(p_{{\alpha}{\beta}})$ is a basis of $\mathcal{P}({\mathbb{C}}^{n})$
because
$p_{{\alpha}{\beta}}(z)=({\alpha}!{\beta}!)^{-\frac{1}{2}}(-z)^{{\alpha}}\overline{z}^{\beta}+$
a linear combination of $z^{{\alpha}^{\prime}}\overline{z}^{{\beta}^{\prime}}$
with ${\alpha}^{\prime}<{\alpha}$ and ${\beta}^{\prime}<{\beta}$. Since
$(\rho_{{\alpha}{\beta}})$ is a basis of $\mathcal{S}({\mathbb{C}}^{n})$,
$(\tilde{\rho}_{{\alpha}{\beta}})$ is a basis of
$\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$ and it follows that
$\operatorname{Op}$ is an isomorphism. This isomorphism preserves the parity,
because $\tilde{\rho}_{{\alpha}{\beta}}$ and $p_{{\alpha}{\beta}}$ have both
the same parity as $|{\alpha}|+|{\beta}|$. For the first equation of (53), it
suffices to prove it for $q=p_{{\alpha},{\beta}}$, and in this case, it
follows from
$\operatorname{tr}\rho_{{\alpha}{\beta}}=\delta_{{\alpha}{\beta}}=p_{{\alpha}{\beta}}(0)$.
To prove the second equation of (53), observe that we recover $q$ from the
Schwartz kernel of $\operatorname{Op}(q)$ by setting $v=0$, that is

$e^{u\cdot\overline{v}-|v|^{2}}q(u-v)\Bigr{|}_{v=0}=q(u).$

Applying this in the Schwartz kernel composition formula leads to (53). ∎

As a last remark, we can replace in the previous definitions
${\mathbb{C}}^{n}$ with any $n$-dimensional Hermitian space $\mathbf{E}$ as we
did in section 2.1. So we denote by $\mathcal{P}(\mathbf{E})$ the space of
polynomial maps $\mathbf{E}\rightarrow{\mathbb{C}}$ and by
$\widetilde{\mathcal{S}}(\mathbf{E})$ the space of endomorphisms of
$\mathcal{P}(\mathbf{E})$ having the form (52), where we interpret
$u\cdot\overline{v}$ as the scalar product of the vectors $u$, $v$ of
$\mathbf{E}$ and $|v|$ as the norm of $v$. Observe as well that the map
$\operatorname{Op}:\mathcal{P}(\mathbf{E})\rightarrow\widetilde{\mathcal{S}}(\mathbf{E})$
is well-defined. Furthermore the restriction from $\mathcal{P}(\mathbf{E})$ to
its subspace $\mathcal{D}(\mathbf{E})$ induces an isomorphism
$\widetilde{\mathcal{S}}(\mathbf{E})\simeq\mathcal{S}(\mathbf{E})$.

##  5 The Schwartz kernels of operators of $\mathcal{L}(A,B)$

###  5.1 The section $E$

An important ingredient in the global Schwartz kernel description of operators
of $\mathcal{L}(A,B)$ is a section $E$ of $L\boxtimes\overline{L}$ satisfying
the following conditions. For any $y\in M$, denote by $E_{y}$ the section of
$L\otimes\overline{L}_{y}$ given by $E_{y}(x)=E(x,y)$ for any $x\in M$. Then
we will assume that for any $y\in M$

$\displaystyle\begin{split}&E_{y}(y)=u\otimes\overline{u},\qquad\forall u\in
L_{y}\text{ with }|u|=1,\\\ &(\nabla E_{y})(y)=0,\\\
&(\nabla_{\xi}\nabla_{\eta}E_{y})(y)=-\bigl{(}\tfrac{i}{2}{\omega}(\xi,\eta)+\tfrac{1}{2}{\omega}(\xi,j\eta)\bigr{)}E_{y}(y),\qquad\forall\;\xi,\eta\in
T_{y}M\end{split}$ (54)

Such a section appeared already in the expansion (38) as follows. Choose a
unitary frame $t$ of $L$ and a coordinate system on the same open set, then
the section

$\displaystyle E(y+\xi,y):=e^{-\varphi(y,\xi)}t(y+\xi)\otimes\overline{t}(y),$
(55)

with $\varphi$ defined as in (38), satisfies (54). From this local
construction, we easily obtain a global section $E$ by using a partition of
unity.

The conditions (54) determine the second-order Taylor expansion of $E$ at
$(y,y)$ in the directions tangent to the first factor of $M^{2}$. Since any
tangent vector of $M^{2}$ at $(y,y)$ is the sum of a vector tangent to the
diagonal and a vector tangent to the first factor, we deduce that $E$ is
uniquely determined modulo a section vanishing to third order along the
diagonal.

The function $\psi_{y}(x)=-2\ln|E_{y}(x)|$ vanishes to second order at $y$ and
for any $\xi,\eta\in T_{y}M$, $(\xi.\eta.\psi_{y})(y)={\omega}(\xi,j\eta)$, so
$\psi_{y}(x)>0$ when $x\neq y$ is sufficiently close to $y$. So modifying $E$
outside the diagonal, we can assume that it satisfies as well

$\displaystyle|E(x,y)|<1,\qquad\forall(x,y)\in M^{2}\text{ such that }x\neq y$
(56)

Another important property of $E$ is the symmetry:

$\displaystyle\overline{E(x,y)}=E(y,x)+\mathcal{O}(|x-y|^{3})$ (57)

For a longer discussion, the reader is referred to [9].

In the sequel we will need the following expression of $E$ in terms of complex
coordinates and a frame of $L$, both normal at a point $p_{0}\in M$. We say
that a function or a section is in $\mathcal{O}_{p_{0}}(m)$ if it vanishes to
order $m$ at $p_{0}$. Let $(\partial_{i})_{i=1}^{n}$ be an orthonormal basis
of $T^{1,0}_{p_{0}}M$, i.e.
$\frac{1}{i}{\omega}_{p_{0}}(\partial_{i},\overline{\partial}_{j})=\delta_{ij}$.
Choose complex valued functions $z_{i}$ on a neighborhood of $p_{0}$ such that

$\displaystyle z_{i}(p_{0})=0,\qquad dz_{i}(\partial_{j})=\delta_{ij},\quad
dz_{i}(\overline{\partial}_{j})=0\quad\text{ at }p_{0}.$ (58)

Then $(\operatorname{Re}z_{i},\operatorname{Im}z_{i})_{i=1}^{n}$ is a
coordinate system on a neighborhood of $p_{0}$ and ${\omega}|_{p_{0}}=i\sum
dz_{i}\wedge d\overline{z}_{i}$. The curvature of $L$ being
$\frac{1}{i}{\omega}$, there exists a unitary frame $t$ of $L$ at $p_{0}$ such
that $\nabla
t=\frac{1}{2}\textstyle{\sum}(z_{i}d\overline{z}_{i}-\overline{z}_{i}dz_{i})\otimes
t+\mathcal{O}_{p_{0}}(2)$. Then

$\displaystyle
E(x,y)=e^{z(x)\cdot\overline{z}(y)-\frac{1}{2}(|z(x)|^{2}+|z(y)|^{2})}t(x)\otimes\overline{t(y)}+\mathcal{O}_{(p_{0},p_{0})}(3).$
(59)

A similar expression already appeared in the description of Schwartz kernel of
the operators of $\widetilde{\mathcal{S}}({\mathbb{C}}^{n})$. Indeed, let
$L_{{\mathbb{C}}^{n}}={\mathbb{C}}^{n}\times{\mathbb{C}}$ be the trivial
holomorphic line bundle equipped with the metric such that the frame
$s(z)=(z,1)$ has a pointwise norm $|s(z)|^{2}=e^{-|z|^{2}}$. Then it is
natural to interpret the elements of $\mathcal{P}({\mathbb{C}}^{n})$ as
sections of $L_{{\mathbb{C}}^{n}}$, because the scalar product (51) is the
integral of the pointwise scalar product
$\bigl{(}f(z)s(z),g(z)s(z)\bigr{)}=f(z)\overline{g}(z)e^{-|z|^{2}}$.
Furthermore, in the integral (52), we can interpret $e^{-|v|^{2}}f(v)$ as the
pointwise scalar product $(f(v)s(v),s(v))$. In other words, the Schwartz
kernel of $V$ is

$(2\pi)^{n}E_{{\mathbb{C}}^{n}}(u,v)q(u-v)\quad\text{ with }\quad
E_{{\mathbb{C}}^{n}}(u,v)=e^{u\cdot\overline{v}}s(u)\otimes\overline{s}(v).$

Now equip $L_{{\mathbb{C}}^{n}}$ with its Chern connection, that the unique
connection compatible with both the holomorphic and Hermitian structures. Then
$\nabla s=-\sum\overline{z}_{i}dz_{i}\otimes s$. So the curvature is
$\frac{1}{i}{\omega}_{{\mathbb{C}}^{n}}$ with
${\omega}_{{\mathbb{C}}^{n}}=i\sum dz_{i}\wedge d\overline{z}_{i}$. And if $t$
is the unitary frame $t(z)=e^{|z|^{2}/2}s(z)$, we have $\nabla
t=\frac{1}{2}\sum(z_{i}d\overline{z}_{i}-\overline{z}_{i}dz_{i})\otimes t$ and

$E_{{\mathbb{C}}^{n}}(u,v)=e^{u\cdot\overline{v}-\frac{1}{2}(|u|^{2}+|v|^{2})}t(u)\otimes\overline{t(v)},$

the same formula as (59).

###  5.2 A global expansion

We consider operator family $(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes B),\;k\in{\mathbb{N}})$
having a smooth Schwartz kernel. Recall the notations introduced in the
beginning of Section 2.2. In particular, $\|P_{k}\|$ is the operator norm
whereas $|P_{k}|$ is the function of $M^{2}$ sending $(x,y)$ into
$|P_{k}(x,y)|$.

Let $E$ be a section of $L\boxtimes\overline{L}$ satisfying (54) and (56) and
$b\in{\mathcal{C}}^{\infty}(M^{2},B\boxtimes\overline{A})$. Then, viewing
$(L^{k}\otimes B)\boxtimes(\overline{L}^{k}\otimes\overline{A})$ as
$(L\boxtimes\overline{L})^{k}\otimes(B\boxtimes\overline{A})$, we introduce
the operator family $(P_{k})$ with Schwartz kernels

$\displaystyle P_{k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)b(x,y)$
(60)

The norms of $P_{k}$ depends in an essential way on the vanishing order of $b$
along the diagonal. If $m\in{\mathbb{N}}$, we write $b=\mathcal{O}(m)$ to say
that all the derivatives of $b$ of order $\leqslant m-1$ are zero at each
point of the diagonal. Recall that

$\psi(x,y)=-2\ln|E(x,y)|$

is positive outside the diagonal, and vanishes to second order along the
diagonal with a Hessian non-degenerate in the transverse direction.

######  Lemma 5.1.

If $P_{k}$ is given by (60) with $b=\mathcal{O}(m)$, then
$|P_{k}|=\mathcal{O}\bigl{(}k^{n-\frac{m}{2}}e^{-k\frac{\psi}{2}}\bigr{)}$ and
$\|P_{k}\|=\mathcal{O}(k^{-\frac{m}{2}})$

###### Proof.

Since $b=\mathcal{O}(m)$, $|b|=\mathcal{O}(\psi^{\frac{m}{2}})$. So

$|P_{k}|\leqslant
Ck^{n}e^{-k\psi}\psi^{\frac{m}{2}}=Ck^{n-\frac{m}{2}}e^{-k\frac{\psi}{2}}e^{-k\frac{\psi}{2}}(k\psi)^{\frac{m}{2}}\leqslant
C^{\prime}k^{n-\frac{m}{2}}e^{-k\frac{\psi}{2}}$

because $t\rightarrow e^{-\frac{t^{2}}{2}}t^{m}$ is bounded on
${\mathbb{R}}_{\geqslant 0}$. This proves the first estimate and can be
written locally in a coordinate system as

$|P_{k}(x,y)|\leqslant Ck^{n-\frac{m}{2}}e^{-k|x-y|^{2}/C}.$

So $\int_{M}|P_{k}(x,y)|\,d\mu_{M}(y)$ and $\int_{M}|P_{k}(x,y)|\,d\mu_{M}(x)$
are both $\leqslant Ck^{-\frac{m}{2}}$ and the operator norm estimate follows
from Schur test. ∎

More generally, instead of $P_{k}$ given by (60), assume that for any
$N\in{\mathbb{N}}$,

$\displaystyle
P_{k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)\sum_{\begin{subarray}{c}\ell\in{\mathbb{Z}},\\\
\ell+m(\ell)\leqslant
N\end{subarray}}k^{-\frac{\ell}{2}}b_{\ell}(x,y)+R_{N,k}(x,y)$ (61)

where

  1. i)

$m:{\mathbb{Z}}\rightarrow{\mathbb{N}}\cup\\{\infty\\}$ is such that for any
$N$, $\\{\ell/\;\ell+m(\ell)\leqslant N\\}$ is finite, and
$\ell+m(\ell)\geqslant 0$ for any $\ell$.

  2. ii)

$(b_{\ell})_{\ell\in{\mathbb{Z}}}$ is a family of
${\mathcal{C}}^{\infty}(M^{2},B\boxtimes\overline{A})$ such that
$b_{\ell}=\mathcal{O}(m(\ell))$ for any $\ell$.

  3. iii)

$|R_{N,k}(x,y)|=\mathcal{O}(k^{n-\frac{N+1}{2}})$ uniformly on $M^{2}$.

By Lemma 5.1,
$|E^{k}(x,y)k^{-\frac{\ell}{2}}b_{\ell}(x,y)|\in\mathcal{O}(k^{-\frac{1}{2}(\ell+m(\ell))})$.
So the expansion (61) is consistent in the sense that passing from $N$ to
$N+1$, we add new terms $k^{-\frac{\ell}{2}}b_{\ell}$ such that
$\ell+m(\ell)=N+1$, which contribute to $P_{k}(x,y)$ with a
$\mathcal{O}(k^{n-\frac{1}{2}(N+1)})$.

######  Lemma 5.2.

If the expansion (61) holds, then for any $q>0$

$|P_{k}|=\mathcal{O}\bigl{(}k^{n}e^{-k\frac{\psi}{2}}\bigr{)}+\mathcal{O}(k^{-q}),\qquad\|P_{k}\|=\mathcal{O}(1)$

Similarly the remainders $R_{N,k}$’s satisfy for any $q>0$,

$|R_{N,k}|=\mathcal{O}\bigl{(}k^{n-\frac{N+1}{2}}e^{-k\frac{\psi}{2}}\bigr{)}+\mathcal{O}(k^{-q}),\qquad\|R_{N,k}\|=\mathcal{O}(k^{-\frac{N+1}{2}}).$

###### Proof.

To prove the first estimate, we use (61) with $N$ sufficiently large so that
$|R_{N,k}|=\mathcal{O}(k^{-q})$, and the result follows from Lemma 5.1 because
$\ell+m(\ell)\geqslant 0$. The operator norm estimate is proved similarly by
choosing $N$ so that $|R_{N,k}|=\mathcal{O}(1)$ which implies that
$\|R_{N,k}\|=\mathcal{O}(1)$. The proof for the $R_{N,k}$ is essentially the
same. ∎

We next show that in the expansion (61), we can choose any section $E$
satisfying the assumptions given in Section 5.1.

######  Lemma 5.3.

Assume (61) holds and let $E^{\prime}$ be a section satisfying (54) and (56).
Then there exists a family $(b^{\prime}_{\ell})$ of
${\mathcal{C}}^{\infty}(M^{2},B\boxtimes\overline{A})$ such that (61) holds
with $E^{\prime}$ and $b^{\prime}_{\ell}$ instead of $E$ and $b_{\ell}$.

###### Proof.

Observe first that (61) holds outside the diagonal if and only $|P_{k}(x,y)|$
is in $\mathcal{O}(k^{-\infty})$ outside the diagonal, and this condition is
clearly independent of the choice of $E$ and the $b_{\ell}$’s. On a
neighborhood of the diagonal, we have $E=e^{g}E^{\prime}$. Since
$g\in\mathcal{O}(3)$, we can assume that $|g|\leqslant\psi/4$. Let us write

$P_{k}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}b_{N,k}+\mathcal{O}(k^{n-\frac{N+1}{2}})\quad\text{with
}\quad b_{N,k}=\sum_{\ell+m(\ell)\leqslant N}k^{-\frac{\ell}{2}}b_{\ell}.$

By Lemma 5.1, $|E|^{k}b_{N,k}=\mathcal{O}(e^{-k\frac{\psi}{2}})$. Using that

$\exp z=\sum_{p=0}^{N}\frac{z^{p}}{p!}+r_{N}(z)\quad\text{ with
}\quad|r_{N}(z)|\leqslant\frac{|z|^{N+1}}{(N+1)!}e^{|\operatorname{Re}z|},$

we deduce from $E^{k}=e^{kg}(E^{\prime})^{k}$ that

$\displaystyle
E^{k}b_{N,k}=(E^{\prime})^{k}b_{N,k}\sum_{p=0}^{N}k^{p}\frac{g^{p}}{p!}+R_{N,k}$
(62)

where

$|R_{N,k}|\leqslant
C_{N}e^{-k\frac{\psi}{2}}|kg|^{N+1}e^{k|\operatorname{Re}g|}\leqslant
C_{N}e^{-k\frac{\psi}{4}}|kg|^{N+1}=\mathcal{O}(k^{-\frac{N+1}{2}})$

because $g=\mathcal{O}(3)$. To conclude now, it suffices to define the
$b^{\prime}_{\ell}$ so that

$\displaystyle(E^{\prime})^{k}\Bigl{[}\sum_{\ell+m(\ell)\leqslant
N}k^{-\frac{\ell}{2}}b_{\ell}\Bigr{]}\Bigl{[}\sum_{p=0}^{N}k^{p}\frac{g^{p}}{p!}\bigr{]}=(E^{\prime})^{k}\sum_{\ell+m^{\prime}(\ell)\leqslant
N}k^{-\frac{\ell}{2}}b^{\prime}_{\ell}+\mathcal{O}(k^{-\frac{N+1}{2}})$ (63)

holds for any $N$. This suggests that each $b^{\prime}_{\ell}$ should be equal
to the infinite sum

$b_{\ell}+b_{\ell+2}\,g+b_{\ell+4}\frac{g^{2}}{2}+b_{\ell+6}\frac{g^{3}}{6}+\ldots$

But by Lemma 5.1, the equality (63) depends only on the class of
$b^{\prime}_{\ell}$ modulo $\mathcal{O}(N-\ell)$, so we can interpret these
infinite sums as sums of Taylor expansions along the diagonal. Since
$b_{\ell+2p}\,g^{p}=\mathcal{O}(m(\ell+2p)+3p)=\mathcal{O}(3p)$, by Borel
lemma, there exists $b^{\prime}_{\ell}$ such that for any $M$

$\displaystyle
b^{\prime}_{\ell}=\sum_{p=0}^{M}b_{\ell+2p}\frac{g^{p}}{p!}+\mathcal{O}(3(M+1))$
(64)

So $b^{\prime}_{\ell}=\mathcal{O}(m^{\prime}(\ell))$ with
$m^{\prime}(\ell):=\operatorname{min}\\{m(\ell+2p)+3p,\;p\in{\mathbb{N}}\\}.$
We easily check that $m^{\prime}$ satisfies the same condition as $m$. We
finally deduce (63) by removing with Lemma 5.1 all the coefficients leading to
a $\mathcal{O}(k^{-\frac{N+1}{2}})$. ∎

Suppose now we have an open set $U$ of $M$, and functions
$u_{i}\in{\mathcal{C}}^{\infty}(U^{2})$, $i=1,\ldots,2n$ vanishing along the
diagonal and such that for any $y\in U$, $(u_{i}(\cdot,y))$ is a coordinate
system on a neighborhood of $y$. Then we can write the Taylor expansions along
the diagonal as follows: any $f\in{\mathcal{C}}^{\infty}(U^{2})$ has a
decomposition

$\displaystyle f(x,y)=\sum_{m=0}^{M}f_{m}(y,u(x,y))+\mathcal{O}(M+1)$ (65)

where each $f_{m}(y,\xi)$ is homogeneous polynomial in $\xi$ with degree $m$.
This can be done also for sections of $B\boxtimes\overline{A}$ by introducing
frames of $A$ and $B$ on $U$, so that
${\mathcal{C}}^{\infty}(U^{2},B\boxtimes\overline{A})\simeq{\mathcal{C}}^{\infty}(U^{2},{\mathbb{C}}^{r})$.

######  Lemma 5.4.

The expansion (61) holds on $U^{2}$ if and only if there exists a sequence
$(a_{p})$ of
${\mathcal{C}}^{\infty}(U\times{\mathbb{R}}^{2n},{\mathbb{C}}^{r})$, each
$a_{p}(x,\xi)$ being polynomial in $\xi$, such that for any $N$

$\displaystyle
P_{k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)\sum_{p=0}^{N}k^{-\frac{p}{2}}a_{p}(y,k^{\frac{1}{2}}u(x,y))+\mathcal{O}(k^{n-\frac{N+1}{2}}).$
(66)

The remainders in (66) satisfy the same pointwise estimates as the $R_{N,k}$
given in Lemma 5.2, the proof is identical.

###### Proof.

If (61) holds on $U^{2}$, writing the Taylor expansion of each $b_{\ell}$ as
in (65), we have by Lemma 5.1

$\displaystyle
E^{k}(x,y)k^{-\frac{\ell}{2}}b_{\ell}(x,y)=E^{k}(x,y)k^{-\frac{\ell}{2}}\sum_{m=m(\ell)}^{N-\ell}b_{\ell,m}(y,u(x,y))+\mathcal{O}(k^{-\frac{N+1}{2}})$
$\displaystyle=E^{k}(x,y)k^{-\frac{\ell+m}{2}}\sum_{m=m(\ell)}^{N-\ell}b_{\ell,m}(y,k^{\frac{1}{2}}u(x,y))+\mathcal{O}(k^{-\frac{N+1}{2}})$

So we obtain (66) with $a_{p}=\sum_{\ell+m(\ell)\leqslant p}b_{\ell,p-\ell}$,
this sum being finite because of the assumption satisfied by $m(\ell)$.

Conversely, starting from the $a_{p}$’s, for each $\ell\in{\mathbb{Z}}$, we
construct by Borel summation a function $b_{\ell}$ such that
$b_{\ell}(x,y)=\sum_{m=0}^{M}a_{\ell+m,m}(y,u(x,y))+\mathcal{O}(M+1)$ for all
$M$, where by convention $a_{p}=0$ for $p<0$, and $a_{m+\ell,m}$ is the degree
$m$ homogeneous component of $a_{m+\ell}$. We readily deduce the expansion
(61) from (66) by using Lemma 5.1 again.

Observe that $b_{\ell}=\mathcal{O}(m(\ell))$ with $m(\ell)$ the smallest $m$
such that $a_{\ell+m,m}\neq 0$. Since $a_{p}=0$ for $p<0$, we have
$\ell+m(\ell)\geqslant 0$. Furthermore, $\ell+m(\ell)\leqslant N$ happens only
if there exists $m\leqslant N-\ell$ such that $a_{\ell+m,m}\neq 0$, that is if
there exists $p\leqslant N$ such that $a_{p,p-\ell}\neq 0$, so necessarily
$p-\ell\leqslant d(p)$ where $d(p)$ is the degree of $a_{p}$. So
$\ell+m(\ell)\leqslant N$ implies that
$\ell\geqslant\min\\{p-d(p)/\;p=0,\ldots,N\\}$. So $\ell+m(\ell)\leqslant N$
only for a finite number of $\ell$. ∎

We can now explain the relation with the space $\mathcal{L}(A,B)$ introduced
in Section 2.2. Identify $U$ with an open convex set of ${\mathbb{R}}^{2n}$,
then the functions $u_{i}(x,y)=x_{i}-y_{i}$ satisfy the above conditions. And
for $(x,y)=(x^{\prime}+\xi^{\prime},x^{\prime})$, we have
$a_{p}(y,k^{\frac{1}{2}}u(x,y))=a_{p}(x^{\prime},k^{\frac{1}{2}}\xi^{\prime})$,
so the expansions (66) and (38) are the same when $E=e^{-\varphi}$.

######  Proposition 5.5.

Let $(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes
A)\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes B),\;k\in{\mathbb{N}})$ be
a family of operators with smooth Schwartz kernels. Then $(P_{k})$ belongs to
$\mathcal{L}(A,B)$ if and only if the Schwartz kernel family of the $P_{k}$’s
has the expansion (61) for some family $(b_{\ell})$ of
${\mathcal{C}}^{\infty}(M^{2},B\boxtimes\overline{A})$.

###### Proof.

This follows from Lemma 5.4, the local version of Lemma 5.3 and the fact that
$E=e^{-\varphi}$ in (38) satisfies the conditions (54). ∎

The parity of an element of $\mathcal{L}(A,B)$ is particularly easy to
understand with the global expansion (61).

######  Proposition 5.6.

For any $(P_{k})$ in $\mathcal{L}(A,B)$, $(P_{k})$ is even (resp. odd) if and
only if its Schwartz kernel has the global expansion (61) with $b_{\ell}=0$
for any odd $\ell\in{\mathbb{Z}}$ (resp. even).

###### Proof.

This follows from the relation between the coefficients $a_{p}$ and the
coefficients $b_{\ell}$ given in the proof of Lemma 5.4. For instance, if
$b_{\ell}=0$ for any odd integer $\ell$, then $b_{\ell,p-\ell}\neq 0$ only for
even $\ell$ and in this case it has the same parity as $p$, so
$a_{p}=\sum_{\ell+m(\ell)\leqslant p}b_{\ell,p-\ell}$ has the same parity as
$p$. Conversely, if $a_{p}$ has the same parity as $p$ for any $p$, then
$a_{\ell+m,m}=0$ for odd $\ell$, so $b_{\ell}$ vanishes to infinite order on
the diagonal for odd $\ell$, so we can assume that $b_{\ell}=0$. The proof for
odd elements is the same. ∎

###  5.3 Filtration and symbol

For an operator $(P_{k})\in\mathcal{L}(A,B)$, we have two different ways of
writing the expansion of its Schwartz kernel: a global one (61) with
coefficients $b_{\ell}$ and a local one (66) with coefficients $a_{p}$. We now
discuss the uniqueness of these coefficients. Recall that
$(P_{k})\in\mathcal{L}_{q}(A,B)$ if in all the local expansions (66), the
coefficients $a_{p}$ are zero for $p<q$.

######  Proposition 5.7.

  1. 1.

In the local expansions (66), the coefficients $a_{p}$ are uniquely determined
by the section $E$, the functions $(u_{i})$ and the frames of $A$ and $B$.

  2. 2.

In the global expansion (61), the Taylor expansions of the coefficients
$b_{\ell}$ along the diagonal are uniquely determined by the section $E$.

  3. 3.

$(P_{k})\in\mathcal{L}_{q}(A,B)$ iff
$\bigl{(}\forall\,\ell\in{\mathbb{Z}},\;b_{\ell}=\mathcal{O}(q-\ell)\bigr{)}$
iff $|P_{k}|=\mathcal{O}(k^{n-\frac{q}{2}})$.

  4. 4.

If $(P_{k})\in\mathcal{L}_{q}(A,B)$, then the coefficient $a_{q}$ of the local
expansion (66), viewed as a section of $\mathcal{P}(TM)\otimes
B\otimes\overline{A}\rightarrow U$ does neither depend on $E$ nor on the
functions $(u_{i})$. Furthermore,

$\displaystyle a_{q}=\sum_{\ell+m(\ell)=q}b_{\ell,q-\ell}$ (67)

where the $b_{\ell}$ are the coefficients of the global expansion (61) and
$b_{\ell,q-\ell}$ is defined as in (65).

###### Proof.

Assertions 1, 2 and 3 follow from the following facts: Let $f_{0}$, …, $f_{q}$
in ${\mathcal{C}}^{\infty}(M^{2})$. Let $\psi=-2\ln|E|$. Then

$\displaystyle
e^{-k\psi}\sum_{\ell=0}^{q}k^{-\frac{\ell}{2}}f_{\ell}=\mathcal{O}(k^{-\frac{q}{2}})\quad\Leftrightarrow\quad
f_{0}\in\mathcal{O}(q),\,\ldots,\;f_{q}\in\mathcal{O}(0).$ (68)

Indeed, recall that $\psi\geqslant 0$, is in $\mathcal{O}(2)$ and its Hessian
is non-degenerate in the direction transverse to the diagonal. The converse of
(68) follows from the same proof as Lemma 5.1. The direct sense of (68)
follows from [9, Proposition 2.4 and Remark 2.5].

From this, we deduce that $|P_{k}|=\mathcal{O}(k^{n-\frac{q}{2}})$ iff
$b_{\ell}=\mathcal{O}(q-\ell)$ for any $\ell$. Since
$a_{p}=\sum_{\ell+m(\ell)\leqslant p}b_{\ell,p-\ell}$ by the proof of Lemma
5.4, $(b_{\ell}=\mathcal{O}(q-\ell)$ for any $\ell$) iff $(a_{p}=0,$ for any
$p<q$). This last condition is the definition of $\mathcal{L}_{q}(A,B)$. We
have just proved Assertion 3. This implies that
$|P_{k}|=\mathcal{O}(k^{-\infty})$ iff ($a_{p}=0$ for any $p$) iff
$(b_{\ell}=\mathcal{O}(\infty)$ for any $\ell$), which proves Assertions 1 and
2.

For the fourth assertion, since $P\in\mathcal{L}_{q}(A,B)$, we have
$b_{\ell}=\mathcal{O}(q-\ell)$ for any $\ell$, so we can assume that
$\ell+m(\ell)\geqslant q$, so

$a_{q}=\sum_{\ell+m(\ell)\leqslant
q}b_{\ell,q-\ell}=\sum_{\ell+m(\ell)=q}b_{\ell,q-\ell}.$

Since $b_{\ell}=\mathcal{O}(q-\ell)$, the $(q-\ell)$-th order term in the
Taylor expansion of $b_{\ell}$ is intrinsically defined as a function

$\xi\in T_{x}M\rightarrow b_{\ell,q-\ell}(x,\xi)\in
B_{x}\otimes\overline{A}_{x},$

so we can view $a_{q}(x,\cdot)$ as an element of $\mathcal{P}(T_{x}M)\otimes
B_{x}\otimes\overline{A}_{x}$.

It remains to prove that for $\ell+m(\ell)=q$, $b_{\ell,q-\ell}$ does not
depend on the choice of $E$. With the notation of the proof of Lemma 5.3, this
amounts to prove that $b^{\prime}_{\ell}=b_{\ell}+\mathcal{O}(q-\ell+1)$. This
follows from (64), because $b_{\ell+2p}g^{p}=\mathcal{O}(m(\ell+2p)+3p)$ and
$m(\ell+2p)+3p\geqslant q-(\ell+2p)+3p=q+p-\ell\geqslant q+1-\ell$ when
$p\geqslant 1$. ∎

We are now ready to define the symbol map

${\sigma}_{q}:\mathcal{L}_{q}(A,B)\rightarrow{\mathcal{C}}^{\infty}(\mathcal{S}(M)\otimes\operatorname{Hom}(A,B)).$

First, for any $x\in M$, $T_{x}M$ is a Hermitian space, so it has an
associated algebra $\widetilde{\mathcal{S}}(T_{x}M)$ with a map
$\operatorname{Op}:\mathcal{P}(T_{x}M)\rightarrow\widetilde{\mathcal{S}}(T_{x}M)$
as in Section 4.

For any $(P_{k})$ in $\mathcal{L}_{q}(A,B)$, by the fourth assertion of
Proposition 5.7, $a_{q}(x,\cdot)\in\mathcal{P}(T_{x}M)\otimes
B_{x}\otimes\overline{A}_{x}$. Identifying $B_{x}\otimes\overline{A}_{x}$ with
$\operatorname{Hom}(A_{x},B_{x})$, we set

$\displaystyle\widetilde{{\sigma}}_{q}(P)(x):=\operatorname{Op}(a_{q}(x,\cdot))\in\widetilde{\mathcal{S}}(T_{x}M)\otimes\operatorname{Hom}(A_{x},B_{x})$
(69)

Recall that we have an isomorphism
$\widetilde{\mathcal{S}}(T_{x}M)\simeq\mathcal{S}(T_{x}M)$ defined by
restriction from $\mathcal{P}(T_{x}M)$ to $\mathcal{D}(T_{x}M)$.

######  Definition 5.8.

${\sigma}_{q}(P)(x)\in\mathcal{S}(T_{x}M)\otimes\operatorname{Hom}(A_{x},B_{x})$
is defined as the restriction of $\widetilde{{\sigma}}_{q}(P)(x)$.

###  5.4 Proofs of the results of Section 2.2

We now give the proof of Proposition 2.1, Theorem 2.2 and Theorem 2.5.

###### Proof of Proposition 2.1.

The first assertion is an easy consequence of the definition of
$\mathcal{L}_{q}(A,B)$ by the local expansions. In the second assertion, the
characterisation in terms of pointwise norm is the third assertion of
Proposition 5.7. By Lemma 5.1 or Lemma 5.2, every
$(P_{k})\in\mathcal{L}_{q}(A,B)$ satisfies
$\|P_{k}\|=\mathcal{O}(k^{-\frac{q}{2}})$. For the converse, it suffices to
show that if ${\sigma}_{0}(P)\neq 0$, then $\|P_{k}\|\geqslant c>0$. This is a
consequence of Corollary 5.10. The third assertion is straightforward. The
fourth assertion is a variation on Borel Lemma, cf. for instance [9,
Proposition 2.1]. ∎

###### Proof of Theorem 2.2.

In Definition 5.8, we have defined a map

${\sigma}_{q}:\mathcal{L}_{q}(A,B)\rightarrow{\mathcal{C}}^{\infty}(M,\mathcal{S}(M)\otimes\operatorname{Hom}(A,B)).$

having kernel $\mathcal{L}_{q+1}(A,B)$ by Proposition 5.7. To prove that it is
surjective, we show that that for any
$c\in{\mathcal{C}}^{\infty}(M,\mathcal{P}(TM)\otimes B\otimes\overline{A})$,
there exists $P\in\mathcal{L}_{q}(A,B)$ such that in the local expansions
(66), $a_{q}=c$. To do this, let $d(q)\in{\mathbb{N}}$ be an upper bound of
the degree of $c(x,\cdot)$ for any $x\in M$. For any $m=0,\ldots,d(q)$, let
$c_{m}(x,\cdot)$ be the homogeneous component with degree $m$ of $c(x,\cdot)$.
Choose a section $b_{q-m}$ of $B\boxtimes\overline{A}$ vanishing to order $m$
along the diagonal and satisfying
$b_{q-m}(x+\xi,x)=c_{m}(x,\xi)+\mathcal{O}(m+1)$. Then we set

$P_{k}(x,y):=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,y)\sum_{\ell=q-d(q)}^{q}k^{-\frac{\ell}{2}}b_{\ell}(x,y).$

Since $b_{\ell}=\mathcal{O}(q-\ell)$ for any $\ell$,
$(P_{k})\in\mathcal{L}_{q}(A,B)$. By (67), we have
$a_{q}=\sum_{\ell=q-d(q)}^{q}b_{\ell,q-\ell}=\sum_{m=0}^{d(q)}c_{m}=c$, as was
to be proved.

Let us prove the remaining assertions of Theorem 2.2. Let
$P\in\mathcal{L}_{q}(A,B)$. Assertion 1, that is
${\sigma}_{q}(P)={\sigma}_{0}(k^{q/2}P)$, follows directly from the local
expansions (66). Let us prove Assertion 2. If
$f\in{\mathcal{C}}^{\infty}(M,\operatorname{Hom}(B,C))$, then the Schwartz
kernel of $P_{k}^{\prime}=f\circ P_{k}$ is
$P_{k}^{\prime}(x,y)=f(x)(P_{k}(x,y))$, so $P_{k}^{\prime}$ has the same
expansion (61) as $P_{k}$ with $b^{\prime}_{\ell}(x,y)=f(x)(b_{\ell}(x,y))$
instead of $b_{\ell}$, which implies that $P^{\prime}_{k}$ belongs to
$\mathcal{L}_{q}(A,B)$ with the same function $\ell\mapsto m(\ell)$.
Furthermore, with the notation (65),
$b^{\prime}_{\ell,m(\ell)}(x,\cdot)=f(x)b_{\ell,m(\ell)}(x,\cdot)$, which
implies by (67) that
${\sigma}_{q}(P^{\prime})(x)=f(x)\circ{\sigma}_{q}(P)(x)$.

Let us prove Assertion 3. Since $P_{k}^{*}(x,y)=\overline{P_{k}(y,x)}$, the
Schwartz kernel of $P_{k}^{*}$ has the expansion (61) with
$E^{\prime}(x,y)=\overline{E(y,x)}$ instead of $E$ and
$b^{\prime}_{\ell}(x,y)=\overline{b_{\ell}(y,x)}$ instead of $b_{\ell}$. By
(57), we deduce that $(P_{k}^{*})\in\mathcal{L}_{q}(B,A)$. Furthermore,
$b^{\prime}_{\ell,m(\ell)}(x,\xi)=\overline{b}_{\ell,m(\ell)}(x,-\xi)$ so
$a_{q}^{\prime}(x,\xi)=\overline{a}_{q}(x,-\xi)$. By (52),
$\operatorname{Op}(q)^{*}=\operatorname{Op}(r)$ with
$r(\xi)=\overline{q}(-\xi)$, so ${\sigma}_{q}(P^{*})={\sigma}_{q}(P)^{*}$.

Let us prove assertion 5. By (66),

$P_{k}(x,x)=\frac{k^{n-q/2}}{(2\pi)^{n}}\bigl{(}a_{q}(x,0)+\mathcal{O}(k^{-\frac{1}{2}})\bigr{)}$

and by the first equation of (53),
$a_{q}(x,0)=\operatorname{tr}({\sigma}_{q}(P)(x))$.

It remains to prove Assertion 4. Let
$(P^{\prime}_{k})\in\mathcal{L}_{q^{\prime}}(B,C)$. We will prove that
$Q_{k}=P^{\prime}_{k}\circ P_{k}$ belong to
$\mathcal{L}_{q^{\prime\prime}}(A,C)$ with $q^{\prime\prime}=q+q^{\prime}$ and
compute its symbol. Since the composition of operators with kernels in
$\mathcal{O}(k^{-p})$ and $\mathcal{O}(k^{-\ell})$ respectively, has a kernel
in $\mathcal{O}(k^{-(p+\ell)})$, we can consider each summand of the
expansions (61) for $P_{k}$ and $P^{\prime}_{k}$ separately. In other words,
we can assume that $P_{k}=\bigl{(}\frac{k}{2\pi}\bigr{)}^{n}E^{k}f$ and
$P^{\prime}_{k}=\bigl{(}\frac{k}{2\pi}\bigr{)}^{n}E^{k}f^{\prime}$ with
$f=\mathcal{O}(q)$ and $f^{\prime}=\mathcal{O}(q^{\prime})$. So

$\displaystyle
Q_{k}(x,z)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{2n}\int_{M}(E(x,y)\cdot
E(y,z))^{k}g(x,y,z)d\mu_{M}(y)$ (70)

with $g(x,y,z)=f(x,y)f^{\prime}(y,z)$. Observe that $g$ vanishes to order
$q^{\prime\prime}$ along ${\Sigma}=\\{(x,y,z)\in M^{3},\;x=y=z\\}$.

By (56), $|E(x,y)\cdot E(y,z)|<1$ if $(x,y,z)\notin{\Sigma}$. This implies
first that the Schwartz kernel of $(Q_{k})$ is in $\mathcal{O}(k^{-\infty})$
outside the diagonal. Furthermore, to compute $Q_{k}$ on a neighborhood of
$(p,p)$ up to a $\mathcal{O}(k^{-\infty})$, we can reduce the integral (70) to
a neighborhood of $p$. So we can work locally.

Introduce a local orthonormal frame $(\partial_{i},i=1,\ldots,n)$ of
$T^{1,0}M$ on an open neighborhood $U$ of $p$ in $M$. Let
${\sigma}_{i}\in{\mathcal{C}}^{\infty}(U^{2})$, $i=1,\ldots n$ be such that

$\displaystyle d{\sigma}_{i}(\partial_{j},0)=\delta_{ij}+\mathcal{O}(1),\qquad
d{\sigma}_{i}(\overline{\partial}_{j},0)=\mathcal{O}(1)$ (71)

for any $i$ and $j$. Observe that if the $z_{i}$ are coordinates as in (58),
then

$\displaystyle{\sigma}_{i}(x,y)=z_{i}(x)-z_{i}(y)+\mathcal{O}_{(p_{0},p_{0})}(2).$
(72)

So we can use the functions $u_{i}=\operatorname{Re}{\sigma}_{i}$ and
$u_{i+n}=\operatorname{Im}{\sigma}_{i}$ when we write the Taylor expansion
(65) and the local expansion (66).

Restricting $U$ if necessary, we can assume that for any $z$, the map $y\in
U\rightarrow({\sigma}_{i}(y,z))\in{\mathbb{C}}^{n}$ is a diffeomorphism onto
its image. Let $\mu_{z}$ be the pull-back of the volume $\mu_{n}$ by this map.
By (72), we have $\mu_{M}(y)=\rho(y,z)\mu_{z}(y)$ with
$\rho\in{\mathcal{C}}^{\infty}(U^{2})$ satisfying $\rho(y,y)=1$.

Now using the expressions (59) and (72), we readily prove that

$E(x,y)\cdot E(y,z)=e^{\varphi(x,y,z)+r(x,y,z)}E(x,z),$

where $r(x,y,z)=\mathcal{O}_{{\Sigma}}(3)$ and
$\varphi(x,y,z)=({\sigma}(x,z)-{\sigma}(y,z))\cdot\overline{{\sigma}}(y,z)$.
Arguing as in the proof of Lemma 5.3, it comes that

$(E(x,y)\cdot
E(y,z))^{k}=E^{k}(x,z)e^{k\varphi(x,y,z)}\sum_{\ell=0}^{N}\frac{k^{\ell}}{\ell!}(r(x,y,z))^{\ell}+\mathcal{O}\bigl{(}k^{-\frac{1}{2}(N+1)}\bigr{)}$

so the integrand of (70) is equal to

$E^{k}(x,z)e^{k\varphi(x,y,z)}\sum_{\ell=0}^{N}k^{\ell}g_{\ell}(x,y,z)\;d\mu_{z}(y)+\mathcal{O}\bigl{(}k^{-\frac{1}{2}(q^{\prime\prime}+N+1)}\bigr{)}$

with
$g_{\ell}(x,y,z)=\rho(y,z)g(x,y,z)(r(x,y,z))^{\ell}/(\ell!)=\mathcal{O}_{{\Sigma}}(q^{\prime\prime}+3\ell)$.

For any $z\in U$, we write the Taylor expansion of $(x,y)\rightarrow
g_{\ell}(x,y,z)$ at $(z,z)$ with the coordinates system

$(x,y)\rightarrow\operatorname{Re}{\sigma}_{i}(x,z),\operatorname{Im}{\sigma}_{i}(x,z),\operatorname{Re}{\sigma}_{i}(y,z),\operatorname{Im}{\sigma}_{i}(y,z).$

We obtain

$g_{\ell}(x,y,z)=\sum_{m=q^{\prime\prime}+3\ell}^{p}h_{\ell,m}(z,{\sigma}(x,z),{\sigma}(y,z))+\mathcal{O}_{{\Sigma}}(p+1)$

with $h_{\ell,m}(z,\xi,\eta)$ homogeneous polynomial in $\xi,\eta$ with degree
$m$. Arguing as in Lemma 5.4, we obtain that

$\displaystyle
Q_{k}(x,z)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x,z)\sum_{\ell=0}^{N}\sum_{m=q^{\prime\prime}+3\ell}^{q^{\prime\prime}+2\ell+N}k^{\ell-\frac{m}{2}}I_{\ell,m}(x,z)+\mathcal{O}\bigl{(}k^{-\frac{1}{2}(q^{\prime\prime}+N+1)}\bigr{)}$
(73)

with

$I_{\ell,m}(x,z)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int_{U}e^{k\varphi(x,y,z)}h_{\ell,m}(z,k^{\frac{1}{2}}{\sigma}(x,z),k^{\frac{1}{2}}{\sigma}(y,z))\;d\mu_{z}(y)$

Set $u_{i}={\sigma}_{i}(x,z)$ and let us use the coordinates
$v_{i}={\sigma}_{i}(y,z)$ for the integration so that
$\varphi(x,y,z)=u\cdot\overline{v}-|v|^{2}$ and
$d\mu_{z}(y)=|dvd\overline{v}|$. It comes that
$I_{\ell,m}(x,z)=J_{\ell,m}(z,k^{\frac{1}{2}}{\sigma}(x,z))$ with

$\displaystyle J_{\ell,m}(z,u)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int
e^{k^{\frac{1}{2}}u\cdot\overline{v}-k|v|^{2}}h_{\ell,m}(z,u,k^{\frac{1}{2}}v)\;d\mu_{n}(v)$
(74)

where we integrate on a neighborhood of the origin in ${\mathbb{C}}^{n}$. We
can actually integrate on ${\mathbb{C}}^{n}$ because this will modify
$E^{k}(x,z)I_{\ell,m}(x,z)$ by a $\mathcal{O}(e^{-k/C})$. Indeed,
$|E(x,z)|=e^{-\frac{1}{2}|u|^{2}}+\mathcal{O}(|u|^{3})$ so
$|E(x,z)|=\mathcal{O}(e^{-\frac{1}{3}|u|^{2}})$ so

$|E(x,z)e^{u\cdot\overline{v}-|v|^{2}}|=\mathcal{O}(e^{-\frac{1}{3}|u|^{2}+|uv|-|v|^{2}})=\mathcal{O}(e^{-\frac{1}{4}|v|^{2}})$

and we conclude by using that
$\int_{|v|\geqslant\epsilon}e^{-\frac{k}{4}|v|^{2}}|v|^{m}\;|dvd\overline{v}|=\mathcal{O}(e^{-k/C})$
for any $\epsilon>0$ and $m\in{\mathbb{N}}$.

Taking the integral (74) over ${\mathbb{C}}^{n}$, it comes that

$\displaystyle
J_{\ell,m}(z,u)=(2\pi)^{-n}\int_{{\mathbb{C}}^{n}}e^{u\cdot\overline{v}-|v|^{2}}h_{\ell,m}(z,u,v)\;d\mu_{n}(v)$
(75)

So $J_{\ell,m}$ does not depend on $k$. Furthermore it is polynomial in $u$.
To see this, it suffices to view $h_{\ell,m}(z,u,v)$ as a polynomial in the
variables $u-v$, $v$ and to compare with the formula (52). So $Q_{k}(x,z)$ has
the local expansion (66), so $(Q_{k})$ belongs to
$\mathcal{L}_{q^{\prime\prime}}(A,C)$. Its symbol is given by the leading
order term in (73) which corresponds to $\ell=0$ and $m=q^{\prime\prime}$,
that is

$\widetilde{{\sigma}}_{q^{\prime\prime}}(Q)(x)=\operatorname{Op}(J_{0,q^{\prime\prime}}(x,\cdot)).$

We can compute it in terms of the symbols of $P$ and $P^{\prime}$ as follows:
by (67), $\widetilde{\sigma}_{q}(P)(x)=\operatorname{Op}(a_{q}(x,\cdot))$
where $\xi\rightarrow a_{q}(x,\xi)$ is the homogeneous polynomial of degree
$q$ such that $f(x,y)=a_{q}(y,{\sigma}(x,y))+\mathcal{O}(q+1)$. Similarly,
$\widetilde{\sigma}_{q^{\prime}}(P^{\prime})(x)=\operatorname{Op}(a^{\prime}_{q^{\prime}}(x,\cdot))$
with
$f^{\prime}(x,y)=a^{\prime}_{q^{\prime}}(y,{\sigma}(x,y))+\mathcal{O}(q^{\prime}+1)$.
Now by (72),
${\sigma}(x,y)={\sigma}(x,z)-{\sigma}(y,z)+\mathcal{O}_{{\Sigma}}(2)$ and it
comes that

$\displaystyle g(x,y,z)$
$\displaystyle=a_{q}(y,{\sigma}(x,z)-{\sigma}(y,z))a^{\prime}_{q^{\prime}}(z,{\sigma}(y,z))+\mathcal{O}_{{\Sigma}}(q^{\prime\prime})$
$\displaystyle=a_{q}(z,{\sigma}(x,z)-{\sigma}(y,z))a^{\prime}_{q^{\prime}}(z,{\sigma}(y,z))+\mathcal{O}_{{\Sigma}}(q^{\prime\prime})$

leading to
$h_{0,q^{\prime\prime}}(z,u,v)=a_{q}(z,u-v)a^{\prime}_{q^{\prime}}(z,v)$ and
using first (52) and then (53), we have that

$\displaystyle\widetilde{{\sigma}}_{q^{\prime\prime}}(Q)(x)$
$\displaystyle=\operatorname{Op}(J_{0,q^{\prime\prime}}(x,\cdot))=\operatorname{Op}\bigl{(}\operatorname{Op}(a_{q}(x,\cdot))a^{\prime}_{q^{\prime}}(x,\cdot)\bigr{)}$
$\displaystyle=\operatorname{Op}(a_{q}(x,\cdot))\circ\operatorname{Op}(a^{\prime}_{q^{\prime}}(x,\cdot))=\widetilde{\sigma}_{q}(P)(x)\circ\widetilde{\sigma}_{q^{\prime}}(P^{\prime})(x)$

as was to be proved. ∎

###### Proof of theorem 2.5.

The first assertion follows from Proposition 5.6. For the composition, it
suffices to consider the case treated in the previous proof: we start from
$(P_{k})$ and $(P^{\prime}_{k})$ both even. Since $h_{\ell,m}(x,\cdot)$ has
degree $m$, $J_{\ell,m}(x,\cdot)$ given by (75) has the same parity as $m$, so
by (73), $(P^{\prime}_{k}\circ P_{k})$ is even. Last assertion is simply the
fact that $\operatorname{Op}(a_{q}(x,\cdot))$ has the same parity as
$a_{q}(x,\cdot)$ by Proposition 4.4. ∎

###  5.5 Peaked sections

In this section, we state and prove a generalisation of Proposition 2.4.
Consider an auxiliary bundle $A$. Let us choose a base point $x\in M$, with a
coordinate chart $U$ centered at $x$, and a trivialisation $A|_{U}\simeq
U\times A_{x}$. To any $f\in\mathcal{P}(T_{x}M)\otimes A_{x}$, we associate
the section of $L^{k}\otimes A$

$\displaystyle\Phi_{k}^{f}(x+\xi)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{\frac{n}{2}}E^{k}(x+\xi,x)\,f(k^{\frac{1}{2}}\xi)\,\psi(x+\xi)$
(76)

where $E$ is chosen as in Section 5.1, and
$\psi\in{\mathcal{C}}^{\infty}_{0}(U)$ is equal to $1$ on a neighborhood of
$x$.

######  Proposition 5.9.

  1. 1.

For any $f$, $g\in\mathcal{P}(T_{x}M)\otimes A_{x}$,
$\bigl{\langle}\Phi_{k}^{f},\Phi_{k}^{g}\bigr{\rangle}=\langle
f,g\rangle+\mathcal{O}(k^{-\frac{1}{2}})$.

  2. 2.

For any $f\in\mathcal{P}(T_{x}M)\otimes A_{x}$ and $Q\in\mathcal{L}(A,B)$,
$Q_{k}\Phi_{k}^{f}=\Phi_{k}^{h}+\mathcal{O}(k^{-\frac{1}{2}})$ where
$h=\tilde{{\sigma}}_{0}(Q)(x)\cdot f\in\mathcal{P}(T_{x}M)\otimes B_{x}$.

In the second part, we used the symbol $\tilde{{\sigma}}_{0}(P)$ defined in
(69), and $\Phi_{k}^{h}$ is defined as $\Phi_{k}^{f}$ with a trivialisation of
$B$.

###### Proof.

Consider the operator $P^{f}\in\mathcal{L}({\mathbb{C}},A)$ with Schwartz
kernel

$P^{f}_{k}(x+\xi,x)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}E^{k}(x+\xi,x)\,f(k^{\frac{1}{2}}\xi)\,\psi(x+\xi)\,\psi(x).$

On one hand, $(\frac{k}{2\pi})^{\frac{n}{2}}\phi^{f}_{k}=P_{k}^{f}(\cdot,x)$.
On the other hand, $\tilde{{\sigma}}_{0}(P^{f})(x)=\operatorname{Op}(f)$. So
we can compute the scalar product of $\phi^{f}_{k}$ and $\phi^{g}_{k}$ as a
composition of Schwartz kernels

$\displaystyle\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\bigl{\langle}\Phi_{k}^{f},\Phi_{k}^{g}\bigr{\rangle}$
$\displaystyle=((P_{k}^{g})^{*}P_{k}^{f})(x,x)$
$\displaystyle=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\operatorname{tr}_{\mathcal{D}(T_{x}M)}(\operatorname{Op}(g)^{*}\operatorname{Op}(f))+\mathcal{O}(k^{n-\frac{1}{2}})$

by the last part of Theorem 2.2. To conclude, we have by (53)

$\operatorname{tr}_{\mathcal{D}(T_{x}M)}(\operatorname{Op}(g)^{*}\operatorname{Op}(f))=(\operatorname{Op}(g)^{*}f)(0)=\langle
f,g\rangle.$

The proof of the second part is similar, we have

$\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{\frac{n}{2}}Q_{k}\Phi^{f}_{k}=(Q_{k}P_{k})(\cdot,x)$

By Theorem 2.2, $Q_{k}P_{k}\in\mathcal{L}({\mathbb{C}},B)$ with symbol at $x$
equal to

$\tilde{{\sigma}}_{0}(Q)(x)\circ\operatorname{Op}(f)=\operatorname{Op}(\tilde{{\sigma}}_{0}(Q)(x)f)=\operatorname{Op}(h)$

by (53). ∎

We deduce the following lower bound for the operator norm of operators of
$\mathcal{L}(A,B)$. If
$\rho\in\mathcal{S}_{x}(M)\otimes\operatorname{Hom}(A_{x},B_{x})$, then we
denote by $\|\rho\|$ the norm

$\|\rho\|=\sup\bigl{\\{}\|\rho f\|/\|f\|,\;f\in\mathcal{D}(T_{x}M)\otimes
A_{x},\;f\neq 0\bigr{\\}}.$

######  Corollary 5.10.

For any $P\in\mathcal{L}(A,B)$, we have

$\liminf_{k\rightarrow\infty}\|P_{k}\|\geqslant\sup_{x\in