# Functional calculus and quantization commutes with reduction for Toeplitz
operators on CR manifolds

Andrea Galasso and Chin-Yu Hsiao111Address: Room 407, Chee-Chun Leung
Cosmology Hall, National Taiwan University; ORCID iD: 0000-0002-5792-1674;
e-mail<EMAIL_ADDRESS>Address: Institute of Mathematics,
Academia Sinica, 6F, Astronomy-Mathematics Building, No.1, Sec.4, Roosevel:
ORCHID iD: 0000-0002-1781-0013; email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Given a CR manifold with non-degenerate Levi form, we show that the operators
of the functional calculus for Toeplitz operators are complex Fourier integral
operators of Szegő type. As an application, we establish semi-classical
spectral dimensions for Toeplitz operators. We then consider a CR manifold
with a compact Lie group action $G$ and we establish quantization commutes
with reduction for Toeplitz operators. Moreover, we also compute semi-
classical spectral dimensions for $G$-invariant Toeplitz operators.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results: Functional calculus for Toeplitz operators
    2. 1.2 Main results: Semi-classical spectral dimensions for Toeplitz operators
    3. 1.3 Main results: Quantization commutes with reduction for Toeplitz operators
  2. 2 Preliminaries
    1. 2.1 Standard notations
    2. 2.2 CR manifolds
  3. 3 Functional calculus for Toeplitz operators
    1. 3.1 Some symbol spaces
      1. 3.1.1 Fourier integral operators of Szegő type
      2. 3.1.2 The spectral projections and functional calculus for Toeplitz operators
    2. 3.2 Group actions on CR manifolds
      1. 3.2.1 Invariant Fourier integral operators of Szegő type
  4. 4 Quantization commutes with reduction for Toeplitz operators
  5. 5 CR manifolds with circle action
    1. 5.1 Some standard notations in semi-classical analysis
    2. 5.2 Functional calculus for Toeplitz operators on CR manifolds with circle action
    3. 5.3 Semi-classical spectral dimensions law for Toeplitz operators on CR manifolds with circle action
    4. 5.4 Quantization commutes with reduction on CR manifolds with circle action
    5. 5.5 Semi-classical spectral dimensions for invariant Toeplitz operators on CR manifolds with circle action

Keywords: CR manifolds, Toeplitz operators, group actions

Mathematics Subject Classification: 32Vxx, 32A25, 53D50

Data Availability Statements: Data sharing not applicable to this article as
no datasets were generated or analysed during the current study.

##  1 Introduction

The study of Toeplitz operators plays an important role in modern mathematical
research, theoretical physics and it is closely related to various domain of
mathematics such as operator theory, spectral theory, analysis, geometry and
geometric quantization theory. Boutet de Monvel and Guillemin [BG] have been
set up as an extension of the classical Toeplitz operators on Hardy spaces.
They consider $\Pi_{\Sigma}P\Pi_{\Sigma}$, where $\Pi_{\Sigma}$ is the Szegő
projection associated to a given symplectic conic subset $\Sigma$ and $P$ is a
pseudodifferential operator. The great interest for us is that the Toeplitz
operators enjoy similar properties as pseudodifferential operators (symbolic
caculus, Weyl law,…). In this work, we will study the functional calculus for
Toeplitz operators on a CR manifold with non-degenerate Levi form which is
fundamental for our study of Toeplitz operators. We show that the operators of
the functional calculus for Toeplitz operators are complex Fourier integral
operators of Szegő type. As an application, we establish semi-classical
spectral dimensions for Toeplitz operators. We then consider a CR manifold
with a compact Lie group action $G$ and we establish quantization commutes
with reduction for Toeplitz operators. Moreover, we also compute semi-
classical spectral dimensions for $G$-invariant Toeplitz operators.

Let us recall some prior literature about micro-local theory of Toeplitz
operators, without the pretense of being complete. Toeplitz operators on
strictly pseudoconvex domain were introduced in the monograph [BG], in the
survey [P] it is reviewed some recent results concerning eigenfunction
asymptotics in this context, largely based on the generalization to the almost
complex symplectic category by Shiffman and Zelditch. In the presence of
symmetries, in [MZ] some Toeplitz operator type properties in semiclassical
analysis was investigated based on the analytic localization techniques
developed by Bismut and Lebeau. We refer to [MM] for more comprehensive
bibliographic notes, our approach is based on the Boutet de Monvel–Sjöstrand
parametrix for the Szegő kernel.

We now formulate the main results. They are based on [Hs], [HH] and [GH] where
it is studied Toeplitz operators for CR manifolds with non-degenerate Levi-
form, in particular in [GH] we studied the properties of the corresponding
star-product inspired by previous results as in [BMS] and [S]. We refer the
reader to Section 2 for some notations and terminology used here.

###  1.1 Main results: Functional calculus for Toeplitz operators

Let $(X,T^{1,0}X)$ be a compact, orientable CR manifold of dimension $2n+1$,
$n\geq 1$, where $T^{1,0}X$ is a CR structure of $X$. Since $X$ is orientable,
there always exists a real non-vanishing $1$-form
$\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ so that
$\langle\,\omega_{0}(x)\,,\,u\,\rangle=0$, for every $u\in T^{1,0}_{x}X\oplus
T^{0,1}_{x}X$, for every $x\in X$. We fix $\omega_{0}$ and let
$\mathcal{L}_{x}$ be the Levi form at $x\in X$ with respect to $\omega_{0}$
(see (9)). In this work, we assume that the Levi form is non-degenerate. The
Reeb vector field $R\in\mathcal{C}^{\infty}(X,TX)$ is defined to be the non-
vanishing vector field determined by

$\omega_{0}(R)=-1\quad\text{ and }\quad\mathrm{d}\omega_{0}(R,\cdot)\equiv 0\
\ \mbox{on $TX$}\,.$

Fix a smooth Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$TX\otimes\mathbb{C}$ so that $T^{1,0}X$ is orthogonal to $T^{0,1}X$,
$\langle\,u\,|\,v\,\rangle$ is real if $u,v$ are real tangent vectors,
$\langle\,R\,|\,R\,\rangle=1$ and $R$ is orthogonal to $T^{1,0}X\oplus
T^{0,1}X$. Let $(\,\cdot\,|\,\cdot\,)$ be the $L^{2}$ inner product on
$\Omega^{0,q}(X)$ induced by $\langle\,\cdot\,|\,\cdot\,\rangle$. Let
$\Box^{q}_{b}$ denote the (Gaffney extension) of the Kohn Laplacian given by
(10). Let

$S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{q}_{b}$ (1)

be the orthogonal projection with respect to the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$. In this work, we assume

######  Assumption 1.1.

The Levi form is non-degenerate of constant signature $(n_{-},n_{+})$, where
$n_{-}$ denotes the number of negative eigenvalues of the Levi form and
$n_{+}$ denotes the number of positive eigenvalues of the Levi form. We always
let $q=n_{-}$ and suppose that $\Box^{q}_{b}$ has $L^{2}$ closed range.

Let $L^{m}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ denote the space
of classical pseudodifferential operators on $X$ of order $m$ from sections of
$T^{*0,q}X$ to sections of $T^{*0,q}X$. Let $P\in L^{m}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$. We write $\sigma^{0}_{P}$ to
denote the principal symbol of $P$.

Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be a self-
adjoint classical pseudodifferential operator on $X$ with scalar principal
symbol. The Toeplitz operator is given by

$T^{(q)}_{P}:=S^{(q)}\circ P\circ S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm
Ker\,}\Box^{q}_{b}$ (2)

and let $T^{(q)}_{P}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution kernel of
$T^{(q)}_{P}$ with respect to $(\,\cdot\,|\,\cdot\,)$.

Given a self-adjoint operator

$A:{\rm Dom\,}A\subset H\rightarrow H$

where $H$ is a Hilbert space, let $\mathrm{Spec}(A)$ denote the spectrum of
$A$. Let $\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ and let $\tau(A)$
denote the functional calculus of $A$. Let

$E_{\tau}(A)={\rm Range\,}(\tau(A))\subset H.$ (3)

Let $I\subset\mathbb{R}$ be an interval. Consider an increasing sequence of
non-negative test functions $\chi_{n}\,:\,\mathbb{R}\rightarrow\mathbb{R}$
such that $\mathrm{supp}(\chi_{n})\Subset I$ converging pointwise to the
characteristic function on $I$. By Theorem $2.5.5$ in [D] the sequence of
operators $\chi_{n}(A)$ converges strongly to a canonically determined
projection $\Pi_{I}(A)$, it is called spectral projection for $I$, its range
space is denoted by $E_{I}(A)$.

For every $k\in\mathbb{Z}$, let $\Psi_{k}(X)$ denote the space of all complex
Fourier integral operators of Szegő type of order $k$ (see Definition 3.5).
For $A\in\Psi_{k}(X)$, let $\sigma^{0}_{A}$ to denote the leading term of the
symbol of $A$, let $\sigma^{0}_{A,\mp}$ to denote the leading term of the
symbol of $A_{\mp}$ (see Definition 3.5). One of the main results of this work
is the following

######  Theorem 1.1.

Let $(X,T^{1,0}X)$ be a compact, orientable CR manifold of dimension $2n+1$,
$n\geq 1$. Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
be a self-adjoint classical pseudodifferential operator on $X$ with scalar
principal symbol and let $\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with
$0\notin{\rm supp\,}\tau$. With the same notations above, under Assumption
1.1, we have $\tau(T^{(q)}_{P})(x,y)\in\Psi_{k}(X)$,

$\mbox{$\sigma^{0}_{\tau(T^{(q)}_{P}),-}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}}$,
for every $x\in X$},\\\ $ (4)

and if $q=n_{-}=n_{+}$,

$\mbox{$\sigma^{0}_{\tau(T^{(q)}_{P}),+}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,\omega_{0}(x)))\pi_{x,n_{+}}$,
for every $x\in X$},$ (5)

where $\pi_{x,n_{-}}$ and $\pi_{x,n_{+}}$ are given by (20) below and ${\rm
det\,}\mathcal{L}_{x}=\lambda_{1}(x)\cdots\lambda_{n}(x)$, $\lambda_{j}(x)$,
$j=1,\ldots,n$, are the eigenvalues of $\mathcal{L}_{x}$ with respect to
$\langle\,\cdot\,|\,\cdot\,\rangle$.

######  Remark 1.1.

It should be mentioned that if $n_{-}=n_{+}$ or $\left|n_{-}-n_{+}\right|>1$,
then $\Box^{q}_{b}$ has $L^{2}$ closed range (see [Kohn]).

###  1.2 Main results: Semi-classical spectral dimensions for Toeplitz
operators

We now assume that $X$ admits a CR and transversal locally free $S^{1}$-action
$e^{i\theta}$. Let $T$ be the vector field on $X$ given by

$(Tu)(x):=\frac{\partial}{\partial\theta}u(e^{i\theta}\circ x)|_{\theta=0},\ \
\forall u\in\mathcal{C}^{\infty}(X).$

We take $\omega_{0}$ so that the associated Reeb vector field $R$ is equal to
$T$. Assume that the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$TX\otimes\mathbb{C}$ is $S^{1}$-invariant. For every $m\in\mathbb{Z}$, set

$\Omega^{0,q}_{m}(X):=\left\\{u\in\Omega^{0,q}(X);\,(e^{i\theta})^{*}u=e^{im\theta}u,\
\ \forall e^{i\theta}\in S^{1}\right\\}.$

Let $L^{2}_{(0,q),m}(X)$ be the completion of $\Omega^{0,q}_{m}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. Put

$({\rm Ker\,}\Box^{q}_{b})_{m}:=({\rm Ker\,}\Box^{q}_{b})\cap
L^{2}_{(0,q),m}(X).$

The $m$-th Szegő projection is the orthogonal projection
$S^{(q)}_{m}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{(q)}_{b})_{m}$ with
respect to $(\,\cdot\,|\,\cdot\,)$. Let $P\in L^{0}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{S^{1}}$ be a self-adjoint
$S^{1}$-invariant classical pseudodifferential operator on $X$ with scalar
principal symbol (see Definition 3.9). Put

$T^{(q)}_{P,m}:=S^{(q)}_{m}\circ P\circ
S^{(q)}_{m}:L^{2}_{(0,q)}(X)\rightarrow L^{2}_{(0,q),m}(X).$

We also obtain an asymptotic expansion for the functional calculus of
$T^{(q)}_{P,m}$ as $m\rightarrow+\infty$ (see Theorem 5.1). Moreover, we have
the following semi-classical spectral dimensions for Toeplitz operators (see
Section 5.3)

######  Theorem 1.2.

With the notations and assumptions above, assume that $X$ is connected and
$X_{{\rm reg\,}}\neq\emptyset$, where $X_{{\rm reg\,}}=\left\\{x\in
X;\,e^{i\theta}x\neq x,\forall\theta\in]0,2\pi[\right\\}$. Let $P\in
L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{S^{1}}$ be a self-
adjoint $S^{1}$-invariant classical pseudodifferential operator on $X$ with
scalar principal symbol. Let $I\subset\mathbb{R}$ be an open bounded interval
with $0\notin\overline{I}$. We have

$\lim_{m\rightarrow+\infty}m^{-n}{\rm
dim\,}E_{I}(T^{(q)}_{P,m})=\frac{1}{2}\pi^{-n-1}\int_{\left\\{x\in
X;\,\sigma^{0}_{P}(x,-\omega_{0}(x))\in I\right\\}}\left|{\rm
det\,}\mathcal{L}_{x}\right|\mathrm{dV}_{X}(x),$ (6)

where $\mathrm{dV}_{X}$ is the volume form on $X$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle$.

###  1.3 Main results: Quantization commutes with reduction for Toeplitz
operators

In geometric quantization given a symplectic manifold $(M,\omega)$ one wants
to associate a Hilbert space. The first step is called pre-quantization: if
$\omega$ defines an integral cohomology class then there always exists a line
bundle $L$ with Hermitian structure $h$ such that the curvature of the
connection compatible with $h^{L}$ is $-2i\,\omega$. The second step consists
in fixing a polarization, we consider complex polarizations: given an almost
complex structure $J$ compatible with $\omega$, we suppose $J$ to be
integrable. When $(M,\omega)$ ia a Kähler manifold, the Hilbert space of the
quantization is the direct sum over $k\in\mathbb{Z}$ of the spaces
$H^{0}(M,L^{k})$ of holomorphic sections of the $k$-th power of the bundle.
Given a smooth function on $M$ one wants to associate an operator acting on
the Hilbert space of quantization, if $(M,\omega)$ is Kähler, in [S] it was
shown that Berezin-Toeplitz operators can be used to define a star product for
the Poisson algebra of smooth functions on $M$. If $\omega$ is non-degenerate
but not positive definite then the dimensions of $H^{0}(M,L^{k})$ as $k$ goes
to infinity does not grow as in the Kähler case, thus we consider spaces of
$(0,q)$-forms. We remark that the spaces of $(0,q)$-forms we are considering
here can be identified with the kernel of the Dolbeault-Dirac operator, see
equation $(2.36)$, pag. $16$ in [D]. In [GH] we generalize the results in [S]
in this setting. In this paper we are interested in studying spectrum spaces
of Toeplitz operators in the presence of an action of a compact connected Lie
group and we aim to generalize quantization commutes with reduction for
Toeplitz operators.

In this subsection, we do not assume that $X$ admits a transversal and CR
$S^{1}$ action but we assume that $X$ admits a $d$-dimensional compact
connected Lie group action $G$. We assume throughout that Assumption 3.1 and
Assumption 3.2 hold. We refer the reader to Section 3.2 for some notations and
terminology used here.

Let $\mathfrak{g}$ denote the Lie algebra of $G$. For any
$\xi\in\mathfrak{g}$, we write $\xi_{X}$ to denote the vector field on $X$
induced by $\xi$. Put $\underline{\mathfrak{g}}={\rm
Span\,}(\xi_{X};\,\xi\in\mathfrak{g})$, Let $HX=\left\\{{\rm Re\,}u;\,u\in
T^{1,0}X\right\\}$ and let $J:HX\rightarrow HX$ be the complex structure map
given by $J(u+\overline{u})=iu-i\overline{u}$, for every $u\in T^{1,0}X$. Let
$Y:=\mu^{-1}(0)$ and let $HY:=HX\cap TY$, where $\mu$ is the momentum map (see
Definition 3.8). Fix a $G$-invariant smooth Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ on $\mathbb{C}TX$ so that $T^{1,0}X$ is
orthogonal to $T^{0,1}X$, $\underline{\mathfrak{g}}$ is orthogonal to $HY\cap
JHY$ at every point of $Y$, $\langle\,u\,|\,v\,\rangle$ is real if $u,v$ are
real tangent vectors, $\langle\,R\,|\,R\,\rangle=1$.

Let $X_{G}:=\mu^{-1}(0)/G$. In [HH], it is proved that $X_{G}$ is a CR
manifold with natural CR structure induced by $T^{1,0}X$ of dimension
$2n-2d+1$. Let $\mathcal{L}_{X_{G}}$ be the Levi form on $X_{G}$ induced
naturally from the Levi form $\mathcal{L}$ on $X$. Since $\mathcal{L}$ is non-
degenerate of constant signature, $\mathcal{L}_{X_{G}}$ is also non-degenerate
of constant signature. We denote by $n_{X_{G},-}$ (respectively $n_{X_{G},+}$)
the number of negative (respectively positive) eigenvalues of $L_{X_{G}}$. The
Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$
induces an Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$ on
$TX_{G}\otimes\mathbb{C}$ and let $(\,\cdot\,|\,\cdot\,)_{X_{G}}$ be the
$L^{2}$ inner product on $L^{2}_{(0,q)}(X_{G})$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$. We write $\mathrm{dV}_{X_{G}}(x)$
to denote the volume form on $X_{G}$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$.

Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ be a
$G$-invariant scalar pseudodifferential operator (see Definition 3.9). The
operator $P$ gives rise to ${P}_{X_{G}}$ on the CR reduction
$X_{G}=\mu^{-1}({0})/G$. Put

$({\rm Ker\,}\Box^{q}_{b})^{G}:={\rm Ker\,}\Box^{q}_{b}\cap
L^{2}_{(0,q)}(X)^{G},$

where $L^{2}_{(0,q)}(X)^{G}$ denote the space of $G$-invariant $L^{2}$ $(0,q)$
forms. The $G$-invariant Szegő projection is the orthogonal projection

$S^{(q)}_{G}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{q}_{b})^{G}$

with respect to $(\,\cdot\,|\,\cdot\,)$. The $G$-invariant Toeplitz operator
is given by

$T^{(q)}_{P,G}:=S^{(q)}_{G}\circ P\circ
S^{(q)}_{G}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{q}_{b})^{G}.$

One of the main theorems of this paper is the following.

######  Theorem 1.3.

. Let $X$ be a compact orientable non-degenerate CR manifold and let $G$ be a
connected compact Lie group acting freely on $\mu^{-1}(0)$ such that
Assumption 3.1 and Assumption 3.2 hold. Let $P\in L^{0}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ be a $G$-invariant scalar
pseudodifferential operator and assume that
$\sigma^{0}_{P}(x,\mp\omega_{0}(x))>0$ for every $x\in X$. Thus,
$\mathrm{Spec}(T^{(n_{\mp})}_{P,G})\subset I\cup\left\\{0\right\\}$ if
$\Box^{n_{\mp}}_{b}$ has closed range,
$\mathrm{Spec}(T^{(n_{X_{G},\mp})}_{P_{X_{G}}})\subset
I\cup\left\\{0\right\\}$, if $\Box^{n_{X_{G},\mp}}_{b,X_{G}}$ has closed
range, for some open bounded interval $I$ with $0\notin\overline{I}$ (see
Lemma 3.6). Recall that we work with Assumption 1.1.

If $n_{-}=n_{+}$, $n_{X_{G},-}=n_{X_{G},+}$ or $n_{-}\neq n_{+}$,
$n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that $\Box^{n_{X_{G},-}}_{b}$ has
closed range. The map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (62) and (65) below is Fredholm.

If $n_{-}=n_{+}$ and $n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that
$\Box^{n_{X_{G},-}}_{b}$ and $\Box^{n_{X_{G},+}}_{b}$ have closed range. The
map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\oplus
E_{I}(T^{(n_{X_{G},+})}_{P_{X_{G}}})$

given by (67) below is Fredholm.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$. Suppose that
$\Box^{n_{+}}_{b}$ and $\Box^{n_{X_{G},-}}_{b}$ have closed range. (Recall
that we always assume that $\Box^{n_{-}}_{b}$ has closed range.) The map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\oplus E_{I}(T^{(n_{+})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (68) below is Fredholm.

We refer the reader to Theorem 4.2 and Theorem 4.3 for more general setting of
Theorem 1.3.

Let $H^{q}_{b}(X)^{G}={\rm Ker\,}\Box^{q}_{b}\cap L^{2}_{(0,q)}(X)^{G}$,
$H^{q}_{b}(X_{G}):={\rm Ker\,}\Box^{q}_{b,X_{G}}$. We have ${\rm
Spec\,}(S^{(q)})\subset\left\\{0,1\right\\}$. Take $\tau(x)=1$ near $x=1$. As
a corollary of Theorem 1.3, we deduce

######  Corollary 1.1.

With the same notations and assumptions above, if $n_{-}=n_{+}$,
$n_{X_{G},-}=n_{X_{G},+}$ or $n_{-}\neq n_{+}$, $n_{X_{G},-}\neq n_{X_{G},+}$,
the map

$\sigma:H^{n_{-}}_{b}(X)^{G}\rightarrow H^{n_{X_{G},-}}_{b}(X_{G})$

given by (62) and (65) below is Fredholm.

If $n_{-}=n_{+}$ and $n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that
$\Box^{n_{X_{G},-}}_{b}$ and $\Box^{n_{X_{G},+}}_{b}$ have closed range. The
map

$\sigma:H^{n_{-}}_{b}(X)^{G}\rightarrow H^{n_{X_{G},-}}_{b}(X_{G})\oplus
H^{n_{X_{G},+}}_{b}(X_{G})$

given by (67) below is Fredholm.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$. Suppose that
$\Box^{n_{+}}_{b}$ and $\Box^{n_{X_{G},-}}_{b}$ have closed range. The map

$\sigma:H^{n_{-}}_{b}(X)^{G}\oplus H^{n_{+}}_{b}(X)^{G}\rightarrow
H^{n_{X_{G},-}}_{b}(X_{G})$

given by (67) below is Fredholm.

Theorem 1.3 has a natural application in geometric quantization. Given a
symplectic $(M,\omega)$ with $[\omega]\in H^{2}(M,\mathbb{Z})$ one can always
consider an almost complex structure $J$ compatible with $\omega$. Suppose
that $J$ is integrable then $(M,\omega,J)$ is a complex manifold. Let
$(L,h^{L})$ be the quantizing holomorphic line bundle over $M$ and let
$(L^{k},h^{L^{k}})$ be the $k$-th power of $(L,h^{L})$, where $h^{L}$ denotes
the Hermitian metric of $L$. Let $R^{L}=-2i\,\omega$ be the curvature of $L$
induced by $h^{L}$. Fix a Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$
on the holomorphic tangent bundle $T^{1,0}M$ of $M$ and let
$(\,\cdot\,|\,\cdot)_{k}$ be the $L^{2}$ inner product of
$\Omega^{0,q}(M,L^{k})$ induced by $\langle\,\cdot\,|\,\cdot\,\rangle$ and
$h^{L^{k}}$, where $\Omega^{0,q}(M,L^{k})$ denotes the space of smooth $(0,q)$
forms of $M$ with values in $L^{k}$. Let

$\Box^{q}_{k}:=\overline{\partial}^{*}\,\overline{\partial}+\overline{\partial}\,\overline{\partial}^{*}:\Omega^{0,q}(M,L^{k})\rightarrow\Omega^{0,q}(M,L^{k})$

be the Kodaira Laplacian, where $\overline{\partial}^{*}$ is the adjoint of
$\overline{\partial}$ with respect to $(\,\cdot\,|\,\cdot)_{k}$. Let

$B^{(q)}_{k}:L^{2}_{(0,q)}(M,L^{k})\rightarrow{\rm Ker\,}\Box^{q}_{k}$

be the orthogonal projection with respect to $(\,\cdot\,|\,\cdot)_{k}$
(Bergman projection). Let $f\in\mathcal{C}^{\infty}(M)$, the Toeplitz operator
is given by

$T_{f,k}^{(q)}:=B^{(q)}_{k}\circ M_{f}\circ
B^{(q)}_{k}:L^{2}_{(0,q)}(M,L^{k})\rightarrow{\rm Ker\,}\Box^{q}_{k},$

where $M_{f}$ denote the operator given by the multiplication $f$. In the
presence of a holomorphic and Hamiltonian action on $(M,\omega)$ of a compact
connected Lie group $G$, suppose that $0\in\mathfrak{g}^{*}$ is a regular
value for the symplectic momentum map and the action of $G$ is free near
$\mu^{-1}(0)$. We can consider the Marsden–Weinstein reduction
$(M_{G},\omega_{G})$ which is a quantizable symplectic manifold. Assume that
the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ is $G$-invariant as
in CR case and the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$T^{1,0}M$ induces a Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle_{M_{G}}$ on $T^{1,0}M_{G}$. Assume that the
action $G$ can be lifted to $L$ and $h^{L}$ is $G$-invariant. Then,
$L_{G}:=L/G$ is a holomorphic line bundle over $M_{G}$. Fix a $G$-invariant
smooth function $f$ on $M$, let

$B^{(q)}_{G,k}:L^{2}_{(0,q)}(M,L^{k})\rightarrow({\rm Ker\,}\Box^{q}_{k})^{G}$

be the orthogonal projection with respect to $(\,\cdot\,|\,\cdot)_{k}$
($G$-invariant Bergman projection), where $({\rm Ker\,}\Box^{q}_{k})^{G}$ is
the space of $G$-invariant holomorphic sections. We can define two Toeplitz
operators. The first is the $G$-invariant Toeplitz operator

$T_{f,G,k}^{(q)}:=B^{(q)}_{G,k}\circ M_{f}\circ
B^{(q)}_{G,k}:L^{2}_{(0,q)}(M,L^{k})\rightarrow({\rm Ker\,}\Box^{q}_{k})^{G}.$

The second is the standard Toeplitz operator $T_{f_{M_{G}},k}^{(q)}$ on the
quotient $M_{G}$. Applying Theorem 1.3 to the circle bundle of
$(L^{*},h^{L^{*}})$, we get

######  Theorem 1.4.

With the same notations and assumptions above, suppose that the curvature
$R^{L}$ is non-degenerate of constant signature $(n_{-},n_{+})$ and that the
curvature $R^{L_{G}}$ is non-degenerate of constant signature
$(n_{M_{G},-},n_{M_{G},+})$, where $R^{L_{G}}$ is the curvature of $L_{G}$
induced by the $G$-invariant Hermitian metric $h^{L}$ on $L$. Let
$f\in\mathcal{C}^{\infty}(M)$ be a $G$-invariant positive function. Then,
$\mathrm{Spec}(T^{(n_{\mp})}_{f,G,k})\subset I\cup\left\\{0\right\\}$,
$\mathrm{Spec}(T^{(n_{M_{G},\mp})}_{f_{M_{G}}})\subset
I\cup\left\\{0\right\\}$, for every $k\in\mathbb{Z}$, for some open bounded
interval $I$ with $0\notin\overline{I}$.

If $n_{-}=n_{+}$ and $n_{M_{G},-}=n_{M_{G},+}$ or $n_{-}\neq n_{+}$ and
$n_{X_{G},-}\neq n_{X_{G},+}$, then

$E_{I}(T^{n_{-}}_{f,G,k})\cong E_{I}(T^{n_{M_{G},-}}_{f_{M_{G}},k})$

for $\left|k\right|\gg 1$.

If $n_{-}=n_{+}$ and $n_{M_{G},-}\neq n_{M_{G},+}$, then

$E_{I}(T^{n_{-}}_{f,G,k})\cong E_{I}(T^{n_{M_{G},-}}_{f_{M_{G}},k})\oplus
E_{I}(T^{n_{M_{G},+}}_{f_{M_{G}},k})$

for $\left|k\right|\gg 1$.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$, then

$E_{I}(T^{n_{-}}_{f,G,k})\oplus E_{I}(T^{n_{+}}_{f,G,k})\cong
E_{I}(T^{n_{M_{G},-}}_{f_{M_{G}},k})$

for $\left|k\right|\gg 1$.

From Theorem 1.2 and Theorem 4.3 (general version of Theorem 1.3), we obtain
the semi-classical spectral dimensions for $G$-invariant Toeplitz operators on
complex manifolds (see Theorem 5.4)

######  Theorem 1.5.

With the same notations and assumptions above, suppose that the curvature
$R^{L}$ is non-degenerate of constant signature $(n_{-},n_{+})$ and that the
curvature $R^{L_{G}}$ is non-degenerate of constant signature
$(n_{M_{G},-},n_{M_{G},+})$, where $R^{L_{G}}$ is the curvature of $L_{G}$
induced by the $G$-invariant Hermitian metric $h^{L}$ on $L$. Let
$I\subset\mathbb{R}$ be an open bounded interval with $0\notin\overline{I}$.
We have

$\lim_{k\rightarrow+\infty}k^{-n+d}{\rm
dim\,}E_{I}(T^{(n_{-})}_{f,G,k})=(2\pi)^{-n+d}\int_{\left\\{x\in
M_{G};\,f(x)\in I\right\\}}\left|{\rm
det\,}R^{L_{G}}\right|\mathrm{dV}_{M_{G}}(x),$ (7)

where $d={\rm dim\,}G$, ${\rm
det\,}R^{L_{G}}(x)=\lambda_{1}(x)\cdots\lambda_{d}(x)$, $\lambda_{j}(x)$,
$j=1,\ldots,d$, are the eigenvalues of $R^{L_{G}}(x)$ with respect to
$\langle\,\cdot\,|\,\cdot\,\rangle_{M_{G}}$ and $\mathrm{dV}_{M_{G}}(x)$ is
the volume form on $M_{G}$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle_{M_{G}}$.

######  Remark 1.2.

(i) In Theorem 1.2 and Theorem 1.5, $I$ is any open interval with
$0\notin\overline{I}$. We do not need the assumptions about $I$ as in Theorem
1.3 and Theorem 1.4. Moreover, in Theorem 1.2 and Theorem 1.5, we do not need
the assumptions that $\sigma^{0}_{P}(x,\mp\omega_{0}(x))>0$ for every $x\in X$
and $f$ is positive.

(ii) In Theorem 5.4, we establish the semi-classical spectral dimensions for
$G$-invariant Toeplitz operators on CR manifolds with locally free $S^{1}$
action. Moreover, in Theorem 5.4, we can replace the $G$-invariant smooth
function $f\in\mathcal{C}^{\infty}(M)$ by any $G\times S^{1}$-invariant scalar
pseudodifferential operator $P$.

##  2 Preliminaries

###  2.1 Standard notations

We use the following notations through this article:
$\mathbb{N}=\\{1,2,\ldots\\}$ is the set of natural numbers,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$, $\mathbb{R}$ is the set of real
numbers, $\mathbb{R}_{+}=\\{x\in\mathbb{R};\,x>0\\}$,
$\overline{\mathbb{R}}_{+}=\\{x\in\mathbb{R};\,x\geq 0\\}$. We write
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}_{0}$ if
$\alpha_{j}\in\mathbb{N}_{0}$, $j=1,\ldots,n$.

Let $M$ be a $\mathcal{C}^{\infty}$ paracompact manifold. We let $TM$ and
$T^{*}M$ denote the tangent bundle of $M$ and the cotangent bundle of $M$,
respectively. The complexified tangent bundle of $M$ and the complexified
cotangent bundle of $M$ will be denoted by $TM\otimes\mathbb{C}$ and
$T^{*}M\otimes\mathbb{C}$, respectively. Write
$\langle\,\cdot\,,\cdot\,\rangle$ to denote the point-wise duality between
$TM$ and $T^{*}M$. We extend $\langle\,\cdot\,,\cdot\,\rangle$ bi-linearly to
$TM\otimes\mathbb{C}\times T^{*}M\otimes\mathbb{C}$. Let $B$ be a
$\mathcal{C}^{\infty}$ vector bundle over $M$. The fibre of $B$ at $x\in M$
will be denoted by $B_{x}$. Let $E$ be a vector bundle over a
$\mathcal{C}^{\infty}$ paracompact manifold $N$. We write $B\boxtimes E^{*}$
to denote the vector bundle over $M\times N$ with fibre over $(x,y)\in M\times
N$ consisting of the linear maps from $E_{y}$ to $B_{x}$. Let $Y\subset M$ be
an open set. From now on, the spaces of distribution sections of $B$ over $Y$
and smooth sections of $B$ over $Y$ will be denoted by
$\mathcal{D}^{\prime}(Y,B)$ and $\mathcal{C}^{\infty}(Y,B)$, respectively. Let
$\mathcal{E}^{\prime}(Y,B)$ be the subspace of $\mathcal{D}^{\prime}(Y,B)$
whose elements have compact support in $Y$. Let
$\mathcal{C}^{\infty}_{c}(Y,B):=\mathcal{C}^{\infty}(Y,B)\cap\mathcal{E}^{\prime}(Y,B)$.
For $m\in\mathbb{R}$, let $H^{m}(Y,B)$ denote the Sobolev space of order $m$
of sections of $B$ over $Y$. Put

$\displaystyle H^{m}_{\rm
loc\,}(Y,B)=\left\\{u\in\mathcal{D}^{\prime}(Y,B);\,\varphi u\in
H^{m}(Y,B),\,\forall\varphi\in\mathcal{C}^{\infty}_{c}(Y)\right\\}\,,$
$\displaystyle H^{m}_{\rm comp\,}(Y,B)=H^{m}_{\rm
loc}(Y,B)\cap\mathcal{E}^{\prime}(Y,B)\,.$

We recall the Schwartz kernel theorem. Let $B$ and $E$ be
$\mathcal{C}^{\infty}$ vector bundles over paracompact orientable
$\mathcal{C}^{\infty}$ manifolds $M$ and $N$, respectively, equipped with
smooth densities of integration. If
$A:\mathcal{C}^{\infty}_{c}(N,E)\rightarrow\mathcal{D}^{\prime}(M,B)$ is
continuous, we write $A(x,y)$ to denote the distribution kernel of $A$. The
following two statements are equivalent

  1. 1.

$A$ is continuous:
$\mathcal{E}^{\prime}(N,E)\rightarrow\mathcal{C}^{\infty}(M,B)$,

  2. 2.

$A(x,y)\in\mathcal{C}^{\infty}(M\times N,B\boxtimes E^{*})$.

If $A$ satisfies (1) or (2), we say that $A$ is smoothing on $M\times N$. Let
$A,\hat{A}:\mathcal{C}^{\infty}_{c}(N,E)\rightarrow\mathcal{D}^{\prime}(M,B)$
be continuous operators. We write

$A\equiv\hat{A}\quad$ on $M\times N$ (8)

if $A-\hat{A}$ is a smoothing operator. If $M=N$, we simply write “on $M$”. We
say that $A$ is properly supported if the restrictions of the two projections
$(x,y)\mapsto x$, $(x,y)\mapsto y$ to ${\rm supp\,}(A(x,y))$ are proper.

Let $H(x,y)\in\mathcal{D}^{\prime}(M\times N,B\boxtimes E^{*})$. We write $H$
to denote the unique continuous operator
$\mathcal{C}^{\infty}_{c}(N,E)\rightarrow\mathcal{D}^{\prime}(M,B)$ with
distribution kernel $H(x,y)$. In this work, we identify $H$ with $H(x,y)$.

###  2.2 CR manifolds

We recall some notations concerning CR and contact geometry. Let
$(X,T^{1,0}X)$ be a compact, orientable CR manifold of dimension $2n+1$,
$n\geq 1$, where $T^{1,0}X$ is a CR structure of $X$. There is a unique sub-
bundle $HX$ of $TX$ such that $HX\otimes\mathbb{C}=T^{1,0}X\oplus T^{0,1}X$,
which is called horizontal tangent bundle. Let $J:HX\rightarrow HX$ be the
complex structure map given by $J(u+\overline{u})=\imath
u-\imath\overline{u}$, for every $u\in T^{1,0}X$; thus by complex linear
extension of $J$ to $TX\otimes\mathbb{C}$, the $\imath$-eigenspace of $J$ is
$T^{1,0}X$.

Since $X$ is orientable, there always exists a real non-vanishing $1$-form
$\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ so that
$\langle\,\omega_{0}(x)\,,\,u\,\rangle=0$, for every $u\in H_{x}X$, for every
$x\in X$. If $d\omega_{0}|_{HX}$ is non-degenerate, $\omega_{0}$ is a contact
form. For each $x\in X$, we define a quadratic form on $HX$ by

$\mathcal{L}_{x}(U,V)=\frac{1}{2}\mathrm{d}\omega_{0}(JU,V),\qquad\forall\
U,V\in H_{x}X.$

Then, we extend $\mathcal{L}$ to $HX\otimes\mathbb{C}$ by complex linear
extension; for $U,V\in T^{1,0}_{x}X$,

$\mathcal{L}_{x}(U,\overline{V})=\frac{1}{2}\,\mathrm{d}\omega_{0}(JU,\overline{V})=-\frac{1}{2i}\,\mathrm{d}\omega_{0}(U,\overline{V}).$
(9)

The Hermitian quadratic form $\mathcal{L}_{x}$ on $T^{1,0}_{x}X$ is called
Levi form at $x$. In this work, we assume that the Levi form is non-degenerate
of constant signature $(n_{-},n_{+})$. The Reeb vector field
$R\in\mathcal{C}^{\infty}(X,TX)$ is defined to be the non-vanishing vector
field determined by

$\omega_{0}(R)=-1,\quad\mathrm{d}\omega_{0}(R,\cdot)\equiv 0\ \ \mbox{on
$TX$}.$

Fix a smooth Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$TX\otimes\mathbb{C}$ so that $T^{1,0}X$ is orthogonal to $T^{0,1}X$,
$\langle\,u\,|\,v\,\rangle$ is real if $u,v$ are real tangent vectors,
$\langle\,R\,|\,R\,\rangle=1$ and $R$ is orthogonal to $T^{1,0}X\oplus
T^{0,1}X$. For $u\in TX\otimes\mathbb{C}$, we write
$|u|^{2}:=\langle\,u\,|\,u\,\rangle$. Denote by $T^{*1,0}X$ and $T^{*0,1}X$
the dual bundles of $T^{1,0}X$ and $T^{0,1}X$, respectively. They can be
identified with sub-bundles of the complexified cotangent bundle
$T^{*}X\otimes\mathbb{C}$. For $q\in\mathbb{N}$, the bundle of $(0,q)$ forms
of $X$ is given by $T^{*0,q}X:=\Lambda^{q}(T^{*0,1}X)$. The Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$ induces a
Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on $T^{*0,q}X$.

Let $D$ be an open set of $X$. Let $\Omega^{0,q}(D)$ denote the space of
smooth sections of $T^{*0,q}X$ over $D$ and let $\Omega^{0,q}_{c}(D)$ be the
subspace of $\Omega^{0,q}(D)$ whose elements have compact support in $D$. Let

$\overline{\partial}_{b}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q+1}(X)$

be the tangential Cauchy-Riemann operator. Let $\mathrm{d}v(x)$ be the volume
form on $X$ induced by the Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$. The $L^{2}$-inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by $\mathrm{d}v(x)$ and
$\langle\,\cdot\,|\,\cdot\,\rangle$ is given by

$(\,u\,|\,v\,):=\int_{X}\langle\,u(x)\,|\,v(x)\,\rangle\,\mathrm{d}v(x)\,,\quad
u,v\in\Omega^{0,q}(X)\,.$

We denote by $L^{2}_{(0,q)}(X)$ the completion of $\Omega^{0,q}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. We extend $(\,\cdot\,|\,\cdot\,)$ to
$L^{2}_{(0,q)}(X)$ in the standard way. For every $f\in L^{2}_{(0,q)}(X)$, we
denote by $\left\|f\right\|^{2}:=(\,f\,|\,f\,)$. We extend
$\overline{\partial}_{b}$ to $L^{2}_{(0,r)}(X)$, $r=0,1,\ldots,n$, by

$\overline{\partial}_{b}:{\rm Dom\,}\overline{\partial}_{b}\subset
L^{2}_{(0,r)}(X)\rightarrow L^{2}_{(0,r+1)}(X)\,,$

where ${\rm Dom\,}\overline{\partial}_{b}:=\\{u\in
L^{2}_{(0,r)}(X);\,\overline{\partial}_{b}u\in L^{2}_{(0,r+1)}(X)\\}$ and, for
any $u\in L^{2}_{(0,r)}(X)$, $\overline{\partial}_{b}u$ is defined in the
sense of distributions. We also write

$\overline{\partial}^{*}_{b}:{\rm Dom\,}\overline{\partial}^{*}_{b}\subset
L^{2}_{(0,r+1)}(X)\rightarrow L^{2}_{(0,r)}(X)$

to denote the Hilbert space adjoint of $\overline{\partial}_{b}$ in the
$L^{2}$ space with respect to the inner product $(\,\cdot\,|\,\cdot\,)$. Let
$\Box^{q}_{b}$ denote the so called Gaffney extension of the Kohn Laplacian
which is given by

$\begin{array}[]{c}{\rm Dom\,}\Box^{q}_{b}=\left\\{s\in
L^{2}_{(0,q)}(X);\,s\in{\rm Dom\,}\overline{\partial}_{b}\cap{\rm
Dom\,}\overline{\partial}^{*}_{b},\,\overline{\partial}_{b}s\in{\rm
Dom\,}\overline{\partial}^{*}_{b},\,\overline{\partial}^{*}_{b}s\in{\rm
Dom\,}\overline{\partial}_{b}\right\\}\,,\\\
\Box^{q}_{b}s=\overline{\partial}_{b}\overline{\partial}^{*}_{b}s+\overline{\partial}^{*}_{b}\overline{\partial}_{b}s\>\>\text{for
$s\in{\rm Dom\,}\Box^{q}_{b}$}\,.\end{array}$ (10)

Eventually, let

$S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{q}_{b}$ (11)

be the orthogonal projection with respect to the $L^{2}$-inner product
$(\,\cdot\,|\,\cdot\,)$ and let

$S^{(q)}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

denote the distribution kernel of $S^{(q)}$, which is called Szegő kernel for
$(0,q)$-forms.

##  3 Functional calculus for Toeplitz operators

In this section, we will show that the functional calculus for Toeplitz
operators are complex Fourier integral operators. In the following section we
introduce some symbol spaces.

###  3.1 Some symbol spaces

First, we recall Hörmander symbol spaces. Let $D\subset X$ be a local
coordinate patch with local coordinates $x=(x_{1},\ldots,x_{2n+1})$.

######  Definition 3.1.

For $m\in\mathbb{R}$, $S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ is the space of all

$a(x,y,t)\in\mathcal{C}^{\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

such that, for all compact $K\Subset D\times D$ and all
$\alpha,\beta\in\mathbb{N}^{2n+1}_{0}$, $\gamma\in\mathbb{N}_{0}$, there is a
constant $C_{\alpha,\beta,\gamma}>0$ such that

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}\partial^{\gamma}_{t}a(x,y,t)\right|\leq
C_{\alpha,\beta,\gamma}(1+\left|t\right|)^{m-\gamma},\ \ \mbox{for every
$(x,y,t)\in K\times\mathbb{R}_{+},t\geq 1$}.$

Furthermore, put

$S^{-\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}):=\bigcap_{m\in\mathbb{R}}S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

We sometimes simply write $S^{m}_{1,0}$ to denote $S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
$m\in\mathbb{R}\cup\left\\{-\infty\right\\}$. Let $a_{j}\in S^{m_{j}}_{1,0}$,
$j=0,1,2,\ldots$ with $m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$. Then
there exists $a\in S^{m_{0}}_{1,0}$ unique modulo $S^{-\infty}$, such that
$a-\sum^{k-1}_{j=0}a_{j}\in S^{m_{k}}_{1,0}$ for $k=0,1,2,\ldots$.

If $a$ and $a_{j}$ have the properties above, we write

$a\sim\sum^{\infty}_{j=0}a_{j}$ in $S^{m_{0}}_{1,0}\left(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\right)\,.$

We write

$s(x,y,t)\in S^{m}_{{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

if $s(x,y,t)\in S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ and

$\displaystyle s(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}(x,y)t^{m-j}\text{ in
}S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\,,$

where $s^{j}(x,y)\in\mathcal{C}^{\infty}(D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ j\in\mathbb{N}_{0}$.

Let $D\subset X$ be an open set. Let $L^{m}_{{\rm
cl\,}}(D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ denote the space of classical
pseudodifferential operators on $D$ of order $m$ from sections of $T^{*0,q}X$
to sections of $T^{*0,q}X$. Let $P\in L^{m}_{{\rm
cl\,}}(D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$. We write $\sigma^{0}_{P}$ to
denote the principal symbol of $P$.

Let $P\in L^{0}_{{\rm cl\,}}(D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be a self-
adjoint operator with positive scalar principal symbol. Let
$\tau\in\mathcal{C}^{\infty}_{c}(I)$, $I\subset\mathbb{R}$ is an open
interval. In the next sections we are going to study $\tau(T^{(q)}_{P})$,
where $\tau(T^{(q)}_{P})$ denotes the functional calculus of $T^{(q)}_{P}$.
Here $T^{(q)}_{P}$ denotes the Toeplitz operator with symbol $P$ (see (24)).
Thus, we need to introduce another symbol spaces. Let $U\Subset\mathbb{C}$ be
an open set such with $U\cap\mathbb{R}\neq\emptyset$. Let $\hat{U}:=\\{z\in
U;\,{\rm Im\,}z\neq 0\\}$.

######  Definition 3.2.

For $m\in\mathbb{R}$, the space $S^{m}_{z}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ is the set of all
$a(x,y,t,z)\in\mathcal{C}^{\infty}(D\times
D\times\mathbb{R}_{+}\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ such
that for every $\alpha,\,\beta\in\mathbb{N}^{2n+1}_{0}$,
$\gamma\in\mathbb{N}_{0}$ there exists
$N(\alpha,\,\beta,\,\gamma)\in\mathbb{N}$ such that for every compact set
$K\Subset D\times D$, we have

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}\partial^{\gamma}_{t}a(x,y,t,z)\right|\leq
C_{K}\left|\mathrm{Im}z\right|^{-N(\alpha,\beta,r)}(1+\left|t\right|)^{m-\gamma},\
\ \mbox{for every $(x,y,t,z)\in K\times\mathbb{R}_{+}\times\hat{U},t\geq 1$},$

where $C_{K}>0$ is a constant independent of $z$.

######  Definition 3.3.

With the notations used above, the space $\hat{\mathcal{C}}^{\infty}(D\times
D\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ is the set of all
$f\in\mathcal{C}^{\infty}(D\times
D\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ such that for every
$\alpha,\,\beta\in\mathbb{N}^{2n+1}_{0}$ there exists
$N(\alpha,\,\beta)\in\mathbb{N}$ such that for every compact set $K\Subset
D\times D$, we have

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}f(x,y,z)\right|\leq
C_{K}\left|\mathrm{Im}z\right|^{-N(\alpha,\beta)},\ \ \mbox{for every
$(x,y,z)\in K\times\hat{U}$},$

where $C_{K}>0$ is a constant independent of $z$.

We define the space $\hat{\mathcal{C}}^{\infty}(X\times
X\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ in the similar way.

######  Remark 3.1.

We identify $D$ with an open set of $\mathbb{R}^{2n+1}$. Let $D^{\mathbb{C}}$
be an open set of $\mathbb{C}^{2n+1}$ with
$D^{\mathbb{C}}\cap\mathbb{R}^{2n+1}=D$. We can also define the space
$\hat{\mathcal{C}}^{\infty}(D^{\mathbb{C}}\times
D^{\mathbb{C}}\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ in the same
way. From now on, for any $f\in\mathcal{C}^{\infty}(D\times
D\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, we always take an almost
analytic extension $\widetilde{f}$ of $f$ so that
$\widetilde{f}\in\hat{\mathcal{C}}^{\infty}(D^{\mathbb{C}}\times
D^{\mathbb{C}}\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ (it is
straightforward to see that this is always possible).

######  Definition 3.4.

With the notations used above, fix $m\in\mathbb{R}$. For every

$a\in S^{m}_{z}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

we say that

$a\in S^{m}_{z,\mathrm{cl}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

if we can find $a_{j}\in\hat{\mathcal{C}}^{\infty}(D\times
D\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $j=0,\,1,\,\dots$, such
that

$a(x,\,y,\,t,\,z)-\sum^{N}_{j=0}a_{j}(x,\,y,\,z)\,t^{m-j}\in
S^{m-N-1}_{z}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ (12)

for all $N\in\mathbb{N}$.

In the same way as for Hörmander symbol spaces, put

$S^{-\infty}_{z}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}):=\bigcap_{m\in\mathbb{R}}S^{m}_{z}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

######  Lemma 3.1.

With the notations used above, let $a_{j}\in
S^{m_{j}}_{z,\,\mathrm{cl}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $j=0,1,2,\ldots$
with $m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$. Then there exists

$a\in S^{m_{0}}_{z,\,\mathrm{cl}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

unique modulo $S^{-\infty}_{z}$, such that

$a-\sum^{k-1}_{j=0}a_{j}\in S^{m_{k}}_{z,\,\mathrm{cl}}\left(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\right)$ (13)

for $k=1,2,\ldots$. If $a$ and $a_{j}$ have the properties above, we write

$a\sim\sum^{\infty}_{j=0}a_{j}$ in $S^{m_{0}}_{z,\,\mathrm{cl}}\left(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\right)$ .

###### Proof.

Let $K_{0}\Subset K_{1}\Subset\cdots\Subset D$, $\cup_{j=1}^{\infty}K_{j}=D$.
For every $j=0,1,2,\dots$, take $0<\epsilon_{j}<1$ small enough and
$N_{j}\in\mathbb{N}$ so that

$\lVert
a_{j}(x,y,t,z)(1-\chi(t\lvert\mathrm{Im}z\rvert^{N_{j}}\epsilon^{N_{j}}_{j}))\rVert_{\mathcal{C}^{j}(K_{j}\times
K_{j})}\leq 2^{-j}\,t^{m_{j}+1-j}\,,$ (14)

for every $z\in\hat{U}$, $t\geq 1$, where
$\chi\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$, $\chi\equiv 1$ on $[-1,\,1]$.
Let

$a(x,y,t,z)=\sum_{j=0}^{+\infty}a_{j}(x,y,t,z)(1-\chi(t\lvert\mathrm{Im}z\rvert^{N_{j}}\epsilon^{N_{j}}_{j}))\,.$

From (14) we can check that $a$ is well defined as a smooth function on
$D\times D\times\mathbb{R}_{+}\times\hat{U}$, $a\in\hat{S}^{m_{0}}_{z,{\rm
cl\,}}$ and satisfies (13). ∎

####  3.1.1 Fourier integral operators of Szegő type

We are now going to recall results about Toeplitz operators relying on [GH].
In this work, we work with Assumption 1.1.

Let us adopt the following notation

$\Sigma=\Sigma^{-}\cup\Sigma^{+},\,\Sigma^{-}=\left\\{(x,\lambda\omega_{0}(x))\in
T^{*}X;\,\lambda<0\right\\},\,\Sigma^{+}=\left\\{(x,\lambda\omega_{0}(x))\in
T^{*}X;\,\lambda>0\right\\}.$

It is known that (see [Hs, Theorem 1.2], [HM, Theorem 4.7]) there exist
continuous operators $S_{-},S_{+}:L^{2}_{(0,q)}(X)\rightarrow{\rm
Ker\,}\Box^{q}_{b}$ such that

$S^{(q)}=S_{-}+S_{+},\ \ S^{*}_{-}=S^{*}_{-},\ \
S^{*}_{+}=S^{*}_{+},\,S_{-}=S^{(q)}S_{-},\ \ S_{+}=S^{(q)}S_{+},$ ${\rm
WF^{\prime}\,}(S_{-})={\rm diag\,}(\Sigma^{-}\times\Sigma^{-}),S_{+}\equiv 0\
\ \mbox{if $q\neq n_{+}$},\,{\rm WF^{\prime}\,}(S_{+})={\rm
diag\,}(\Sigma^{+}\times\Sigma^{+})\ \ \mbox{if $q=n_{-}=n_{+}$},$

where $S^{*}_{\mp}$ is the adjoint of $S_{\mp}$, ${\rm WF\,}(S_{\mp})$ is the
wave front set of $S_{\mp}$ in the sense of Hörmander. We have the following
(see [Hs, Theorem 1.2], [HM, Theorem 4.7]) about micro-local properties of the
Szegő kernel.

######  Theorem 3.1.

Let $D\subset X$ be any local coordinate patch with coordinates
$x=(x_{1},\ldots,x_{2n+1})$, then $S_{-}(x,y)$, $S_{+}(x,y)$ satisfy

$S_{\mp}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{\mp}(x,y)t}s_{\mp}(x,y,t)\mathrm{d}t\
\ \mbox{on $D$},$

with

$s_{\mp}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{\mp}(x,y)t^{n-j}\in
S^{n}_{1,0},\,\text{ where }s^{j}_{\mp}(x,y)\in\mathcal{C}^{\infty}(D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),$ (15)

$s_{+}(x,y,t)=0$ if $q\neq n_{+}$ and $s^{0}_{-}(x,x)\neq 0$, for all $x\in
D$. The phase functions $\varphi_{-}$, $\varphi_{+}$ satisfy

$\begin{array}[]{ll}&\varphi_{+},\varphi_{-}\in\mathcal{C}^{\infty}(D\times
D),\ \ {\rm Im\,}\varphi_{\mp}(x,y)\geq 0,\varphi_{-}(x,x)=0,\ \
\varphi_{-}(x,y)\neq 0\ \ \mbox{if}\ \ x\neq y,\\\
&\mathrm{d}_{x}\varphi_{-}(x,y)\big{|}_{x=y}=-\omega_{0}(x),\ \
\mathrm{d}_{y}\varphi_{-}(x,y)\big{|}_{x=y}=\omega_{0}(x),-\overline{\varphi}_{+}(x,y)=\varphi_{-}(x,y).\end{array}$

The following result describes the phase function in local coordinates (see
chapter 8 of part I in [Hs]).

######  Theorem 3.2.

For a given point $p\in X$, let $\\{W_{j}\\}_{j=1}^{n}$ be an orthonormal
frame of $T^{1,0}X$ in a neighborhood of $p$ such that the Levi form is
diagonal at $p$, i.e.
$\mathcal{L}_{p}(W_{j},\overline{W}_{s})=\delta_{j,s}\mu_{j}$,
$j,s=1,\ldots,n$. We take local coordinates $x=(x_{1},\ldots,x_{2n+1})$,
$z_{j}=x_{2j-1}+ix_{2j}$, $j=1,2,\ldots,n$, defined on some neighborhood of
$p$ such that $\omega_{0}(p)=\mathrm{d}x_{2n+1}$, $x(p)=0$, and, for some
$c_{j}\in\mathbb{C}$, $j=1,\ldots,n$ ,

$R=-\frac{\partial}{\partial x_{2n+1}},\quad W_{j}=\frac{\partial}{\partial
z_{j}}-i\mu_{j}\overline{z}_{j}\frac{\partial}{\partial
x_{2n+1}}-c_{j}x_{2n+1}\frac{\partial}{\partial
x_{2n+1}}+\sum^{2n}_{k=1}a_{j,k}(x)\frac{\partial}{\partial
x_{k}}+O(\left|x\right|^{2})\,,$ (16)

where $j=1,\ldots,n$, $a_{j,k}\in\mathcal{C}^{\infty}(X)$,
$a_{j,k}(x)=O(\left|x\right|)$, for every $j=1,\ldots,n$, $k=1,\ldots,2n$. Set
$y=(y_{1},\ldots,y_{2n+1})$, $w_{j}=y_{2j-1}+iy_{2j}$, $j=1,2,\ldots,n$. Then,
for $\varphi_{-}$ in Theorem 3.1, we have

${\rm Im\,}\varphi_{-}(x,y)\geq c\sum^{2n}_{j=1}\left|x_{j}-y_{j}\right|^{2},\
\ c>0,$ (17)

in some neighborhood of $(0,0)$ and

$\begin{array}[]{l}\varphi_{-}(x,y)=-x_{2n+1}+y_{2n+1}+i\sum^{n}_{j=1}\left|\mu_{j}\right|\left|z_{j}-w_{j}\right|^{2}\\\
\
+\sum^{n}_{j=1}\left(i\mu_{j}(\overline{z}_{j}w_{j}-z_{j}\overline{w}_{j})+c_{j}(-z_{j}x_{2n+1}+w_{j}y_{2n+1})+\overline{c}_{j}(-\overline{z}_{j}x_{2n+1}+\overline{w}_{j}y_{2n+1})\right)\\\
\ +(x_{2n+1}-y_{2n+1})f(x,y)+O(\left|(x,y)\right|^{3}),\end{array}$ (18)

where $f$ is smooth and satisfies $f(0,0)=0$.

We pause and introduce some notations in order to given an explicit
description for the leading term of the symbol of Szegő kernel. For a given
point $p\in X$, let $\\{W_{j}\\}_{j=1}^{n}$ be an orthonormal frame of
$(T^{1,0}X,\langle\,\cdot\,|\,\cdot\,\rangle)$ near $p$, for which the Levi
form is diagonal at $p$. We will denote by

$\mathcal{L}_{p}(W_{j},\overline{W}_{\ell})=\mu_{j}(p)\delta_{j,\ell}\,,\;\;j,\ell=1,\ldots,n\,\quad\text{
and }\quad\det\mathcal{L}_{p}=\prod_{j=1}^{n}\mu_{j}(p)\,.$ (19)

Let $\\{e_{j}\\}_{j=1}^{n}$ denote the basis of $T^{*0,1}X$, dual to
$\\{\overline{W}_{j}\\}^{n}_{j=1}$. We assume that, if $1\leq j\leq n_{-}$,
$\mu_{j}(p)<0$ and if $n_{-}+1\leq j\leq n$ we have $\mu_{j}(p)>0$. Put

$\mathcal{N}(p,n_{-}):=\left\\{ce_{1}(p)\wedge\ldots\wedge
e_{n_{-}}(p);\,c\in\mathbb{C}\right\\},\,\mathcal{N}(p,n_{+}):=\left\\{ce_{n_{-}+1}(p)\wedge\ldots\wedge
e_{n}(p);\,c\in\mathbb{C}\right\\}$

and let

$\pi_{p,n_{-}}:T^{*0,q}_{p}X\rightarrow\mathcal{N}(p,n_{-})\,,\quad\pi_{p,n_{+}}:T^{*0,q}_{p}X\rightarrow\mathcal{N}(p,n_{+})\,,$
(20)

be the orthogonal projections onto $\mathcal{N}(p,n_{-})$ and
$\mathcal{N}(p,n_{+})$ with respect to $\langle\,\cdot\,|\,\cdot\,\rangle$,
respectively. For $J=(j_{1},\ldots,j_{q})$, $1\leq j_{1}<\cdots<j_{q}\leq n$,
let $e_{J}:=e_{j_{1}}\wedge\cdots\wedge e_{j_{q}}$. For
$\left|I\right|=\left|J\right|=q$, $I$, $J$ are strictly increasing, let
$e_{I}\otimes(e_{J})^{*}$ be the linear transformation from $T^{*0,q}X$ to
$T^{*0,q}X$ given by

$(e_{I}\otimes(e_{J})^{*})(e_{K})=\delta_{J,K}e_{I},$

for every $\left|K\right|=q$, $K$ is strictly increasing, where
$\delta_{J,K}=1$ if $J=K$, $\delta_{J,K}=0$ if $J\neq K$. For any $f\in
T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}$, we have

$f=\sideset{}{{}^{\prime}}{\sum}_{\left|I\right|=\left|J\right|=q}c_{I,J}e_{I}\otimes(e_{J})^{*},$

$c_{I,J}\in\mathbb{C}$, for all $\left|I\right|=\left|J\right|=q$, $I$, $J$
are strictly increasing, where $\sum^{\prime}$ means that the summation is
performed only over strictly increasing multi-indices. We call
$c_{I,J}e_{I}\otimes(e_{J})^{*}$ the component of $f$ in the direction
$e_{I}\otimes(e_{J})^{*}$. Let $I_{0}=(1,2,\ldots,q)$. We can check that

$\pi_{p,n_{-}}=e_{I_{0}}(p)\otimes(e_{I_{0}}(p))^{*}.$

The following formula for the leading term $s^{0}_{-}$ on the diagonal follows
from [Hs, §9]. The formula for the leading term $s^{0}_{+}$ on the diagonal
follows similarly.

######  Theorem 3.3.

For the leading term $s^{0}_{-}(x,y)$ of the expansion (15) of $s_{-}(x,y,t)$,
we have

$s^{0}_{-}(x_{0},x_{0})=\frac{1}{2}\pi^{-n-1}\left|\det\mathcal{L}_{x_{0}}\right|\pi_{x_{0},n_{-}}\,,\>\>x_{0}\in
D.$

We need to recall the following definition from [GH].

######  Definition 3.5.

Let $H:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be a continuous operator
with distribution kernel $H(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$. We say that $H$ is a complex Fourier
integral operator of Szegő type of order $k\in\mathbb{Z}$ if $H$ is smoothing
away the diagonal on $X$ and for every local coordinate patch $D\subset X$
with local coordinates $x=(x_{1},\ldots,x_{2n+1})$, we have on $D$

$H(x,y)\equiv H_{-}(x,y)+H_{+}(x,y)\,,\quad
H_{\mp}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{\mp}(x,y)t}a_{\mp}(x,y,t)\mathrm{d}t\,,$

where $a_{-},a_{+}\in S^{k}_{{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $a_{+}=0$ if $q\neq
n_{+}$, $\varphi_{-}$, $\varphi_{+}$ are as in Theorem 3.1 and Theorem 3.2. We
write $\sigma^{0}_{H,-}(x,y)$ to denote the leading term of the expansion (15)
of $a_{-}(x,y,t)$. If $q=n_{+}$, we write $\sigma^{0}_{H,+}(x,y)$ to denote
the leading term of the expansion (15) of $a_{+}(x,y,t)$. If $n_{-}\neq
n_{+}$, we sometimes write $\sigma^{0}_{H}(x,y):=\sigma^{0}_{H,-}(x,y)$. Note
that $\sigma^{0}_{H,-}(x,y)$ and $\sigma^{0}_{H,+}(x,y)$ depend on the choices
of the phases $\varphi_{-}$ and $\varphi_{+}$ but $\sigma^{0}_{H,-}(x,x)$ and
$\sigma^{0}_{H,+}(x,x)$ are independent of the choices of the phases
$\varphi_{-}$ and $\varphi_{+}$.

Let us denote by $\Psi_{k}(X)$ the space of all complex Fourier integral
operators of Szegő type of order $k$.

Let $U\Subset\mathbb{C}$ be an open set with $U\cap\mathbb{R}\neq\emptyset$.
Let $\hat{U}:=\\{z\in U;\,{\rm Im\,}z\neq 0\\}$. We need

######  Definition 3.6.

With the notations used above, let
$H_{z}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be a $z$-dependent
continuous operator with distribution kernel
$H_{z}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $z\in\hat{U}$. We write

$H_{z}\equiv 0\mod\hat{\mathcal{C}}^{\infty}$

or

$H_{z}(x,y)\equiv 0\mod\hat{\mathcal{C}}^{\infty}$

if

$H_{z}(x,y)\in\hat{\mathcal{C}}^{\infty}(X\times
X\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

If $H_{z}\equiv 0\mod\hat{\mathcal{C}}^{\infty}$, we say that $H_{z}$ is
$z$-smoothing.

We say that $H_{z}$ is $z$-smoothing away the diagonal if for every
$\chi_{1},\chi_{2}\in\mathcal{C}^{\infty}_{c}(X)$ with

${\rm supp\,}\chi_{1}\cap{\rm supp\,}\chi_{2}=\emptyset,$

we have $\chi_{1}H_{z}\chi_{2}\equiv 0\mod\hat{\mathcal{C}}^{\infty}$.

Let $A_{z}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be another $z$-dependent
continuous operator with distribution kernel
$A_{z}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $z\in\hat{U}$. We write

$H_{z}\equiv A_{z}\mod\hat{\mathcal{C}}^{\infty}$

or

$H_{z}(x,y)\equiv A_{z}(x,y)\mod\hat{\mathcal{C}}^{\infty}$

if $H_{z}-A_{z}\equiv 0\mod\hat{\mathcal{C}}^{\infty}$. In this definition, we
can replace $X$ by any open set $D$ of $X$.

Similarly we define

######  Definition 3.7.

With the notations used above, let
$H_{z}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be a $z$-dependent
continuous operator with distribution kernel
$H_{z}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $z\in\hat{U}$. We say that $H_{z}$ is a
complex Fourier integral operator of $z$-Szegő type of order $k\in\mathbb{Z}$
if $H_{z}$ is $z$-smoothing away the diagonal on $X$ and for every local
coordinate patch $D\subset X$ with local coordinates
$x=(x_{1},\ldots,x_{2n+1})$, we have on $D$

$H_{z}(x,y)\equiv
H_{z,-}(x,y)+H_{z,+}(x,y)\mod\hat{\mathcal{C}}^{\infty},\quad
H_{z,\mp}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{\mp}(x,y)t}a_{\mp}(x,y,t,z)\mathrm{d}t\,,$

where $a_{-},a_{+}\in S^{k}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $a_{+}=0$ if $q\neq
n_{+}$, $\varphi_{-}$, $\varphi_{+}$ are as in Theorem 3.1 and Theorem 3.2. We
write $\sigma^{0}_{H_{z},-}(x,y)$ to denote the leading term of the expansion
(12) of $a_{-}(x,y,t,z)$. If $q=n_{+}$, we write $\sigma^{0}_{H_{z},+}(x,y)$
to denote the leading term of the expansion (12) of $a_{+}(x,y,t,z)$. If
$n_{-}\neq n_{+}$, we sometimes write
$\sigma^{0}_{H_{z}}(x,y):=\sigma^{0}_{H_{z},-}(x,y)$.

Let us denote by $\Psi_{k,z}(X)$ the space of all complex Fourier integral
operators of $z$-Szegő type of order $k$.

The following is straightforward. We omit the details.

######  Lemma 3.2.

With the notations used above, let $D$ be a local coordinate patch of $X$ with
local coordinates $x=(x_{1},\ldots,x_{2n+1})$. Let

$A_{z}(x,y)=\int^{\infty}_{0}e^{i\varphi_{-}(x,y)t}t^{k}b(x,y,z)\mathrm{d}t,$

where $k\in\mathbb{Z}$, $b(x,y,z)\in\hat{\mathcal{C}}^{\infty}(D\times
D\times\hat{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$. If $b$ vanishes to
infinite order at $x=y$, then

$A_{z}\equiv 0\mod\hat{\mathcal{C}}^{\infty}.$

From Lemma 3.2, we can repeat the proof of Lemma $3.1$ in [GH] and get

######  Lemma 3.3.

Let $B_{z}\in\Psi_{k,z}(X)$ with $S^{(q)}B_{z}\equiv
B_{z}\mod\hat{\mathcal{C}}^{\infty}$, $B_{z}S^{(q)}\equiv
B_{z}\mod\hat{\mathcal{C}}^{\infty}$. If

$\pi_{x,n_{-}}\sigma^{0}_{B_{z},-}(x,x)\pi_{x,n_{-}}=0\text{ and
}\pi_{x,n_{+}}\sigma^{0}_{B_{z},+}(x,x)\pi_{x,n_{+}}=0\text{ if
}q=n_{-}=n_{+}\,,$

for every $x\in X$, then $B_{z}\in\Psi_{k-1,z}(X)$.

We need

######  Lemma 3.4.

Let $A_{z}\in\Psi_{n-1,z}(X)$.There exist $N\in\mathbb{N}$ and $c>0$ such that

$\left\|A_{z}u\right\|\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N}}\left\|u\right\|,$

for every $z\in\hat{U}$ and every $u\in\Omega^{0,q}(X)$.

###### Proof.

Let $A^{*}_{z}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be the adjoint of
$A_{z}$ with respect to $(\,\cdot\,|\,\cdot\,)$. For every
$u\in\Omega^{0,q}(X)$, we have

$\begin{split}&\left\|A_{z}u\right\|^{2}=(\,A^{*}_{z}A_{z}u\,|\,u\,)\leq\left\|A^{*}_{z}A_{z}u\right\|\left\|u\right\|\\\
&\leq\left\|(A^{*}_{z}A_{z})^{2}u\right\|^{\frac{1}{2}}\left\|u\right\|^{\frac{3}{2}}\leq\cdots\leq\left\|(A^{*}_{z}A_{z})^{2^{k}}u\right\|^{\frac{1}{2^{k}}}\left\|u\right\|^{2-\frac{1}{2^{k}}},\end{split}$
(21)

for every $k\in\mathbb{N}$ and every $z\in\hat{U}$. From complex stationary
phase formula, we see that

$(A^{*}_{z}A_{z})^{2^{k}}\in\Psi_{n-2^{k+1},z}(X).$

We take $k\gg 1$ so that

$\left\|(A^{*}_{z}A_{z})^{2^{k}}v\right\|\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N_{1}}}\left\|v\right\|,$ (22)

for every $z\in\hat{U}$ and every $v\in\Omega^{0,q}(X)$, where $c>0$ and
$N_{1}\in\mathbb{N}$ are constants. From (21) and (22), the lemma follows. ∎

######  Lemma 3.5.

Let $A_{z}\in\Psi_{n,z}(X)$. For every $s\in\mathbb{Z}$, there exist
$N_{s}\in\mathbb{N}$ and $c>0$ such that

$\left\|A_{z}u\right\|_{s-1}\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N_{s}}}\left\|u\right\|_{s},$ (23)

for every $z\in\hat{U}$ and every $u\in\Omega^{0,q}(X)$, where
$\left\|\cdot\right\|_{s}$ denotes the Sobolev norm of order $s$ on $X$.

Thus, $A_{z}$ can be extended to a continuous operator:

$A_{z}:H^{s}(X,T^{*0,q}X)\rightarrow H^{s-1}(X,T^{*0,q}X)$

and we have the estimates (23), for every $s\in\mathbb{Z}$, every
$z\in\hat{U}$ and every $u\in H^{s}(X)$.

###### Proof.

For every $s\in\mathbb{Z}$, let $\Lambda_{s}\in L^{s}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be a classical elliptic
pseudodifferential operator on $X$ of order $s$ from sections of $T^{*0,q}X$
to sections of $T^{*0,q}X$. From complex stationary phase formula, we see that

$\Lambda_{s-1}A_{z}\Lambda_{-s}\in\Psi_{n-1,z}(X).$

From this observation and Lemma 3.4, the lemma follows. ∎

Let $P\in L^{\ell}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ with
scalar principal symbol, $\ell\leq 0$, $\ell\in\mathbb{Z}$. We recall that a
Toeplitz operator is given by

$T^{(q)}_{P}:=S^{(q)}\circ P\circ S^{(q)}:L^{2}(X)\rightarrow{\rm
Ker\,}\Box^{q}_{b}.$ (24)

Let $f\in\mathcal{C}^{\infty}(X)$ and let $M_{f}$ denote the operator given by
the multiplication $f$. When $P=M_{f}$, we write $T^{(q)}_{f}:=T^{(q)}_{P}$.

The following follows from the standard calculus of Fourier integral operator
of complex type (see the calculation after Theorem $3.4$ in [GH]).

######  Theorem 3.4.

Let $P\in L^{\ell}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ with
scalar principal symbol, $\ell\leq 0$, $\ell\in\mathbb{Z}$. We have
$T^{(q)}_{P}\in\Psi_{n+\ell}(X)$ and

$\mbox{$\sigma^{0}_{T^{(q)}_{P,-}}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\sigma^{0}_{P}(x,-\omega_{0}(x))\pi_{x,n_{-}}$,
for every $x\in X$}.$

If $q=n_{-}=n_{+}$, then

$\mbox{$\sigma^{0}_{T^{(q)}_{P,+}}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\sigma^{0}_{P}(x,\omega_{0}(x))\pi_{x,n_{+}}$, for
every $x\in X$}.$

####  3.1.2 The spectral projections and functional calculus for Toeplitz
operators

Given a self-adjoint operator

$A:{\rm Dom\,}A\subset H\rightarrow H$

where $H$ is a Hilbert space, let $\mathrm{Spec}(A)$ denote the spectrum of
$A$. Let $\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ and let $\tau(A)$
denote the functional calculus of $A$. Let

$E_{\tau}(A)={\rm Range\,}(\tau(A))\subset H.$ (25)

Let $I\subset\mathbb{R}$ be an interval. Consider an increasing sequence of
non-negative test functions $\chi_{n}\,:\,\mathbb{R}\rightarrow\mathbb{R}$
such that $\mathrm{supp}(\chi_{n})\Subset I$ converging point-wise to the
characteristic function on $I$. By Theorem $2.5.5$ in [D] the sequence of
operators $\chi_{n}(A)$ converges strongly to a canonically determined
projection $\Pi_{I}(A)$, it is called spectral projection for $I$, its range
space is denoted by $E_{I}(A)$.

Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be a self-
adjoint classical pseudodifferential operator on $X$ with scalar principal
symbol. Then, it is not difficult to see that the Toeplitz operator
$T^{(q)}_{P}$ is self-adjoint. In this section, we will show that the
functional calculus for Toeplitz operators is a complex Fourier integral
operators of Szegő type of order $0$.

######  Lemma 3.6.

Suppose that $\sigma^{0}_{P}(x,\mp\omega_{0}(x))\neq 0$ for every $x\in X$. We
have $\mathrm{Spec}(T^{(q)}_{P})\subset I\cup\left\\{0\right\\}$, for some
open interval $I$ with $0\notin\overline{I}$.

###### Proof.

The statement is equivalent to show that $a<\lVert T^{(q)}_{P}u\rVert<b$, for
every $u\in L^{2}_{(0,q)}(X)$, $u\perp{\rm Ker\,}T^{(q)}_{P}$, $\lVert
u\rVert=1$. Since $P$ is of order zero, $T^{(q)}_{P}$ is $L^{2}$ bounded,
there is a $b>0$ such that $\|T^{(q)}_{P}u\rVert<b$, for all $u\in
L^{2}_{(0,q)}(X)$, $\lVert u\rVert=1$. We only need to show that there is a
$c>0$ such that $\|T^{(q)}_{P}u\rVert\geq c$ for all $u\in L^{2}_{(0,q)}(X)$,
$u\perp{\rm Ker\,}T^{(q)}_{P}$, $\lVert u\rVert=1$. If this is not true, we
can find $u_{j}\in L^{2}_{(0,q)}(X)$, $u_{j}\perp{\rm Ker\,}T_{P}$, $\lVert
u_{j}\rVert=1$, $j=1,2,\dots$, such that $\|T^{(q)}_{P}u_{j}\rVert<1/j$,
$j=1,2,\dots$. Let $v_{j}:=T^{(q)}_{P}u_{j}$, $j=1,2,\ldots$, then
$\lim_{j}\lVert v_{j}\rVert=0$. Since $\sigma^{0}_{P}(x,\mp\omega_{0}(x))\neq
0$ for every $x\in X$, by using the same computation as in [GH] we can find a
pseudodifferential operator $Q$ of order $0$ such that

$T^{(q)}_{Q}T^{(q)}_{P}=S^{(q)}+F,$

where $F$ is smoothing. Thus,
$T^{(q)}_{Q}T^{(q)}_{P}u_{j}=T^{(q)}_{Q}v_{j}=S^{(q)}u_{j}+Fu_{j}$,
$j=1,2,\dots$. Since $u_{j}\perp{\rm Ker\,}T^{(q)}_{P}$, $u_{j}=S^{q}u_{j}$,
hence $T^{(q)}_{Q}v_{j}=u_{j}+Fu_{j}$, $j=1,2,\dots$. Since $T^{(q)}_{Q}$ is
$L^{2}$ bounded, $\lim_{j}\lVert T^{(q)}_{Q}v_{j}\rVert=0$. Fix $\epsilon>0$.
By Rellich’s Lemma, there is a subsequence $1\leq j_{1}<j_{2}<\dots$ such that
$u_{j_{s}}\rightarrow u$ in $H^{-\epsilon}(X)$ as $s\rightarrow+\infty$. Since
$F$ is smoothing, $F:H^{-\epsilon}(X)\rightarrow L^{2}(X)$ is continuous,
$\lim_{s}Fu_{j_{s}}=u$ in $L^{2}_{(0,q)}(X)$. Thus, $\lim_{s}u_{j_{s}}=u$ in
$L^{2}_{(0,q)}(X)$. Since $u_{j}\perp{\rm Ker\,}T^{(q)}_{P}$ for each $j$,
then $u\perp{\rm Ker\,}T^{(q)}_{P}$; but
$T^{(q)}_{P}u=\lim_{s}T^{(q)}_{P}u_{j_{s}}=0$ and thus we get a contradiction.
∎

Now, take $\tau\in\mathcal{C}^{\infty}_{c}(I)$, where $I\subset\mathbb{R}_{+}$
is an open interval with $\overline{I}$ is a compact subset of
$\mathbb{R}_{+}$. Let
$\widetilde{\tau}\in\mathcal{C}^{\infty}_{c}(I^{\mathbb{C}})$ be an almost
analytic extension of $\tau$, where $I^{\mathbb{C}}$ is a bounded interval of
$\mathbb{C}$ with $I^{\mathbb{C}}\cap\mathbb{R}=I$. The explicit expression
for the operator $\tau(T^{(q)}_{P})$ due to Heffler and Sjöstrand is the
following:

$\tau(T^{(q)}_{P})=-\frac{1}{\pi}\int_{\mathbb{C}}\frac{\partial\tilde{\tau}(z)}{\partial\bar{z}}(z-T^{(q)}_{P})^{-1}\mathrm{d}x\,\mathrm{d}y,$
(26)

where $z=x+iy$. From (26), we shall study the distributional kernel of
$(z-T_{P})^{-1}$.

For simplicity, from now on, until further notice, we always assume that
$n_{-}\neq n_{+}$. We will always assume that $z\in I^{\mathbb{C}}$ with
$\mathrm{Im}z\neq 0$.

######  Theorem 3.5.

We can find $B_{z}\in\Psi_{n,z}(X)$ so that

$(z-T^{(q)}_{P})B_{z}\equiv S^{(q)}\mod\hat{\mathcal{C}}^{\infty},$

$B_{z}\equiv S^{(q)}B_{z}\equiv B_{z}S^{(q)}$ and

$\displaystyle\pi_{x,n_{-}}\sigma^{0}_{B_{z}}(x,x,z)\pi_{x,n_{-}}=\frac{1}{z-\sigma^{0}_{P}(x,-\omega_{0}(x))}\cdot
s_{-}^{0}(x,x)\,,$ (27)

for every $x\in X$, where $s^{0}_{-}(x,x)$ is as in Theorem 3.3.

###### Proof.

Let $x=(x_{1},\ldots,x_{2n+1})$ be local coordinates of $X$ defined on an open
set $D\subset X$. We will first work on $D$. Let
$\hat{B}_{0,z}(x,y):=\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}\hat{b}_{0}(x,y,z,t)\mathrm{d}t$,
where

$\hat{b}_{0}(x,y,z,t):=\frac{s^{0}_{-}(x,x)}{z-\sigma^{0}_{P}(x,-\omega_{0}(x))}t^{n}\in
S^{n}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

Put $B_{0,z}:=S^{(q)}\circ\hat{B}_{0,z}\circ S^{(q)}$. From complex stationary
phase formula of Melin-Sjöstrand, we have $B_{0,z}\in\Psi_{n,z}(X)$ and

$\begin{split}&B_{0,z}(x,y)\equiv\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}b_{0}(x,y,z,t)\mathrm{d}t\mod\hat{\mathcal{C}}^{\infty},\\\
&\mbox{$b_{0}(x,y,z,t)\sim\sum^{+\infty}_{j=0}b_{0,j}(x,y,z)t^{n-j}$ in
$S^{n}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$}\end{split}$

and

$b_{0,0}(x,x,z)=\frac{s^{0}_{-}(x,x)}{z-\sigma^{0}_{P}(x,-\omega_{0}(x))}t^{n}.$
(28)

From complex stationary phase formula of Melin-Sjöstrand and (28), it is
straightforward to check that

$\begin{split}&(z-T^{(q)}_{P})B_{0,z}-S^{(q)}\equiv\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}\hat{r}_{0}(x,y,z,t)\mathrm{d}t\mod\hat{\mathcal{C}}^{\infty},\\\
&\mbox{$\hat{r}_{0}(x,y,z,t)\sim\sum^{+\infty}_{j=0}\hat{r}_{0,j}(x,y,z)t^{n-j}$
in $S^{n}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\
&\mbox{$\hat{r}_{0,0}(x,x,z)=0$, for every $x\in D$}.\end{split}$ (29)

From Lemma 3.3 and (29), we see that
$(z-T^{(q)}_{P})B_{0,z}-S^{(q)}\in\Psi_{n-1,z}(D)$. Thus,

$\begin{split}&(z-T^{(q)}_{P})B_{0,z}-S^{(q)}\equiv\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}r_{0}(x,y,z,t)\mathrm{d}t\mod\hat{\mathcal{C}}^{\infty},\\\
&\mbox{$r_{0}(x,y,z,t)\sim\sum^{+\infty}_{j=0}r_{0,j}(x,y,z)t^{n-1-j}$ in
$S^{n-1}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\
&\mbox{$r_{0,0}(x,x,z)=0$, for every $x\in D$}.\end{split}$ (30)

Let

$\begin{split}&\hat{G}_{z}(x,y):=\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}\hat{b}_{1}(x,y,z,t)\mathrm{d}t\in\Psi_{n-1,z}(D),\\\
&\hat{b}_{1}(x,y,z,t):=-\pi_{x,n_{-}}r_{0,0}(x,x,z)\pi_{x,n_{-}}\Bigr{(}\frac{1}{2}\left|{\rm
det\,}\mathcal{L}_{x}\right|\pi^{-n-1}\Bigr{)}^{-1}\frac{t^{n-1}}{z-\sigma^{0}_{P}(x,-\omega_{0}(x))}.\end{split}$

Let $G_{z}:=S^{(q)}\circ\hat{G}_{z}\circ S^{(q)}$. From complex stationary
phase formula, we have

$\begin{split}&G_{z}(x,y)\equiv\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}b_{1}(x,y,z,t)\mathrm{d}t\mod\hat{\mathcal{C}}^{\infty},\\\
&b_{1}(x,y,z,t)=-\pi_{x,n_{-}}r_{0,0}(x,x,z)\pi_{x,n_{-}}\frac{t^{n-1}}{z-\sigma^{0}_{P}(x,-\omega_{0}(x))}.\end{split}$
(31)

Let $B_{1,z}:=B_{0,z}+G_{z}$. From (30) and (31), we have

$\begin{split}&(z-T^{(q)}_{P})B_{1,z}-S^{(q)}\equiv\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}r_{1}(x,y,z,t)\mathrm{d}t\mod\hat{\mathcal{C}}^{\infty},\\\
&\mbox{$r_{1}(x,y,z,t)\sim\sum^{+\infty}_{j=0}r_{1,j}(x,y,z)t^{n-1-j}$ in
$S^{n-1}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\
&\mbox{$\pi_{x,n_{-}}r_{1,0}(x,x,z)\pi_{x,n_{-}}=0$, for every $x\in
D$}.\end{split}$ (32)

From Lemma 3.3 and (32), we see that
$(z-T^{(q)}_{P})B_{1,z}-S^{(q)}\in\Psi_{n-2,z}(D)$. Continuing in this way, we
get $b_{j}(x,y,z,t)\in S^{n-j}_{z,{\rm cl\,}}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $j=0,1,\ldots$,
such that for every $N\in\mathbb{N}$, we have

$(z-T^{(q)}_{P})B^{N}_{1,z}-S^{(q)}\in\Psi_{n-N-1,z}(D),$

where
$B^{N}_{z}:=\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}\sum^{N}_{j=0}b_{j}(x,y,z,t)\mathrm{d}t$.
Let

$b(x,y,z,t)\sim\sum^{+\infty}_{j=0}b_{j}(x,y,z,t)$ in $S^{n}_{z,{\rm
cl\,}}(D\times D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

and put

$\widetilde{B}_{z,D}:=\int^{+\infty}_{0}e^{it\varphi_{-}(x,y)}b(x,y,z,t)\mathrm{d}t\in\Psi_{n,z}(D).$

We have

$(z-T^{(q)}_{P})\widetilde{B}_{z,D}\equiv
S^{(q)}\mod\hat{\mathcal{C}}^{\infty}.$

We now assume that $b(x,y,z,t)$ is properly supported on $D\times D$.

Assume that $X=\bigcup^{N}_{j=1}D_{j}$, $D_{j}$ is an open set of $X$ as
above, $j=1,\ldots,N$. Let $\chi_{j}\in\mathcal{C}^{\infty}_{c}(D_{j})$,
$j=1,\ldots,N$, $\sum^{N}_{j=1}\chi_{j}\equiv 1$ on $X$. Let
$\widetilde{\chi}_{j}\in\mathcal{C}^{\infty}_{c}(D_{j})$,
$\widetilde{\chi}_{j}\equiv 1$ near ${\rm supp\,}\chi_{j}$, $j=1,\ldots,N$.
Put

$B_{z}:=\sum^{N}_{j=1}S^{(q)}\widetilde{\chi}_{j}\widetilde{B}_{z,D_{j}}\chi_{j}S^{(q)}.$

Then, $B_{z}\in\Psi_{n,z}(X)$ and $(z-T^{(q)}_{P})B_{z}\equiv
S^{(q)}\mod\hat{\mathcal{C}}^{\infty}$. The theorem follows. ∎

We need

######  Lemma 3.7.

For every $s\in\mathbb{Z}$, there exist $N_{s}\in\mathbb{N}$ and $c>0$ such
that

$\left\|(z-T^{(q)}_{P})^{-1}S^{(q)}u\right\|_{s-1}\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N_{s}}}\left\|u\right\|_{s},$ (33)

for every $z\in I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every
$u\in\Omega^{0,q}(X)$.

Thus, $(z-T^{(q)}_{P})^{-1}S^{(q)}$ can be extended to a continuous operator:

$(z-T^{(q)}_{P})^{-1}S^{(q)}:H^{s}(X,T^{*0,q}X)\rightarrow
H^{s-1}(X,T^{*0,q}X)$

and we have the estimates (33), for every $s\in\mathbb{Z}$, every $z\in
I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every $u\in H^{s}(X,T^{*0,q}X)$.

###### Proof.

Fix $s\in\mathbb{Z}$, $s\leq 1$. From Theorem 3.5, we write

$(z-T^{(q)}_{P})B_{z}\equiv S^{(q)}+S^{(q)}R_{z},$ (34)

where $R_{z}$ is $z$-smoothing. From (34), we have

$B_{z}\equiv(z-T^{(q)}_{P})^{-1}S^{(q)}+(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z}.$
(35)

Note that

$\left\|(z-T^{(q)}_{P})^{-1}u\right\|\leq\frac{1}{\left|{\rm
Im\,}z\right|}\left\|u\right\|,$ (36)

for every $z\in I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every $u\in
L^{2}_{(0,q)}(X)$. Since $R_{z}$ is $z$-smoothing, we have

$\left\|R_{z}u\right\|\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N_{s}}}\left\|u\right\|_{s},$ (37)

for every $z\in I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every $u\in
H^{s}(X,T^{*0,q}X)$, where $c>0$ and $N_{s}\in\mathbb{N}$. From (36) and (37),
we deduce that

$\left\|(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z}u\right\|_{s-1}\leq\left\|(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z}u\right\|\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N_{s}}}\left\|u\right\|_{s},$ (38)

for every $z\in I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every $u\in
H^{s}(X,T^{*0,q}X)$, where $c>0$ and $N_{s}\in\mathbb{N}$. From Lemma 3.5,(35)
and (38), we get (33) for $s\leq 1$. Since $(z-T^{(q)}_{P})^{-1}S^{(q)}$ is
self-adjoint, by taking adjoint in (33), we get (33) for all $s\in\mathbb{Z}$.
The lemma follows. ∎

######  Lemma 3.8.

We have

$(z-T^{(q)}_{P})^{-1}S^{(q)}\equiv B_{z}\mod\hat{\mathcal{C}}^{\infty},$ (39)

where $B_{z}$ is as in Theorem 3.5.

###### Proof.

From (35), we have

$B_{z}\equiv(z-T^{(q)}_{P})^{-1}S^{(q)}+(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z},$
(40)

where $R_{z}$ is $z$-smoothing. From Lemma 3.7, we see that for every
$s_{1},s_{2}\in\mathbb{Z}$, there exist $c>0$ and $N>0$ such that

$\left\|(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z}u\right\|_{s_{2}}\leq\frac{c}{\left|{\rm
Im\,}z\right|^{N}}\left\|u\right\|_{s_{1}},$ (41)

for every $z\in I^{\mathbb{C}}$ with ${\rm Im\,}z\neq 0$ and every
$u\in\Omega^{0,q}(X)$. From (41) and Sobolev embedding theorem, we get

$(z-T^{(q)}_{P})^{-1}S^{(q)}R_{z}\equiv 0\mod\hat{\mathcal{C}}^{\infty}.$ (42)

From (40) and (42), the lemma follows. ∎

We can now prove one of the main results of this work

######  Theorem 3.6.

With the notations and assumptions above, assume that $0\notin{\rm
supp\,}\tau$. Then, $\tau(T^{(q)}_{P})\in\Psi_{n}(X)$ with

$\mbox{$\sigma^{0}_{\tau(T^{(q)}_{P})}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}}$,
for every $x\in X$}.$ (43)

###### Proof.

Since $0\notin{\rm supp\,}\tau$, it is not difficult to see that

$\tau(T^{(q)}_{P})=\tau(T^{(q)}_{P})\circ
S^{(q)}=S^{(q)}\circ\tau(T^{(q)}_{P}).$

From this observation and (26), we have

$\tau(T^{(q)}_{P})=-\frac{1}{\pi}\int_{\mathbb{C}}\frac{\partial\tilde{\tau}(z)}{\partial\bar{z}}(z-T^{(q)}_{P})^{-1}\circ
S^{(q)}\mathrm{d}x\,\mathrm{d}y.$ (44)

From Lemma 3.8 and (44), we have

$\tau(T^{(q)}_{P})\equiv-\frac{1}{\pi}\int_{\mathbb{C}}\frac{\partial\tilde{\tau}(z)}{\partial\bar{z}}B_{z}\mathrm{d}x\,\mathrm{d}y.$
(45)

From (27), (45) and Cauchy integral formula, we get (43). ∎

If $q=n_{-}=n_{+}$, we can repeat the proof of Theorem 43 with minor changes
and deduce that

######  Theorem 3.7.

With the notations and assumptions above, assume that $0\notin{\rm
supp\,}\tau$. Suppose $q=n_{-}=n_{+}$. Then, $\tau(T^{(q)}_{P})\in\Psi_{n}(X)$
with

$\begin{split}&\mbox{$\sigma^{0}_{\tau(T^{(q)}_{P}),-}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}}$,
for every $x\in X$},\\\
&\mbox{$\sigma^{0}_{\tau(T^{(q)}_{P}),+}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,\omega_{0}(x)))\pi_{x,n_{+}}$,
for every $x\in X$}.\end{split}$ (46)

###  3.2 Group actions on CR manifolds

We recall that $J:HX\rightarrow HX$ is the complex structure map. In this
section we assume that $X$ admits an actions of a $d$-dimensional connected
compact Lie group $G$. We assume throughout that

######  Assumption 3.1.

The group $G$ acts via contactomorphisms and holomorphisms. This means that
$g^{\ast}\omega_{0}=\omega_{0}$ on $X$ and also $g_{\ast}J=Jg_{\ast}$ on $HX$,
for every $g\in G$, where $g^{*}$ and $g_{*}$ denote the pull-back map and
push-forward map of $G$, respectively.

We denote the Lie algebra of the group $G$ by $\mathfrak{g}$. Furthermore for
every $\xi\in\mathfrak{g}$, we write $\xi_{X}$ to denote the infinitesimal
vector field on $X$ induced by $\xi$, which is given by

$(\xi_{X}u)(x)=\frac{\partial}{\partial t}\left(u(\exp_{G}(t\xi)\circ
x)\right)|_{t=0}\,,$

for any smooth function $u\in\mathcal{C}^{\infty}(X)$.

######  Definition 3.8.

The momentum map associated to the contact form $\omega_{0}$ is a map
$\mu:X\to\mathfrak{g}^{*}$ such that

$\langle\mu(x),\xi\rangle=\omega_{0}(\xi_{X}(x))$ (47)

for all $x\in X$ and $\xi\in\mathfrak{g}$.

Let us denote by $b$ the non-degenerate bi-linear form

$b(\cdot,\cdot)=\mathrm{d}\omega_{0}(\cdot,J\cdot)\text{ on }HX\,,$ (48)

and

$\underline{\mathfrak{g}}^{\perp_{b}}=\left\\{v\in HX;\,b(\xi_{X},v)=0,\ \
\forall\xi_{X}\in\underline{\mathfrak{g}}\right\\}$

where $\underline{\mathfrak{g}}={\rm Span\,}(\xi_{X};\,\xi\in\mathfrak{g})$.

We need also to assume that

######  Assumption 3.2.

$0\in\mathfrak{g}^{*}$ is a regular value for the moment map $\mu$, the action
$G$ on the submanifold $\mu^{-1}(0)$ is free and

$\mbox{$\underline{\mathfrak{g}}_{x}\cap\underline{\mathfrak{g}}^{\perp_{b}}_{x}=\left\\{0\right\\}$
},$ (49)

at every point $x\in\mu^{-1}(0)$.

By assumptions 3.1 and 3.2 above, $Y:=\mu^{-1}(0)$ is a submanifold of $X$ a
co-dimension $d$. Let $HY:=HX\cap TY$. Note that if the Levi form is positive
at $Y$, then (49) holds. Fix a $G$-invariant smooth Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$ so that
$T^{1,0}X$ is orthogonal to $T^{0,1}X$, furthermore $\underline{\mathfrak{g}}$
is orthogonal to $HY\cap JHY$ at every point of $Y$,
$\langle\,u\,|\,v\,\rangle$ is real if $u,v$ are real tangent vectors and
$\langle\,R\,|\,R\,\rangle=1$.

####  3.2.1 Invariant Fourier integral operators of Szegő type

We first recall some results about invariant Fourier integral operators of
Szegő type from [GH]. For a given $g\in G$ let

$g^{*}:\Lambda^{r}_{x}(T^{*}X\otimes\mathbb{C})\rightarrow\Lambda^{r}_{g^{-1}\circ
x}(T^{*}X\otimes\mathbb{C})$

be the pull-back map. Since the action is holomorphic, we have that
$g^{*}:T^{*0,q}_{x}X\rightarrow T^{*0,q}_{g^{-1}\circ x}X$, for all $x\in X.$
Thus, for $u\in\Omega^{0,q}(X)$, we have $g^{*}u\in\Omega^{0,q}(X)$.
Furthermore, put

$\Omega^{0,q}(X)^{G}:=\left\\{u\in\Omega^{0,q}(X);\,g^{*}u=u,\ \ \forall g\in
G\right\\}.$

Since the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$TX\otimes\mathbb{C}$ is $G$-invariant, the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle$ is $G$-invariant. Let $u\in
L^{2}_{(0,q)}(X)$ and $g\in G$, we can also define $g^{*}u$ in the standard
way. Put

$L^{2}_{(0,q)}(X)^{G}:=\left\\{u\in L^{2}_{(0,q)}(X);\,g^{*}u=u,\ \ \forall
g\in G\right\\}$

and

$({\rm Ker\,}\Box^{q}_{b})^{G}:={\rm Ker\,}\Box^{q}_{b}\cap
L^{2}_{(0,q)}(X)^{G}.$

The $G$-invariant Szegő projection is the orthogonal projection

$S^{(q)}_{G}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{q}_{b})^{G}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let
$S^{(q)}_{G}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution kernel of
$S^{(q)}_{G}$. Let

$Q_{G}:L^{2}_{(0,q)}(X)\rightarrow L^{2}_{(0,q)}(X)^{G}$

be the orthogonal projection with respect to $(\,\cdot\,|\,\cdot\,)$. Let

$S^{G}_{\mp}:=Q_{G}\circ S_{\mp}:L^{2}_{(0,q)}(X)\rightarrow({\rm
Ker\,}\Box^{q}_{b})^{G}\,,$ (50)

where $S_{-}$, $S_{+}$ are as in Theorem 3.1. We have
$S^{(q)}_{G}=S^{G}_{-}+S^{G}_{+}$. If $q\neq n_{+}$, then $S^{G}_{+}\equiv 0$.
Recall that we work with the assumption that $q=n_{-}$.

######  Definition 3.9.

Let $P\in L^{\ell}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
$\ell\in\mathbb{Z}$, $\ell\leq 0$. We say that $P$ is in $L^{\ell}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ if $g^{*}(Pu)=P(g^{*}u)$, for
every $u\in L^{2}_{(0,q)}(X)$ and every $g\in G$.

######  Definition 3.10.

Let $P\in L^{\ell}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ with
scalar principal symbol, $\ell\leq 0$, $\ell\in\mathbb{Z}$. The $G$-invariant
Toeplitz operator is given by

$T^{(q)}_{P,G}:=S^{(q)}_{G}\circ P\circ
S^{(q)}_{G}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{q}_{b})^{G},\,$ (51)

Let $f\in\mathcal{C}^{\infty}(X)^{G}$, we write
$T^{(q)}_{f,G}:=T^{(q)}_{M_{f},G}$.

We assume that the action is free near $\mu^{-1}(0)$. Fix $x\in\mu^{-1}(0)$,
consider the linear map

$R_{x}:\underline{\mathfrak{g}}_{x}\rightarrow\underline{\mathfrak{g}}_{x},\,u\mapsto
R_{x}u,\ \
\langle\,R_{x}u\,|\,v\,\rangle=\langle\,\mathrm{d}\omega_{0}(x)\,,\,Ju\wedge
v\,\rangle.$

Let $\det R_{x}=\lambda_{1}(x)\cdots\lambda_{d}(x)$, where $\lambda_{j}(x)$,
$j=1,2,\ldots,d$, are the eigenvalues of $R_{x}$. Fix $x\in\mu^{-1}(0)$, put
$Y_{x}=\left\\{g\circ x;\,g\in G\right\\}$. $Y_{x}$ is a $d$-dimensional
submanifold of $X$. The $G$-invariant Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ induces a volume form
$\mathrm{dV}_{Y_{x}}$ on $Y_{x}$. Put

$V_{{\rm eff\,}}(x):=\int_{Y_{x}}\mathrm{dV}_{Y_{x}}.$ (52)

As before, for simplicity, until further notice, we always assume that
$n_{-}\neq n_{+}$.

We recall Theorem $4.1$ in [GH].

######  Theorem 3.8.

Let $P\in L^{\ell}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ with
scalar principal symbol, $\ell\leq 0$, $\ell\in\mathbb{Z}$. Let $D$ be an open
set in $X$ such that the intersection $\mu^{-1}(0)\cap D=\emptyset$. Then
$T^{(q)}_{P,G}\equiv 0$ on $D$.

Let $p\in\mu^{-1}(0)$ and let $U$ be a local neighborhood of $p$ with local
coordinates $(x_{1},\dots\,x_{2n+1})$. Then, the distributional kernel of
$T_{P,G}^{(q)}$ satisfies

$T_{P,G}^{(q)}(x,y)\equiv\int_{0}^{\infty}e^{\imath
t\,\Phi_{-}(x,y)}a_{P}(x,\,y,\,t)\,\mathrm{d}t\ \ \ \mbox{on $U\times U$}.$
(53)

The phase $\Phi_{-}$ is described in local coordinates below and it is equal
to the phase of $S_{-}^{G}(x,\,y)$ in [HH], the symbol $a_{P}$ satisfies the
following properties

$a_{P}(x,\,y,\,t)\sim\sum_{j=0}^{+\infty}a_{P}^{j}(x,\,y)\,t^{n-d/2-j}$

in $S^{n-d/2}_{1,0}(U\times
U\times\mathbb{R},\,T^{*\,0,q}X\boxtimes(T^{*\,0,q}X)^{*})$,

$a_{P}^{j}(x,\,y)\in\mathcal{C}^{\infty}(U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j\in\mathbb{N}_{0},$

and for every $x\in\mu^{-1}(0)$,

$a_{P}^{0}(x,\,x)=2^{d-1}\frac{1}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\left|\det
R_{x}^{-\frac{1}{2}}\right|\left|\det\mathcal{L}_{x}\right|\sigma^{0}_{P}(x,-\omega_{0}(x))\pi_{x,n_{-}}.$
(54)

We do make use of local coordinates defined in [HH] to write the phase in
local coordinates, we will not recall it here explicitly, we refer to Theorem
3.7 in [HH]. For convenience of the reader we just mention that the phase
function $\Phi_{-}(x,y)\in\mathcal{C}^{\infty}(U\times U)$ is $G$-invariant
which results in the fact that it is independent of $(x_{1},\ldots,x_{d})$ and
$(y_{1},\ldots,y_{d})$. Hence, we have that
$\Phi_{-}(x,y)=\Phi_{-}((0,x^{\prime\prime}),(0,y^{\prime\prime})):=\Phi_{-}(x^{\prime\prime},y^{\prime\prime})$,
$x^{\prime\prime}=(x_{d+1},\ldots,x_{2n+1})$,
$y^{\prime\prime}=(y_{d+1},\ldots,y_{2n+1})$. Moreover, there is a constant
$c>0$ such that

${\rm Im\,}\Phi_{-}(x^{\prime\prime},y^{\prime\prime})\geq
c\Bigr{(}\left|\hat{x}^{\prime\prime}\right|^{2}+\left|\hat{y}^{\prime\prime}\right|^{2}+\left|\mathring{x}^{\prime\prime}-\mathring{y}^{\prime\prime}\right|^{2}\Bigr{)},\
\ \forall((0,x^{\prime\prime}),(0,y^{\prime\prime}))\in U\times U,$ (55)

where $\hat{x}^{\prime\prime}=(x_{d+1},\ldots,x_{2d})$,
$\hat{y}^{\prime\prime}=(y_{d+1},\ldots,y_{2d})$,
$\mathring{x}^{\prime\prime}=(x_{d+1},\ldots,x_{2n})$,
$\mathring{y}^{\prime\prime}=(y_{d+1},\ldots,y_{2n})$.

Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ be a
$G$-invariant self-adjoint classical pseudodifferential operator on $X$ with
scalar principal symbol. Then, it is not difficult to see that the Toepliotz
operator $T^{(q)}_{P,G}$ is self-adjoint. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$. We have

$\tau(T^{(q)}_{P,G})=\tau(T^{(q)}_{P,G})\circ Q_{G}.$ (56)

From Theorem 3.6, (43) and (56), we can repeat the proof of Theorem $4.1$ in
[GH] and get

######  Theorem 3.9.

Recall that we work with the assumption that $n_{-}\neq n_{+}$. With the
notations and assumptions above, let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ with
scalar principal symbol. Let $D$ be an open set in $X$ such that the
intersection $\mu^{-1}(0)\cap D=\emptyset$. Then $\tau(T^{(q)}_{P,G})\equiv 0$
on $D$.

Let $p\in\mu^{-1}(0)$ and let $U$ be a local neighborhood of $p$ with local
coordinates $(x_{1},\dots\,x_{2n+1})$. Then, the distributional kernel of
$\tau(T_{P,G}^{(q)})$ satisfies

$\tau(T_{P,G}^{(q)})(x,y)\equiv\int_{0}^{\infty}e^{\imath
t\,\Phi_{-}(x,y)}a_{\tau}(x,\,y,\,t)\,\mathrm{d}t\ \ \ \mbox{on $U\times U$}.$
(57)

The phase $\Phi_{-}$ is as in Theorem 3.8, $a_{\tau}$ satisfies the following
properties

$a_{\tau}(x,\,y,\,t)\sim\sum_{j=0}^{+\infty}a_{\tau}^{j}(x,\,y)\,t^{n-d/2-j}$

in $S^{n-d/2}_{1,0}(U\times
U\times\mathbb{R},\,T^{*\,0,q}X\boxtimes(T^{*\,0,q}X)^{*})$,

$a_{\tau}^{j}(x,\,y)\in\mathcal{C}^{\infty}(U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j\in\mathbb{N}_{0},$

and for every $x\in\mu^{-1}(0)$,

$a_{\tau}^{0}(x,\,x)=2^{d-1}\frac{1}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\left|\det
R_{x}^{-\frac{1}{2}}\right|\left|\det\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}}.$
(58)

Similarly, if $n_{-}=n_{+}$, we have

######  Theorem 3.10.

With the notations and assumptions above, assume that $n_{-}=n_{+}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
Let $P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ with
scalar principal symbol. Let $D$ be an open set in $X$ such that the
intersection $\mu^{-1}(0)\cap D=\emptyset$. Then $\tau(T^{(q)}_{P,G})\equiv 0$
on $D$.

Let $p\in\mu^{-1}(0)$ and let $U$ be a local neighborhood of $p$ with local
coordinates $(x_{1},\dots\,x_{2n+1})$. Then, the distributional kernel of
$\tau(T_{P,G}^{(q)})$ satisfies

$\tau(T_{P,G}^{(q)})(x,y)\equiv\int_{0}^{\infty}e^{\imath
t\,\Phi_{-}(x,y)}a_{\tau,-}(x,\,y,\,t)\mathrm{d}t+\int_{0}^{\infty}e^{\imath
t\,\Phi_{+}(x,y)}a_{\tau,+}(x,\,y,\,t\,)\mathrm{d}t\ \ \ \mbox{on $U\times
U$}.$ (59)

The phases $\Phi_{\mp}$ are equal to the phases of $S_{\mp}^{G}(x,\,y)$ in
[HH], $a_{\tau,\mp}$ satisfies the following properties

$a_{\tau,\,p}(x,\,y,\,t)\sim\sum_{j=0}^{+\infty}a_{\tau,\mp}^{j}(x,\,y)\,t^{n-d/2-j}$

in $S^{n-d/2}_{1,0}(U\times
U\times\mathbb{R},\,T^{*\,0,q}X\boxtimes(T^{*\,0,q}X)^{*})$,

$a_{\tau,\mp}^{j}(x,\,y)\in\mathcal{C}^{\infty}(U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j\in\mathbb{N}_{0},$

and for every $x\in\mu^{-1}(0)$,

$\begin{split}&a_{\tau,-}^{0}(x,\,x)=2^{d-1}\frac{1}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\left|\det
R_{x}^{-\frac{1}{2}}\right|\left|\det\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}},\\\
&a_{\tau,+}^{0}(x,\,x)=2^{d-1}\frac{1}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\left|\det
R_{x}^{-\frac{1}{2}}\right|\left|\det\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,\omega_{0}(x)))\pi_{x,n_{+}}.\end{split}$
(60)

##  4 Quantization commutes with reduction for Toeplitz operators

We will use the same notations and assumptions as Section 3.2. Let
$X_{G}:=\mu^{-1}(0)/G$. In [HH], it was proved that $X_{G}$ is a CR manifold
with natural CR structure induced by $T^{1,0}X$ of dimension $2n-2d+1$. Let
$P\in L^{0}_{{\rm cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G}$ be a scalar
pseudodifferential operator. The operator $P$ gives rise to ${P}_{X_{G}}$ on
the CR reduction $X_{G}=\mu^{-1}({0})/G$. The Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$ induces an
Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$ on
$TX_{G}\otimes\mathbb{C}$ and let $(\,\cdot\,|\,\cdot\,)_{X_{G}}$ be the
$L^{2}$ inner product on $L^{2}_{(0,q)}(X_{G})$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$. We write $\mathrm{dV}_{X_{G}}$ to
denote the volume form on $X_{G}$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$. Let $\Box^{q}_{b,X_{G}}$ be the
Kohn Laplacian for $(0,q)$ forms on $X_{G}$ defined with respect to
$\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$ and
$(\,\cdot\,|\,\cdot\,)_{X_{G}}$. We can define the standard Toeplitz operator
associated to ${P}_{X_{G}}$:

$T^{(q)}_{P_{X_{G}}}:=S^{(q)}_{X_{G}}\circ{P}_{X_{G}}\circ
S^{(q)}_{X_{G}}:L^{2}_{(0,q)}(X_{G})\rightarrow L^{2}_{(0,q)}(X_{G}),$

where
$S^{(q)}_{X_{G}}:L^{2}_{(0,q)}(X_{G})\rightarrow\mathrm{Ker}\,\Box^{q}_{b,X_{G}}$
is the Szegő projection on $X_{G}$.

Let $\mathcal{L}_{X_{G}}$ be the Levi form on $X_{G}$ induced naturally from
the Levi form $\mathcal{L}$ on $X$. Since $\mathcal{L}$ is non-degenerate of
constant signature, $\mathcal{L}_{X_{G}}$ is also non-degenerate of constant
signature. We denote by $n_{X_{G},-}$ (respectively $n_{X_{G},+}$) the number
of negative (respectively positive) eigenvalues of $\mathcal{L}_{X_{G}}$. For
simplicity, until further notice, we assume that $n_{X_{G},-}\neq
n_{X_{G},+}$. We introduce the map $\sigma$. Since
$\underline{\mathfrak{g}}_{x}$ is orthogonal to $H_{x}Y\cap JH_{x}Y$ and
$H_{x}Y\cap JH_{x}Y\subset\underline{\mathfrak{g}}^{\perp_{b}}_{x}$, we can
find a $G$-invariant orthonormal basis $\left\\{Z_{1},\ldots,Z_{n}\right\\}$
of $T^{1,0}X$ on $Y$ such that

$\begin{array}[]{ll}&\mathcal{L}_{x}(Z_{j}(x),\overline{Z}_{k}(x))=\delta_{j,k}\lambda_{j}(x),\
\ j,k=1,\ldots,n\,,\\\
&Z_{j}(x)\in\underline{\mathfrak{g}}_{x}+iJ\underline{\mathfrak{g}}_{x},\ \
j=1,2,\ldots,d\end{array}$

and

$Z_{j}(x)\in H_{x}Y\otimes\mathbb{C}\cap J(H_{x}Y\otimes\mathbb{C}),\ \
j=d+1,\ldots,n\,.$

Let $\left\\{e_{1},\ldots,e_{n}\right\\}$ denote the orthonormal basis of
$T^{*0,1}X$ on $Y$, dual to
$\left\\{\overline{Z}_{1},\ldots,\overline{Z}_{n}\right\\}$. Fix
$s=0,1,2,\ldots,n-d$. For $x\in Y$, put

$B^{*0,s}_{x}X=\left\\{\sum_{d+1\leq j_{1}<\cdots<j_{s}\leq
n}a_{j_{1},\ldots,j_{s}}e_{j_{1}}\wedge\cdots\wedge
e_{j_{s}};\,a_{j_{1},\ldots,j_{s}}\in\mathbb{C},\ \forall d+1\leq
j_{1}<\cdots<j_{s}\leq n\right\\}$

and let $B^{*0,s}X$ be the vector bundle of $Y$ with fiber $B^{*0,s}_{x}X$,
$x\in Y$. Let $\mathcal{C}^{\infty}(Y,B^{*0,s}X)^{G}$ denote the set of all
$G$-invariant sections of $Y$ with values in $B^{*0,s}X$. Let

$\iota_{G}:\mathcal{C}^{\infty}(Y,B^{*0,s}X)^{G}\rightarrow\Omega^{0,s}(Y_{G})$

be the natural identification.

Assume that $\lambda_{1}<0,\ldots,\lambda_{r}<0$, and
$\lambda_{d+1}<0,\ldots,\lambda_{n_{-}-r+d}<0$, where $n_{-}-r=n_{X_{G},-}$.
For $x\in Y$, put

$\hat{\mathcal{N}}(x,n_{-})=\left\\{ce_{d+1}\wedge\cdots\wedge
e_{n_{-}-r+d};\,c\in\mathbb{C}\right\\},$

and let

$\begin{array}[]{c}\hat{p}_{-}=\hat{p}_{x,-}:\mathcal{N}(x,n_{-})\rightarrow\hat{\mathcal{N}}(x,n_{-}),\\\
u=ce_{1}\wedge\cdots\wedge e_{r}\wedge e_{d+1}\wedge\cdots\wedge
e_{n_{-}-r+d}\rightarrow ce_{d+1}\wedge\cdots\wedge e_{n_{-}-r+d}.\end{array}$

We can define $\hat{p}_{+}$ in the similar way. Let $\iota:Y\rightarrow X$ be
the natural inclusion and let
$\iota^{*}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(Y)$ be the pull-back of
$\iota$. Let

$f(x)=\sqrt{V_{{\rm
eff\,}}(x)}\left|\det\,R_{x}\right|^{-\frac{1}{4}}\in\mathcal{C}^{\infty}(Y)^{G}.$

Let $E\in L^{-\frac{d}{4}}_{{\rm
cl\,}}(X_{G},T^{*0,n_{X_{G},-}}X_{G}\boxtimes(T^{*0,n_{X_{G},-}}X_{G})^{*})$
be a scalar elliptic pseudodifferential operator with principal symbol
$\sigma^{0}_{E}(x,\xi)=\left|\xi\right|^{-\frac{d}{4}}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$. Let

$\begin{split}\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\cap\Omega^{0,n_{-}}(X)&\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\cap\Omega^{0,n_{X_{G},-}}(X_{G}),\\\
u&\rightarrow\tau(T^{(n_{X_{G},-})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}_{-}\circ\pi_{x,n_{-}}\circ f\circ\iota^{*}\circ
u.\end{split}$ (61)

We can extend $\sigma$ to $\Omega^{0,n_{-}}(X)$ by

$\begin{split}\sigma:\Omega^{0,n_{-}}(X)&\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\cap\Omega^{0,n_{X_{G},-}}(X_{G}),\\\
u&\rightarrow\tau(T^{(n_{X_{G},-})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}_{-}\circ\pi_{x,n_{-}}\circ
f\circ\iota^{*}\circ\tau(T^{(n_{-})}_{P,G})u.\,.\end{split}$ (62)

Let
$\sigma^{\,*}:\Omega^{0,n_{X_{G},-}}(X_{G})\rightarrow\mathcal{D}^{\prime}(X,T^{*0,n_{-}}X)$
be the formal adjoint of $\sigma$. From Theorem 3.6 and Theorem 3.9, we can
repeat the method in Section 4.2 of [HMM] and deduce that

######  Theorem 4.1.

With the same notations and assumptions above, suppose that
$\Box^{n_{X_{G},-}}_{b}$ has closed range. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
Then, $\sigma^{\,*}$ maps $\Omega^{0,n_{X_{G},-}}(X_{G})$ continuously to
$\Omega^{0,n_{-}}(X)$. Hence, $\sigma^{*}\,\sigma$ is a well-defined
continuous operator

$\sigma^{*}\,\sigma:\Omega^{0,n_{-}}(X)\rightarrow\Omega^{0,n_{-}}(X).$

Furthermore, the distribution kernel of $\sigma^{*}\,\sigma$ satisfies

$\sigma^{*}\,\sigma=\tau(T^{(q)}_{P,G})^{2}+\Gamma,$ (63)

where $\Gamma:\Omega^{0,n_{-}}(X)\rightarrow\Omega^{0,n_{-}}(X)$ is a
continuous map and the distribution kernel of $\Gamma$ satisfies the
following: Let $D$ be an open set in $X$ such that the intersection
$\mu^{-1}(0)\cap D=\emptyset$. Then $\Gamma\equiv 0$ on $D$. Let
$p\in\mu^{-1}(0)$ and let $U$ be a local neighborhood of $p$ with local
coordinates $(x_{1},\dots\,x_{2n+1})$. Then, the distributional kernel of
$\Gamma$ satisfies

$\Gamma(x,y)\equiv\int_{0}^{\infty}e^{\imath
t\,\Phi_{-}(x,y)}r(x,\,y,\,t)\,\mathrm{d}t\ \ \ \mbox{on $U\times U$}.$ (64)

The phase $\Phi_{-}$ is as in Theorem 3.8, $r$ satisfies the following
properties

$r(x,\,y,\,t)\sim\sum_{j=0}^{+\infty}r_{j}(x,\,y)\,t^{n-d/2-j}$

in $S^{n-d/2}_{1,0}(U\times
U\times\mathbb{R},\,T^{*\,0,q}X\boxtimes(T^{*\,0,q}X)^{*})$,

$r_{j}(x,\,y)\in\mathcal{C}^{\infty}(U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j\in\mathbb{N}_{0},$

and for every $x\in\mu^{-1}(0)$, $r_{0}(x,x)=0$.

Since the leading term of $\Gamma$ vanishes, we can repeat the method in
Section 4.1 of [HMM] and deduce that

######  Lemma 4.1.

With the same notations and assumptions above, suppose that
$\Box^{n_{X_{G},-}}_{b}$ has closed range. Let $\Gamma$ be as in (63). There
is a $\varepsilon>0$ such that $\Gamma$ is continuous:

$\Gamma:H^{s}(X,T^{*0,n_{-}}X)\rightarrow H^{s+\varepsilon}(X,T^{*0,n_{-}}X),$

for every $s\in\mathbb{Z}$.

From (63) and Lemma 4.1, we see that $\sigma$ can be extended to
$E_{\tau}(T^{(n_{-})}_{P,G})$ by density and thus,

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}}).$ (65)

We can now prove

######  Theorem 4.2.

With the same notations and assumptions above, suppose that
$\Box^{n_{X_{G},-}}_{b}$ has closed range. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
The map $\sigma$ given by (62) and (65) has the following properties: (i) For
every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv 1$ on
$I$, we have that ${\rm Ker\,}\sigma\cap E_{I}(T^{(n_{-})}_{P,G})$ is a finite
dimensional subspace of $\Omega^{0,n_{-}}(X)$. (ii) For every open bounded
interval $I$, $0\notin\overline{I}$, with $\tau\equiv 1$ on $I$, we have that
${\rm Coker\,}\sigma\cap E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$ is a finite
dimensional subspace of $\Omega^{0,n_{X_{G},-}}(X_{G})$.

###### Proof.

Fix an open bounded interval $I$, $0\notin\overline{I}$ with $\tau\equiv 1$ on
$I$. Let $u\in{\rm Ker\,}\sigma\cap E_{I}(T^{(n_{-})}_{P,G})$. From (63), we
have

$0=\sigma^{*}\,\sigma u=(\tau(T^{(q)}_{P,G})^{2}+\Gamma)u=u+\Gamma u.$ (66)

From Lemma 4.1 and (66), we deduce that $u\in
H^{\varepsilon}(X,T^{*0,n_{-}}X)$. By using Lemma 4.1 and (66) again, we get
$u\in H^{2\varepsilon}(X,T^{*0,n_{-}}X)$. Continuing in this way, we conclude
that $u\in\Omega^{0,n_{-}}(X)$.

Suppose that ${\rm Ker\,}\sigma\cap E_{I}(T^{(n_{-})}_{P,G})$ is not a finite
dimensional subspace of $\Omega^{0,n_{-}}(X)$. We can find $u_{j}\in{\rm
Ker\,}\sigma\cap E_{I}(T^{(n_{-})}_{P,G})$, $j=1,2,\ldots$,
$(\,u_{j}\,|\,u_{k}\,)=\delta_{j,k}$, for every $j,k=1,\ldots$. From Lemma 4.1
and (66), we see that $\left\\{u_{j}\right\\}^{+\infty}_{j=1}$ is a bounded
set in $H^{\varepsilon}(X,T^{*0,n_{-}}X)$. By Rellich’s lemma, there is a
subsequence $1\leq j_{1}<j_{2}<\cdots$,
$\lim_{s\rightarrow+\infty}j_{s}=+\infty$, such that $u_{j_{s}}\rightarrow u$
in $L^{2}_{(0,n_{-})}(X)$ as $s\rightarrow+\infty$, for some $u\in
L^{2}_{(0,n_{-})}(X)$. Since $(\,u_{j}\,|\,u_{k}\,)=\delta_{j,k}$, for every
$j,k=1,\ldots$, we get a contradiction. Thus, ${\rm Ker\,}\sigma\cap
E_{I}(T^{(n_{-})}_{P,G})$ is a finite dimensional subspace of
$\Omega^{0,n_{-}}(X)$.

We can repeat the procedure above with minor change and deduce that

${\rm Coker\,}\sigma\cap E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

is a finite dimensional subspace of $\Omega^{0,n_{X_{G},-}}(X_{G})$. ∎

Let $H^{q}_{b}(X)^{G}={\rm Ker\,}\Box^{q}_{b}\cap L^{2}_{(0,q)}(X)^{G}$,
$H^{q}_{b}(X_{G}):={\rm Ker\,}\Box^{q}_{b,X_{G}}$ and take $\tau(x)=1$ near
$x=1$. As a corollary of Theorem 4.2, we deduce

######  Corollary 4.1.

With the same notations and assumptions above, suppose that
$\Box^{n_{X_{G},-}}_{b}$ has closed range. We have

$\sigma:H^{n_{-}}_{b}(X)^{G}\rightarrow H^{n_{X_{G},-}}(X_{G})$

is a Fredholm, that is, ${\rm Ker\,}\sigma$ and ${\rm Coker\,}\sigma$ are
finite dimensional subspaces of $\Omega^{0,n_{-}}(X_{G})$ and
$\Omega^{0,n_{X_{G},-}}(X_{G})$ respectively.

Our method works well if $n_{-}=n_{+}$ or $n_{X_{G},-}=n_{X_{G},+}$. If
$n_{-}=n_{+}$, $n_{X_{G},-}=n_{X_{G},+}$, we define $\sigma$ as (62) and (65).
If $n_{-}=n_{+}$, $n_{X_{G},-}\neq n_{X_{G},+}$, we define

$\begin{split}\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\cap\Omega^{0,n_{-}}(X)&\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\cap\Omega^{0,n_{X_{G},-}}(X_{G})\oplus
E_{\tau}(T^{(n_{X_{G},+})}_{P_{X_{G}}})\cap\Omega^{0,n_{X_{G},-}}(X_{G}),\\\
u&\rightarrow v_{1}\oplus v_{2},\end{split}$ (67)

where

$\displaystyle v_{1}:$ $\displaystyle=\tau(T^{(n_{X_{G},-})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}\circ\pi_{x,n_{-}}\circ f\circ\iota^{*}\circ u,$
$\displaystyle v_{2}:$ $\displaystyle=\tau(T^{(n_{X_{G},+})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}_{+}\circ\pi_{x,n_{+}}\circ f\circ\iota^{*}\circ
u\,$

and $\sigma$ can be extended to $E_{\tau}(T^{(n_{-})}_{P,G})$ by density and
thus,

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\oplus
E_{\tau}(T^{(n_{X_{G},+})}_{P_{X_{G}}}).$

If $n_{-}\neq n_{+}$, $n_{X_{G},-}=n_{X_{G},+}$, we define

$\begin{split}\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\cap\Omega^{0,n_{-}}(X)\oplus
E_{\tau}(T^{(n_{+})}_{P,G})\cap\Omega^{0,n_{+}}(X)&\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\cap\Omega^{0,n_{X_{G},-}}(X_{G}),\\\
u_{1}\oplus u_{2}&\mapsto v=v_{1}+v_{2}\end{split}$ (68)

where

$\displaystyle v_{1}:=\tau(T^{(n_{X_{G},-})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}\circ\pi_{x,n_{-}}\circ f\circ\iota^{*}\circ
u_{1},$ $\displaystyle v_{2}:=\tau(T^{(n_{X_{G},+})}_{P_{X_{G}}})\circ
E\circ\iota_{G}\circ\hat{p}_{+}\circ\pi_{x,n_{+}}\circ f\circ\iota^{*}\circ
u_{2}\,$

and $\sigma$ can be extended to $E_{\tau}(T^{(n_{-})}_{P,G})\oplus
E_{\tau}(T^{(n_{+})}_{P,G})$ by density and thus,

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\oplus
E_{\tau}(T^{(n_{+})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}}).$

Repeating the proof of Theorem 4.1 with minor changes, we deduce

######  Theorem 4.3.

With the same notations and assumptions above, let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
If $n_{-}=n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$. Suppose that
$\Box^{n_{X_{G},-}}_{b}$ has closed range. The map

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (62) and (65) has the following properties:

  * (i)

For every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Ker\,}\sigma\cap E_{I}(T^{(n_{-})}_{P,G})$

is a finite dimensional subspace of $\Omega^{0,n_{-}}(X)$;

  * (ii)

for every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Coker\,}\sigma\cap E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

is a finite dimensional subspace of $\Omega^{0,n_{X_{G},-}}(X_{G})$.

If $n_{-}=n_{+}$ and $n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that
$\Box^{n_{X_{G},-}}_{b}$ and $\Box^{n_{X_{G},+}}_{b}$ have closed range. The
map

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\oplus
E_{\tau}(T^{(n_{X_{G},+})}_{P_{X_{G}}})$

given by (67) has the following properties:

  * (i)

For every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Ker\,}\sigma\,\cap E_{I}(T^{(n_{-})}_{P,G})$

is a finite dimensional subspace of $\Omega^{0,n_{-}}(X)$;

  * (ii)

For every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Coker\,}\sigma\,\cap\Bigr{(}E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\oplus
E_{I}(T^{(n_{X_{G},+})}_{P_{X_{G}}}\Bigr{)}$

is a finite dimensional subspace of
$\Omega^{0,n_{X_{G},-}}(X_{G})\oplus\Omega^{0,n_{X_{G},+}}(X_{G})$.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$. Suppose that
$\Box^{n_{+}}_{b}$, $\Box^{n_{X_{G},-}}_{b}$ have closed range. Recall that we
always assume that $\Box^{n_{-}}_{b}$ has closed range. The map

$\sigma:E_{\tau}(T^{(n_{-})}_{P,G})\oplus
E_{\tau}(T^{(n_{+})}_{P,G})\rightarrow
E_{\tau}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (68) has the following properties:

  * (i)

For every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Ker\,}\sigma\,\cap\Bigr{(}E_{I}(T^{(n_{-})}_{P,G})\oplus
E_{I}(T^{(n_{+})}_{P,G})\Bigr{)}$

is a finite dimensional subspace of
$\Omega^{0,n_{-}}(X)\oplus\Omega^{0,n_{+}}(X)$.

  * (ii)

For every open bounded interval $I$, $0\notin\overline{I}$, with $\tau\equiv
1$ on $I$, we have that

${\rm Coker\,}\sigma\,\cap E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

is a finite dimensional subspace of $\Omega^{0,n_{X_{G},-}}(X_{G})$.

From Lemma 3.6, we see that if $\sigma^{0}_{P}(x,\mp\omega_{0}(x))\neq 0$ for
every $x\in X$, then $\mathrm{Spec}(T^{(q)}_{P})\subset
I\cup\left\\{0\right\\}$, for some open interval $I$ with
$0\notin\overline{I}$. From this observation and Theorem 4.3, we get

######  Theorem 4.4.

With the same notations and assumptions above, assume that
$\sigma^{0}_{P}(x,\mp\omega_{0}(x))>0$ for every $x\in X$. Thus,
$\mathrm{Spec}(T^{(n_{\mp})}_{P,G})\subset I\cup\left\\{0\right\\}$ if
$\Box^{n_{\mp}}_{b}$ has closed range,
$\mathrm{Spec}(T^{(n_{X_{G},\mp})}_{P_{X_{G}}})\subset
I\cup\left\\{0\right\\}$, if $\Box^{n_{X_{G},\mp}}_{b,X_{G}}$ has closed
range, for some open bounded interval $I$ with $0\notin\overline{I}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$
and $\tau\equiv 1$ near $I$.

If $n_{-}=n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$ or $n_{-}\neq n_{+}$ and
$n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that $\Box^{n_{X_{G},-}}_{b}$ has
closed range. The map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (62) and (65) is Fredholm.

If $n_{-}=n_{+}$ and $n_{X_{G},-}\neq n_{X_{G},+}$. Suppose that
$\Box^{n_{X_{G},-}}_{b}$ and $\Box^{n_{X_{G},+}}_{b}$ have closed range. The
map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})\oplus
E_{I}(T^{(n_{X_{G},+})}_{P_{X_{G}}})$

given by (67) is Fredholm.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$. Suppose that
$\Box^{n_{+}}_{b}$ and $\Box^{n_{X_{G},-}}_{b}$ have closed range. Recall that
we always assume that $\Box^{n_{-}}_{b}$ has closed range. The map

$\sigma:E_{I}(T^{(n_{-})}_{P,G})\oplus E_{I}(T^{(n_{+})}_{P,G})\rightarrow
E_{I}(T^{(n_{X_{G},-})}_{P_{X_{G}}})$

given by (68) is Fredholm.

##  5 CR manifolds with circle action

###  5.1 Some standard notations in semi-classical analysis

Let $W_{1}$ be an open set in $\mathbb{R}^{N_{1}}$ and let $W_{2}$ be an open
set in $\mathbb{R}^{N_{2}}$. Let $E$ and $F$ be vector bundles over $W_{1}$
and $W_{2}$, respectively. An $m$-dependent continuous operator
$A_{m}:\mathcal{C}^{\infty}_{c}(W_{2},F)\rightarrow\mathcal{D}^{\prime}(W_{1},E)$
is called $m$-negligible on $W_{1}\times W_{2}$ if, for $m$ large enough,
$A_{m}$ is smoothing and, for any $K\Subset W_{1}\times W_{2}$, any multi-
indices $\alpha$, $\beta$ and any $N\in\mathbb{N}$, there exists
$C_{K,\alpha,\beta,N}>0$ such that

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}A_{m}(x,y)\right|\leq
C_{K,\alpha,\beta,N}m^{-N}\>\>\text{on $K$},\ \ \forall m\gg 1.$

In that case we write

$A_{m}(x,y)=O(m^{-\infty})\>\>\text{on $W_{1}\times
W_{2}$,}\quad\text{or}\quad A_{m}=O(m^{-\infty})\>\>\text{on $W_{1}\times
W_{2}$.}$

If
$A_{m},B_{m}:\mathcal{C}^{\infty}_{c}(W_{2},F)\rightarrow\mathcal{D}^{\prime}(W_{1},E)$
are $m$-dependent continuous operators, we write $A_{m}=B_{m}+O(m^{-\infty})$
on $W_{1}\times W_{2}$ or $A_{m}(x,y)=B_{m}(x,y)+O(m^{-\infty})$ on
$W_{1}\times W_{2}$ if $A_{m}-B_{m}=O(m^{-\infty})$ on $W_{1}\times W_{2}$.
When $W=W_{1}=W_{2}$, we sometime write “on $W$”.

Let $X$ and $M$ be smooth manifolds and let $E$ and $F$ be vector bundles over
$X$ and $M$, respectively. Let
$A_{m},B_{m}:\mathcal{C}^{\infty}(M,F)\rightarrow\mathcal{C}^{\infty}(X,E)$ be
$m$-dependent smoothing operators. We write $A_{m}=B_{m}+O(m^{-\infty})$ on
$X\times M$ if on every local coordinate patch $D$ of $X$ and local coordinate
patch $D_{1}$ of $M$, $A_{m}=B_{m}+O(m^{-\infty})$ on $D\times D_{1}$. When
$X=M$, we sometime write on $X$.

We recall the definition of the semi-classical symbol spaces

######  Definition 5.1.

Let $W$ be an open set in $\mathbb{R}^{N}$. Let

$S^{0}_{{\rm
loc\,}}(1;W):=\Big{\\{}(a(\cdot,m))_{m\in\mathbb{R}}\,;\,\forall\alpha\in\mathbb{N}^{N}_{0},\forall\chi\in\mathcal{C}^{\infty}_{c}(W)\,:\>\sup_{m\in\mathbb{R},m\geq
1}\sup_{x\in W}\left|\partial^{\alpha}(\chi a(x,m))\right|<\infty\Big{\\}}\,.$

For $k\in\mathbb{R}$, let

$S^{k}_{{\rm loc}}(1):=S^{k}_{{\rm
loc}}(1;W)=\Big{\\{}(a(\cdot,m))_{m\in\mathbb{R}}\,;\,(m^{-k}a(\cdot,m))\in
S^{0}_{{\rm loc\,}}(1;W)\Big{\\}}\,.$

Hence $a(\cdot,m)\in S^{k}_{{\rm loc}}(1;W)$ if for every
$\alpha\in\mathbb{N}^{N}_{0}$ and $\chi\in\mathcal{C}^{\infty}_{c}(W)$, there
exists $C_{\alpha}>0$ independent of $m$, such that
$\left|\partial^{\alpha}(\chi a(\cdot,m))\right|\leq C_{\alpha}m^{k}$ holds on
$W$.

Consider a sequence $a_{j}\in S^{k_{j}}_{{\rm loc\,}}(1)$,
$j\in\mathbb{N}_{0}$, where $k_{j}\searrow-\infty$, and let $a\in
S^{k_{0}}_{{\rm loc\,}}(1)$. We say

$a(\cdot,m)\sim\sum\limits^{\infty}_{j=0}a_{j}(\cdot,m)\>\>\text{in
$S^{k_{0}}_{{\rm loc\,}}(1)$},$

if, for every $\ell\in\mathbb{N}_{0}$, we have $a-\sum^{\ell}_{j=0}a_{j}\in
S^{k_{\ell+1}}_{{\rm loc\,}}(1)$ . For a given sequence $a_{j}$ as above, we
can always find such an asymptotic sum $a$, which is unique up to an element
in $S^{-\infty}_{{\rm loc\,}}(1)=S^{-\infty}_{{\rm
loc\,}}(1;W):=\cap_{k}S^{k}_{{\rm loc\,}}(1)$.

Similarly, we can define $S^{k}_{{\rm loc\,}}(1;Y,E)$ in the standard way,
where $Y$ is a smooth manifold and $E$ is a vector bundle over $Y$.

###  5.2 Functional calculus for Toeplitz operators on CR manifolds with
circle action

We now assume that $X$ admits a CR and transversal locally free circle action
$e^{i\theta}$. Let $T$ be the vector field on $X$ given by

$(Tu)(x):=\frac{\partial}{\partial\theta}u(e^{i\theta}\circ x)|_{\theta=0},\ \
\forall u\in\mathcal{C}^{\infty}(X).$

We take $\omega_{0}$ so that the associated Reeb vector field $R$ is equal to
$T$. Assume that the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$TX\otimes\mathbb{C}$ is $S^{1}$-invariant. For every $m\in Z$, set

$\Omega^{0,q}_{m}(X):=\left\\{u\in\Omega^{0,q}(X);\,(e^{i\theta})^{*}u=e^{im\theta}u,\
\ \forall e^{i\theta}\in S^{1}\right\\}.$

Let $L^{2}_{(0,q),m}(X)$ be the completion of $\Omega^{0,q}_{m}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. Put

$({\rm Ker\,}\Box^{q}_{b})_{m}:=({\rm Ker\,}\Box^{q}_{b})\cap
L^{2}_{(0,q),m}(X).$

The $m$-th Szegő projection is the orthogonal projection
$S^{(q)}_{m}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{(q)}_{b})_{m}$ with
respect to $(\,\cdot\,|\,\cdot\,)$. Let $P\in L^{0}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{S^{1}}$ be a self-adjoint
$S^{1}$-invariant classical pseudodifferential operator on $X$ with scalar
principal symbol. Put

$T^{(q)}_{P,m}:=S^{(q)}_{m}\circ P\circ
S^{(q)}_{m}:L^{2}_{(0,q)}(X)\rightarrow L^{2}_{(0,q),m}(X).$

Let $\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$. It is straightforward to
see that

$\tau(T^{(q)}_{P,m})(x,y)=\frac{1}{2\pi}\int^{2\pi}_{0}\tau(T^{(q)}_{P})(x,e^{i\theta}y)e^{im\theta}\mathrm{d}\theta.$
(69)

For $p\in X$, let

$N_{p}:=\left\\{g\in S^{1};\,g\circ p=p\right\\}.$

Let

$X_{{\rm reg\,}}=\left\\{x\in X;\,e^{i\theta}x\neq
x,\forall\theta\in]0,2\pi[\right\\}.$

From Theorem 3.6 and Theorem 3.7 and (69), we can repeat the proof of Theorem
4.4 in [GH] and get

######  Theorem 5.1.

With the notations and assumptions above, recall that we let $q=n_{-}$. We do
not assume that $n_{-}\neq n_{+}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
Let $p\in X$ and assume that

$N_{p}=\left\\{e^{\frac{\ell 2\pi}{k}i};\,\ell=0,1,\ldots,k-1\right\\}.$

Let $D$ be an open local coordinate patch of $p$ with local coordinates
$x=(x_{1},\ldots,x_{2n+1})$. Then, as $m\gg 1$,

$\begin{split}&\tau(T^{(q)}_{P,m})(x,y)=\sum^{k-1}_{\ell=0}e^{\frac{im2\pi\ell}{k}}e^{im\Psi(x,e^{\frac{2\pi\ell}{k}i}y)}a_{P}(x,e^{\frac{2\pi\ell}{k}i}y,m)+O(m^{-\infty})\
\ \mbox{on $D\times D$},\\\ &\Psi\in\mathcal{C}^{\infty}(D\times D),\ \
\Psi(x,x)=0,\ \ d_{x}\Psi(x,x)=-d_{y}\Psi(x,x)=-\omega_{0}(x),\ \ \forall x\in
D,\\\ &{\rm Im\,}\Psi(x,y)\geq c\inf_{e^{i\theta}\in
S^{1}}d(e^{i\theta}x,y)^{2},\ \ \forall(x,y)\in D\times D,\ \ \mbox{$c>0$ is a
constant},\\\ &a_{P}(x,y,m)\in S^{n}_{{\rm loc\,}}(1;D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&\mbox{$a_{P}(x,y,m)\sim\sum^{\infty}_{j=0}m^{n-j}a_{P,j}(x,y)$ in
$S^{n}_{{\rm loc\,}}(1;D\times D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\
&a_{P,j}(x,y)\in\mathcal{C}^{\infty}(D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j=0,1,2,\ldots,\end{split}$ (70)

and

$\mbox{$a_{P,0}(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}}$,
for every $x\in D$}.$ (71)

From Theorem 5.1, we deduce

######  Corollary 5.1.

With the notations and assumptions above, recall that we let $q=n_{-}$. We do
not assume that $n_{-}\neq n_{+}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$.
Then, there is a constant $C>0$ such that for $m\gg 1$,

$\left|\tau(T^{(q)}_{P,m})(x,y)\right|\leq Cm^{n},\ \ \mbox{for all $(x,y)\in
X\times X$}.$ (72)

Assume that $X$ is connected and $X_{{\rm reg\,}}\neq\emptyset$. Then,
$X_{{\rm reg\,}}$ is dense in $X$. We have

$\lim_{m\rightarrow+\infty}m^{-n}\tau(T^{(q)}_{P,m})(x,x)=\frac{1}{2}\pi^{-n-1}\left|{\rm
det\,}\mathcal{L}_{x}\right|\tau(\sigma^{0}_{P}(x,-\omega_{0}(x)))\pi_{x,n_{-}},\
\ \mbox{for every $x\in X$}.$ (73)

###  5.3 Semi-classical spectral dimensions law for Toeplitz operators on CR
manifolds with circle action

Let $I\subset\mathbb{R}$ be an open bounded interval with
$0\notin\overline{I}$. We are going to study the asymptotic limit of
$\lim_{m\rightarrow+\infty}m^{-n}{\rm dim\,}E_{I}(T^{(q)}_{P,m})$. For every
$x\in X$, let $f_{1}(x),\ldots,f_{r}(x)\in T^{*0,q}_{x}X$ be an orthonormal
basis for $T^{*0,q}_{x}X$, $r={\rm dim\,}T^{*0,q}_{x}X$. Let
$A:T^{*0,q}_{y}X\rightarrow T^{*0,q}_{x}X$ be a linear transformation, $x,y\in
X$. We write

$A=\sum^{r}_{i,j=1}c_{i,j}f_{i}(x)\otimes f_{k}(y)\,,\quad
c_{i,j}\in\mathbb{R}\,,\quad\forall i,j,$ (74)

where $f_{i}(x)\otimes f_{k}(y):T^{*0,q}_{y}X\rightarrow T^{*0,q}_{x}X$ is the
linear transformation given by

$(f_{i}(x)\otimes f_{k}(y))(f_{s}(y))=\delta_{k,s}f_{i}(x)\,,$

for every $i,k,s=1,\ldots,r$. When $x=y$, we put

${\rm Tr\,}A:=\sum^{r}_{j=1}\langle\,Af_{j}(x)\,|\,f_{j}(x)\,\rangle.$

We need

######  Lemma 5.1.

With the notations and assumptions above, let $I\subset\mathbb{R}$ be an open
bounded interval with $0\notin\overline{I}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$ with $0\notin{\rm supp\,}\tau$
and $\tau\equiv 1$ on some neighborhood of $I$. We have

${\rm Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x))\leq{\rm
Tr\,}(\tau(T^{(q)}_{P,m})^{2}(x,x)),\ \ \mbox{for every $x\in X$}.$ (75)

###### Proof.

Since $\tau\equiv 1$ on some neighborhood of $I$, we have

$\Pi_{I}(T^{(q)}_{P,m})=\tau(T^{(q)}_{P,m})\Pi_{I}(T^{(q)}_{P,m}).$ (76)

From (76), we have

$\Pi_{I}(T^{(q)}_{P,m})(x,x)=\int_{X}\tau(T^{(q)}_{P,m})(x,y)\Pi_{I}(T^{(q)}_{P,m})(y,x)\mathrm{dV}_{X}(y).$
(77)

For every $x\in X$, let $f_{1}(x),\ldots,f_{r}(x)\in T^{*0,q}_{x}X$ be an
orthonormal basis for $T^{*0,q}_{x}X$, $r={\rm dim\,}T^{*0,q}_{x}X$. As (74),
we write

$\begin{split}\Pi_{I}(T^{(q)}_{P,m})(x,y)=\sum^{r}_{i,j=1}a_{i,j}(x,y)f_{i}(x)\otimes
f_{k}(y),\ \ a_{i,j}(x,y)\in\mathbb{R},\ \ \forall i,j,\\\
\tau(T^{(q)}_{P,m})(x,y)=\sum^{r}_{i,j=1}b_{i,j}(x,y)f_{i}(x)\otimes
f_{k}(y),\ \ b_{i,j}(x,y)\in\mathbb{R},\ \ \forall i,j.\end{split}$ (78)

From (77), we have

$\begin{split}&{\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x))=\sum^{r}_{i=1}a_{i,i}(x,x)=\sum^{r}_{i,j=1}\int
b_{i,j}(x,y)a_{j,i}(y,x)\mathrm{dV}_{X}(y)\\\
&\leq\sum^{r}_{i,j=1}\sqrt{\int\left|b_{i,j}(x,y)\right|^{2}\mathrm{dV}_{X}(y)\int\left|a_{j,i}(y,x)\right|^{2}\mathrm{dV}_{X}(y)}\\\
&\leq\sqrt{\sum^{r}_{i,j=1}\int\left|b_{i,j}(x,y)\right|^{2}\mathrm{dV}_{X}(y)}\sqrt{\int\left|a_{j,i}(y,x)\right|^{2}\mathrm{dV}_{X}(y)}.\end{split}$
(79)

Write

$\tau(T^{(q)}_{P,m})^{2}(x,y)=\sum^{r}_{i,j=1}d_{i,j}(x,y)f_{i}(x)\otimes
f_{k}(y),\ \ d_{i,j}(x,y)\in\mathbb{R},\ \ \forall i,j.$ (80)

It is not difficult to see that

$\begin{split}&\sum^{r}_{i,j=1}\int\left|b_{i,j}(x,y)\right|^{2}\mathrm{dV}_{X}(y)=\sum^{r}_{i=1}d_{i,i}(x,x)={\rm
Tr\,}(\tau(T^{(q)}_{P,m})^{2}(x,x)),\\\
&\sum^{r}_{i,j=1}\int\left|a_{i,j}(x,y)\right|^{2}\mathrm{dV}_{X}(y)=\sum^{r}_{i=1}a_{i,i}(x)={\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x)).\end{split}$ (81)

From (79) and (81), we get (75). ∎

We can repeat the proof of Lemma 5.1 with minor change and deduce

######  Lemma 5.2.

With the notations and assumptions above, let $I\subset\mathbb{R}$ be an open
bounded interval with $0\notin\overline{I}$. Let
$\tau\in\mathcal{C}^{\infty}_{c}(I)$ with $0\notin{\rm supp\,}\tau$. We have

${\rm Tr\,}(\tau(T^{(q)}_{P,m})(x,x))\leq\sqrt{{\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x))}\sqrt{{\rm
Tr\,}(\tau(T^{(q)}_{P,m})^{2}(x,x))},\ \ \mbox{for every $x\in X$}.$ (82)

Hence,

$2{\rm Tr\,}(\tau(T^{(q)}_{P,m})(x,x))-{\rm
Tr\,}(\tau(T^{(q)}_{P,m})^{2}(x,x))\leq{\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x)),\ \ \mbox{for every $x\in X$}.$ (83)

We can now prove the following semi-classical spectral dimensions for Toeplitz
operators

######  Theorem 5.2.

With the notations and assumptions above, let $I\subset\mathbb{R}$ be an open
bounded interval with $0\notin\overline{I}$. Suppose that $X$ is connected and
$X_{{\rm reg\,}}\neq\emptyset$. We have

$\lim_{m\rightarrow+\infty}m^{-n}{\rm
dim\,}E_{I}(T^{(q)}_{P,m})=\frac{1}{2}\pi^{-n-1}\int_{\left\\{x\in
X;\,\sigma^{0}_{P}(x,-\omega_{0}(x))\in I\right\\}}\left|{\rm
det\,}\mathcal{L}_{x}\right|\mathrm{dV}_{X}(x).$ (84)

###### Proof.

Let
$\left\\{\tau_{j}\right\\}^{+\infty}_{j=1}\subset\mathcal{C}^{\infty}_{c}(\mathbb{R})$
with $0\notin{\rm supp\,}\tau_{j}$, $\tau_{j}\equiv 1$ on some neighborhood of
$I$, for every $j$ and $\lim_{j\rightarrow+\infty}\tau_{j}=1_{I}$ pointwise,
where $1_{I}(x)=1$ if $x\in I$, $1_{I}(x)=0$ if $x\notin I$. Let
$\left\\{\hat{\tau}_{j}\right\\}^{+\infty}_{j=1}\subset\mathcal{C}^{\infty}_{c}(I)$
with $0\notin{\rm supp\,}\hat{\tau}_{j}$, for every $j$ and
$\lim_{j\rightarrow+\infty}\hat{\tau}_{j}=1_{I}$ pointwise. From (75) and
(83), we have for every $j=1,2,\ldots$,

$\begin{split}&m^{-n}\int_{X}\Bigr{(}2{\rm
Tr\,}(\hat{\tau}_{j}(T^{(q)}_{P,m})(x,x))-{\rm
Tr\,}(\hat{\tau}_{j}(T^{(q)}_{P,m})^{2}(x,x))\Bigr{)}\mathrm{dV}_{X}(x)\\\
&\leq m^{-n}\int_{X}{\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x))\mathrm{dV}_{X}(x)\\\ &\leq
m^{-n}\int_{X}{\rm
Tr\,}(\tau_{j}(T^{(q)}_{P,m})^{2}(x,x))\mathrm{dV}_{X}(x).\end{split}$ (85)

Note that

${\rm dim\,}E_{I}(T^{(q)}_{P,m})=\int_{X}{\rm
Tr\,}(\Pi_{I}(T^{(q)}_{P,m})(x,x))\mathrm{dV}_{X}(x).$

From this observation, (73), (85) and take $j\rightarrow+\infty$ in (85), the
theorem follows. ∎

###  5.4 Quantization commutes with reduction on CR manifolds with circle
action

Now, we assume that $X$ admits a compact Lie group action $G$ of dimension
$d$. We assume that Assumption 3.1 and Assumption 3.2 hold. We assume further
that

######  Assumption 5.1.

The Reeb vector field $R$ is transversal to the space
$\underline{\mathfrak{g}}$ at every point $p\in\mu^{-1}(0)$. Furthermore, we
assume that the circle action commutes with the one of $G$ on $X$:

$e^{i\theta}\circ g\circ x=g\circ e^{i\theta}\circ x,\ \ \mbox{for every $x\in
X$, $\theta\in[0,2\pi[$, $g\in G$}\,,$ (86)

Eventually, we assume

$\mbox{$G\times S^{1}$ acts locally free near $\mu^{-1}(0)$}\,.$

Fix a $S^{1}\times G$-invariant smooth Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$ so that
$T^{1,0}X$ is orthogonal to $T^{0,1}X$, $\underline{\mathfrak{g}}$ is
orthogonal to $HY\cap JHY$ at every point of $Y$, $\langle\,u\,|\,v\,\rangle$
is real if $u,v$ are real tangent vectors and $\langle\,R\,|\,R\,\rangle=1$.
For every $m\in\mathbb{Z}$, we set

$\Omega^{0,q}_{m}(X)^{G}:=\left\\{u\in\Omega^{0,q}(X)^{G};\,(e^{i\theta})^{*}u=e^{im\theta}u,\
\ \forall e^{i\theta}\in S^{1}\right\\}.$

Let $L^{2}_{(0,q),m}(X)^{G}$ be the completion of $\Omega^{0,q}_{m}(X)^{G}$
with respect to $(\,\cdot\,|\,\cdot\,)$. Put $({\rm
Ker\,}\Box^{q}_{b})^{G}_{m}:=({\rm Ker\,}\Box^{q}_{b})^{G}\cap
L^{2}_{(0,q),m}(X).$ Let

$H^{q}_{b,m}(X)^{G}:=({\rm Ker\,}\Box^{q}_{b})^{G}_{m}.$

From Corollary 1.1, we get

######  Theorem 5.3.

With the same notations and assumptions above, if $n_{-}=n_{+}$,
$n_{X_{G},-}=n_{X_{G},+}$ or $n_{-}\neq n_{+}$, $n_{X_{G},-}\neq n_{X_{G},+}$,
then

$H^{n_{-}}_{b,m}(X)^{G}\cong H^{n_{X_{G},-}}_{b,m}(X_{G})$

for $\left|m\right|\gg 1$.

If $n_{-}=n_{+}$ and $n_{X_{G},-}\neq n_{X_{G},+}$, then

$H^{n_{-}}_{b,m}(X)^{G}\cong H^{n_{X_{G},-}}_{b,m}(X_{G})\oplus
H^{n_{X_{G},+}}_{b,m}(X_{G})$

for $\left|m\right|\gg 1$.

If $n_{-}\neq n_{+}$ and $n_{X_{G},-}=n_{X_{G},+}$, then

$H^{n_{-}}_{b,m}(X)^{G}\oplus H^{n_{+}}_{b,m}(X)^{G}\cong
H^{n_{X_{G},-}}_{b,m}(X_{G})$

for $\left|m\right|\gg 1$.

###  5.5 Semi-classical spectral dimensions for invariant Toeplitz operators
on CR manifolds with circle action

The $m$-th $G$-invariant Szegő projection is the orthogonal projection

$S^{(q)}_{G,m}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{(q)}_{b})^{G}_{m}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let $P\in L^{0}_{{\rm
cl\,}}(X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})^{G\times S^{1}}$ be a self-adjoint
$G\times S^{1}$-invariant pseudodifferential operator on $X$ with scalar
principal symbol. Put

$T^{(q)}_{P,G,m}:=S^{(q)}_{G,m}\circ P\circ
S^{(q)}_{G,m}:L^{2}_{(0,q)}(X)\rightarrow L^{2}_{(0,q),m}(X)^{G}.$

We can now prove the following semi-classical spectral dimensions for
$G$-invariant Toeplitz operators

######  Theorem 5.4.

With the notations and assumptions above, let $I\subset\mathbb{R}$ be an open
bounded interval with $0\notin\overline{I}$. Suppose that $X$ is connected and
$X_{{\rm reg\,}}\neq\emptyset$.. We have

$\lim_{m\rightarrow+\infty}m^{-n+d}{\rm
dim\,}E_{I}(T^{(q)}_{P,G,m})=\frac{1}{2}\pi^{-n+d-1}\int_{\left\\{x\in
X_{G};\,\sigma^{0}_{P_{X_{G}}}(x,-\omega_{0}(x))\in I\right\\}}\left|{\rm
det\,}\mathcal{L}_{X_{G},x}\right|\mathrm{dV}_{X_{G}}(x).$ (87)

###### Proof.

Let $I_{1}$ be an open bounded interval with $0\notin\overline{I_{1}}$,
$I\subset I_{1}$, $I\neq I_{1}$. Let $\tau\in\mathcal{C}^{\infty}_{c}(I_{1})$
with $0\notin{\rm supp\,}\tau$ and $\tau\equiv 1$ on some neighborhood of $I$.
From Theorem 4.4 and notice that ${\rm
dim\,}E_{\tau}(T^{(n_{+})}_{P,G,m})={\rm
dim\,}E_{\tau}(T^{(n_{X_{G},+})}_{P,G,m})=0$ if $m\gg 1$, we conclude that if
$m\gg 1$, we have

${\rm dim\,}E_{I}(T^{(q)}_{P,G,m})\leq{\rm
dim\,}E_{\tau}(T^{(q)}_{P,G,m})\leq{\rm
dim\,}E_{I_{1}}(T^{(n_{X_{G},-})}_{P,G,m})\,.$ (88)

From (84) and (88), we get

$\begin{split}&\lim_{m\rightarrow+\infty}m^{-n+d}{\rm
dim\,}E_{I}(T^{(q)}_{P,G,m})\\\ &\leq\lim_{m\rightarrow+\infty}m^{-n+d}{\rm
dim\,}E_{I_{1}}(T^{(n_{X_{G},-})}_{P,G,m})=\frac{1}{2}\pi^{-n+d-1}\int_{\widetilde{X_{G}}}\left|{\rm
det\,}\mathcal{L}_{X_{G},x}\right|\mathrm{dV}_{X_{G}}(x)\end{split}$ (89)

where

$\widetilde{X_{G}}:=\left\\{x\in
X_{G};\,\sigma^{0}_{P_{X_{G}}}(x,-\omega_{0}(x))\in I_{1}\right\\}\,.$

Let $\hat{I}_{1}$ be an open interval with $0\notin\overline{\hat{I}_{1}}$,
$\hat{I}_{1}\subset I$, $\hat{I}_{1}\neq I$. Let
$\hat{\tau}\in\mathcal{C}^{\infty}_{c}(I)$ with $0\notin{\rm
supp\,}\hat{\tau}$ and $\hat{\tau}\equiv 1$ on some neighborhood of
$\hat{I}_{1}$. From Theorem 4.4 again, if $m\gg 1$, we have

${\rm dim\,}E_{I}(T^{(q)}_{P,G,m})\geq{\rm
dim\,}E_{\hat{\tau}}(T^{(q)}_{P,G,m})\geq{\rm
dim\,}E_{\hat{I}_{1}}(T^{(n_{X_{G},-})}_{P,G,m}).$ (90)

From (84) and (90), we get

$\begin{split}&\lim_{m\rightarrow+\infty}m^{-n+d}{\rm
dim\,}E_{I}(T^{(q)}_{P,G,m})\\\ &\geq\lim_{m\rightarrow+\infty}m^{-n+d}{\rm
dim\,}E_{\hat{I}_{1}}(T^{(n_{X_{G},-})}_{P,G,m})=\frac{1}{2}\pi^{-n+d-1}\int_{\widehat{X_{G}}}\left|{\rm
det\,}\mathcal{L}_{X_{G},x}\right|\mathrm{dV}_{X_{G}}(x)\end{split}$ (91)

where

$\widehat{X_{G}}:=\left\\{x\in
X_{G};\,\sigma^{0}_{P_{X_{G}}}(x,-\omega_{0}(x))\in\hat{I}_{1}\right\\}$

From (89) and (91), we get (87). ∎

Acknowledgements: This project was started during the first author’s
postdoctoral fellowship at the National Center for Theoretical Sciences in
Taiwan; we thank the Center for the support. Chin-Yu Hsiao was partially
supported by Taiwan Ministry of Science and Technology projects
108-2115-M-001-012-MY5, 109-2923-M-001-010-MY4.

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