#  Weighted periodic and discrete Pseudo-Differential Operators

Aparajita Dasgupta  Aparajita Dasgupta: Department of Mathematics Indian
Institute of Technology, Delhi, Hauz Khas New Delhi-110016 India E-mail
address<EMAIL_ADDRESS>,  Lalit Mohan  Lalit Mohan: Department of
Mathematics Indian Institute of Technology, Delhi India E-mail address
<EMAIL_ADDRESS>and  Shyam Swarup Mondal  Shyam Swarup Mondal:
Department of Mathematics Indian Institute of Technology, Delhi India E-mail
address<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study elements of symbolic calculus for pseudo-differential
operators associated with the weighted symbol class
$M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$ (associated to a suitable
weight function $\Lambda$ on $\mathbb{Z}$) by deriving formulae for the
asymptotic sums, composition, adjoint, transpose. We also construct the
parametrix of $M$-elliptic pseudo-differential operators on $\mathbb{T}$.
Further, we prove a version of Gohberg’s lemma for pseudo-differetial
operators with weighted symbol class
$M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ and as an application, we
provide a sufficient and necessary condition to ensure that the corresponding
pseudo-differential operator is compact on $L^{2}(\mathbb{T})$. Finally, we
provide Gårding’s and Sharp Gårding’s inequality for $M$-elliptic operators on
$\mathbb{Z}$ and $\mathbb{T}$, respectively, and present an application in the
context of strong solution of the pseudo-differential equation $T_{\sigma}u=f$
in $L^{2}\left(\mathbb{T}\right)$.

###### Key words and phrases:

$M$-ellipticity, Symbolic Calculus, Compact Operators, Gohberg’s Lemma,
Gårding’s Inequality, Strong Solution

###### 1991 Mathematics Subject Classification:

Primary 35S05, 47G30; Secondary 43A85, 35A17

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Symbolic calculus and parametrix for $M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$
  4. 4 Compact $M$-elliptic pseudo-differential operators
  5. 5 Gårding’s and sharp Gårding’s inequalities on $\mathbb{T}$ and $\mathbb{Z}$
  6. 6 Applications

##  1\. Introduction

The theory of pseudo-differential operators plays an important role in modern
mathematics due to the fact that it has drawn a significant motivation from
partial differential equations, signal processing, and time-frequency
analysis, see [20, 28, 13]. Pseudo-differential operators acting on functions
defined on smooth manifolds are an essential generalization of differential
operators. The study of pseudo-differential operators originated in 1960s with
the works of Kohn and Nirenberg [22] and Hörmander [20] in the study of
singular integral differential operators, mainly for inverting differential
operators to solve elliptic differential equations. Ever since the theory is a
key tool, mostly for its connections with mathematical physics and in many
areas of harmonic analysis, quantum field theory, and the index theory.

In general, using the Mikjlin-Hörmander theorem for Fourier multipliers,
pseudo-differential operators associated to the class $S_{1,0}^{0}$ are
$L^{p}$-bounded. It is also well known that the $L^{2}$-boundedness property
holds for pseudo-differential operators associated with the symbols,
$S_{\rho,\delta}^{0}$ with $0\leq\delta<\rho\leq 1$ (see [2, 20]). However,
the situation becomes completely different for the case $p\neq 2$. Fefferman
[12] proved that the pseudo-differential operators whose symbols belong to the
class $S_{\rho,0}^{0}$ with $0<\rho<1$ are not in general $L^{p}$-bounded for
$p\neq 2$. To avoid the difficulty described above, Taylor in [34] introduced
a suitable symbol subclass $M_{\rho,0}^{m}$, of $S_{\rho,0}^{0}$ and developed
symbolic calculus for the associated pseudo-differential operators. Further,
Garello and Morando [15, 16] introduced subclass $M_{\rho,\Lambda}^{m}$ of
$S_{\rho,\Lambda}^{m}$, which are just a weighted version of the symbol class
introduced by Taylor and developed the symbolic calculus for the associated
pseudo-differential operators with many applications to study the regularity
of multi-quasi-elliptic operators.

Considerable attention has been devoted in the past sixteen years to study
various properties of pseudo-differential operators associated with the symbol
class $M_{\rho,\Lambda}^{m}$ on $\mathbb{R}^{n}$ in various directions by
several researchers. For instance, the symbolic calculus, parametrix, and
$L^{p},1<p<\infty,$ boundedness of pseudo-differential operators with symbol
in $M_{\rho,\Lambda}^{m},m>0$ has been studied by Wong [36]. Further, Kalleji
[21] constructed a weighted symbol class
$M_{\rho,\Lambda}^{m}(\mathbb{T}^{n}\times\mathbb{Z}^{n}),m\in\mathbb{R}$
associated to a suitable weight function $\Lambda$ on $\mathbb{Z}^{n}$ and
study minimal and maximal extensions, among some other results for pseudo-
differential operators associated with symbol in
$M_{\rho,\Lambda}^{m}(\mathbb{Z}^{n}\times\mathbb{T}^{n}).$ We also note that
recently, the authors in [23] constructed and studied $M$-elliptic pseudo-
differential operators on $\mathbb{Z}^{n}$ with symbol in
$M_{\rho,\Lambda}^{m}(\mathbb{Z}^{n}\times\mathbb{T}^{n})$ which is just a
weighted version of the Hörmander symbol class
$S_{\rho}^{m}\left(\mathbb{Z}^{n}\times\mathbb{T}^{n}\right)$ on
$\mathbb{Z}^{n}$, introduced by Botchway, Kabiti and Ruzhansky [3]. More
details about discrete pseudo-differential operators can be found in [4, 5, 8,
9]. This paper investigates $M$-elliptic pseudo-differential operators on
$\mathbb{T}$. In particular, we investigate elements of symbolic calculus for
pseudo-differential operators associated with the class
$M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$ by deriving formulae for
the asymptotic sums of symbols, composition, adjoint, transpose. We also
construct the parametrix of $M$-elliptic pseudo-differential operators on
$\mathbb{T}$. e can extend this result for $\mathbb{T}^{n}$.

Another important result in analysis is the Gohberg lemma due to its
application in spectral theory and the singular integral equations. Gohberg
lemma was first obtained by Gohberg [17] to investigate integral operators. In
1970, the Gohberg lemma for pseudo-differential operators with bounded symbols
was obtained by Grušhin [18]. Later, for symbols in the Hörmander class
$S^{0}_{1,0}(\mathbb{S}^{1}\times\mathbb{Z})$, an analogue of Gohberg’s lemma
has been proved in [24] to prove the spectral invariance. Further, this theory
has been extended to compact Lie group by the first author and Ruzhansky in
[11]. Recently, for symbols in the Hörmander class
$S^{0}_{1,0}(\Omega\times\mathcal{I})$, Ruzhansky and Publo [27] investigated
a “non-harmonic version” of Gohberg’s lemma, and provided a sufficient and
necessary condition to ensure that the corresponding pseudo-differential
operator is a compact operator in $L^{2}(\Omega)$. In this manuscript, we also
establish a version of the Gohberg lemma but for the weighted symbol class
$M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$. As an application of
Gohberg’s lemma, we also provide a sufficient and necessary condition to
ensure that the corresponding pseudo-differential operator is compact on
$L^{2}(\mathbb{T})$. Particularly, we evaluate the norm of $T_{\sigma}-K$,
where $K$ is a compact operator and $\sigma\in
M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$, and give estimates for the
essential spectrum of such operator $T_{\sigma}$. Using the relation between
the lattice quantization and the toroidal quantization developed in [3], we
prove Gohberg’s lemma for the weighted symbol class
$M_{\rho,\Lambda}^{0}(\mathbb{Z}\times\mathbb{T})$ on $\mathbb{Z}.$

Gårding’s type inequality plays a crucial role in the study of several
problems related to initial value problem of parabolic type. Investigation of
Gårding’s inequality for strongly elliptic operators was first proved by
Gårding [14] to derive the existence of solutions of the Dirichlet problem for
elliptic operators as well as to study the distribution of the eigenvalues.
After that, considerable attention has been devoted by several researchers to
studying Gårding’s inequality for pseudo-differential operators associated
with the Hörmander symbols with applications to PDE in different contexts. For
example, Gårding’s inequality for pseudo-differential operators associated
with the Hörmander symbols on $\mathbb{R}^{n}$ with $0\leq\delta\leq\rho\leq
1$, on compact Lie group with matrix-valued symbols, and in context of non-
harmonic analysis on general smooth manifolds can be found in [34, 31, 10, 6].
Here we would like to note that, recently, the first and second authors proved
the Gårding’s inequality for SG $M$-elliptic operators to obtain results about
the existence and uniqueness of solutions of the parabolic type IVP. On the
other hand, in order to deal with non-elliptic problems, Hörmander proved in
[19] sharp Gårding’s inequality for operators with symbols having nonnegative
real part. The sharp Gårding’s inequality and its generalizations become an
essential tool to investigate the existence of solutions to a wide class of
boundary value problems and to analyze the global solvability and the local
well-posedness of the Cauchy problem for evolution equations. The sharp
Gårding’s inequality on $\mathbb{R}^{n}$ is one of the most important tools of
the microlocal analysis with numerous applications in the theory of PDE [19].
Notably, the sharp Gårding’s inequality requires the condition imposed on the
full symbol. Further, the sharp Gårding’s inequality for the Kohn-Nirenberg
classes $S^{m}_{1,0}(G)$ proved in [29]. Recently, the authors in [7] extended
these inequality for the Hörmander classes $S^{m}_{\rho,\delta}(G)$ for all
$0\leq\delta<\rho\leq 1$. In this paper, we prove the Gårding’s and sharp
Gårding’s inequality for $M$-elliptic pseudo-differential operator with symbol
in $M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ on $\mathbb{T}$ and
$M_{\rho,\Lambda}^{0}(\mathbb{Z}\times\mathbb{T})$ on $\mathbb{Z}$,
respectively. We also present an application of Gårding’s inequality in the
context of strong solution of the pseudo-differential equation $T_{\sigma}u=f$
in $L^{2}\left(\mathbb{T}\right)$.

The presentation of this manuscript is divided into six sections, including
the introduction as follows:

  * •

In Section 2 we first recall some of the basics of Fourier analysis and
important properties of periodic pseudo-differential operators on
$\mathbb{T}$. We also recall the weighted symbol class
$M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z}),m\in\mathbb{R}$ associated
to a suitable weight function $\Lambda$ on $\mathbb{Z}$ from [21].

  * •

In Section 3 we study elements of symbolic calculus for pseudo-differential
operators associated with symbol in the weighted class
$M_{\rho,\Lambda}^{m}(\mathbb{Z}\times\mathbb{T})$ by deriving formulae for
the asymptotic sums, composition, adjoint, transpose. By recalling the
definition of $M$-ellipticity for symbols we construct the parametrix of
$M$-elliptic pseudo-differential operators.

  * •

In Section 4 we study compact $M$-elliptic pseudo-differential operators on
$\mathbb{T}$. We prove a version of Gohberg’s lemma for pseudo-differential
operators on $\mathbb{T}$ and $\mathbb{Z}$ with symbol in the weighted symbol
class $M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ and
$M_{\rho,\Lambda}^{0}(\mathbb{Z}\times\mathbb{T})$, respectively. Further, we
provide a sufficient and necessary condition to ensure the compactness of a
pseudo-differential operator on $L^{2}(\mathbb{T})$ (also on
$\ell^{2}(\mathbb{Z})$) with symbol in
$M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ (respectively in
$M_{\rho,\Lambda}^{0}(\mathbb{Z}\times\mathbb{T})$).

  * •

In Section 5 we prove Gårding’s and Sharp Gårding’s inequality for
$M$-elliptic operators on $\mathbb{Z}$ and $\mathbb{T}$, respectively.

  * •

In Section 6 we discuss an application of Gårding’s inequality in the context
of strong solution of the pseudo-differential equation $T_{\sigma}u=f$ in
$L^{2}\left(\mathbb{T}\right)$.

Note that, these results can be extended easily from $\mathbb{T}$ to the
$n$-dimensional torus $\mathbb{T}^{n}$ given by
$\mathbb{T}^{n}=\underbrace{\mathbb{T}\times\cdots\times\mathbb{T}}_{n\text{
times }},$ and similarly from $\mathbb{Z}$ to $\mathbb{Z}^{n}.$

##  2\. Preliminaries

In this section, we first recall some notation and basic properties of
periodic Fourier analysis and pseudo-differential operators on $\mathbb{T}$.
We also recall the toroidal symbol class
$S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ in the view of
Ruzhansky-Turunen theory [28] as well as the weighted symbol class
$M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$ from [21]. We refer [28,
26, 24, 21, 25] for more details and the study of various operator theoretical
properties of pseudo-differential operators on $\mathbb{T}.$

The Fourier transform $\hat{f}$ of a function $f\in
L^{1}\left(\mathbb{T}\right)$ is defined by

$\widehat{f}(k)=\int_{x\in\mathbb{T}}e^{2\pi ik\cdot x}f(x)\;dx,\quad
k\in\mathbb{Z},$

where $dx$ is the normalized Haar measure on $\mathbb{T}$. The above periodic
Fourier transform can be extended to $L^{2}\left(\mathbb{T}\right)$ using the
standard density arguments. We normalize the Haar measures on $\mathbb{T}$ in
such a manner so that the following Plancherel formula holds:

$\int_{\mathbb{T}}|\widehat{f}(x)|^{2}dx=\sum_{k\in\mathbb{Z}}|f(k)|^{2}.$

The inverse of the periodic Fourier transform is given by

$f(x)=\sum_{k\in\mathbb{Z}}e^{-2\pi ik\cdot x}\widehat{f}(k),\quad
x\in\mathbb{T},$

where $f$ belongs to a suitable function space, namely, the Schwartz space of
$\mathbb{Z}$, $\mathcal{S}(\mathbb{Z}),$ the space of rapidly decaying
functions from $\mathbb{Z}\to\mathbb{C}$.

######  Definition 2.1.

(Forward and backward differences $\triangle_{k}$ and $\bar{\triangle}_{k}$ )
Let $\sigma:\mathbb{Z}\rightarrow\mathbb{C}$. We define the forward and
backward partial difference operators $\triangle_{k}$ and
$\bar{\triangle}_{k}$, respectively, by

$\triangle_{k}\,\sigma(k):=\sigma\left(k+1\right)-\sigma(k),$
$\bar{\triangle}_{k}\,\sigma(k):=\sigma(k)-\sigma\left(k-1\right).$

Let us now recall the Hörmander symbol class,
$S_{\rho}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$, on $\mathbb{T}$, which
is same as defined in [28].

######  Definition 2.2.

Let $m\in\mathbb{R}$ and $\rho>0.$ We say that a function
$\sigma:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$ belongs to
$S_{\rho}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ if $\sigma(x,k)$ is
smooth in $x$ for all $k\in\mathbb{Z},$ and for all
$\alpha,\beta\in\mathbb{N}_{0}$, there exists a positive constant
$C_{\alpha,\beta}$ such that

$\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\sigma(x,k)\right|\leq
C_{\alpha,\beta}(1+|k|)^{m-\rho|\alpha|},\quad(x,k)\in\mathbb{T}\times\mathbb{Z}.$

For the symbol $\sigma\in
S_{\rho}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$, the corresponding
pseudo-differential operator associated with $\sigma$ is given by

$(T_{\sigma}f)(x):=\sum_{k\in\mathbb{Z}}e^{-2\pi ik\cdot
x}\sigma(x,k)\widehat{f}(k),\quad x\in\mathbb{T}.$

We denote $\mathrm{OP}S_{\rho}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ be
the set of all operators corresponding to the symbol class
$S_{\rho}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.

######  Definition 2.3.

Let $\Lambda$ be a positive function. We say that $\Lambda$ is a weight
function if there exist suitable positive constants, $\mu_{0}\leq\mu_{1}$ and
$C_{0},C_{1}$ such that

$C_{0}(1+|k|)^{\mu_{0}}\leq\Lambda(k)\leq C_{1}(1+|k|)^{\mu_{1}},\quad
k\in\mathbb{Z}.$

Furthermore, we assume that there exists a real constant $\mu$ such that
$\mu\geq\mu_{1}$ and for all $\alpha$,$\gamma$ $\in\mathbb{N}_{0}$ with
$\gamma\in\\{0,1\\}$, we can find a positive constant $C_{\alpha,\gamma}$ such
that

$\displaystyle\left|k^{\gamma}\Delta_{k}^{\alpha+\gamma}\Lambda(k)\right|\leq
C_{\alpha,\gamma}\Lambda(k)^{1-\frac{1}{\mu}\alpha},\quad k\in\mathbb{Z}.$
(2.1)

######  Definition 2.4.

Let $m\in\mathbb{R}$ and $\rho\in\left(0,\frac{1}{\mu}\right]$. Then the
toroidal symbol class
$S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ is the set of
all functions $\sigma:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$ which
are smooth in $x$ for all $k\in\mathbb{Z}$, and for all
$\alpha,\beta\in\mathbb{N}_{0}$, there exists a positive constant
$C_{\alpha,\beta}$ such that

$\displaystyle\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\sigma(x,k)\right|\leq
C_{\alpha,\beta}\Lambda(k)^{m-\rho\alpha},\quad(x,k)\in\mathbb{T}\times\mathbb{Z}.$
(2.2)

As usual we set

$S_{\rho,\Lambda}^{\infty}\left(\mathbb{T}\times\mathbb{Z}\right):=\bigcup_{m\in\mathbb{R}}S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$

and

$S_{\rho,\Lambda}^{-\infty}\left(\mathbb{T}\times\mathbb{Z}\right):=\bigcap_{m\in\mathbb{R}}S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)\text{.
}$

Let $\sigma\in S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Define a pseudo-differential operator $T_{\sigma}$ associated with symbol
$\sigma$ by

$\displaystyle
T_{\sigma}f(x)=(2\pi)^{-1}\sum_{k\in\mathbb{Z}}\int_{\mathbb{T}}e^{i(x-y)\cdot
k}\sigma(x,k)f(y)dy,\quad x\in\mathbb{T}$ (2.3)

for every $f\in C^{\infty}\left(\mathbb{T}\right)$. We write
$\operatorname{Op}(S_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z}))$ for the
class of pseudo-differential operators associated with the symbol class
$S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.

Now, we will describe the main ingredient of this paper, namely the symbol
class $M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.

######  Definition 2.5.

For $m\in\mathbb{R}$ and $\rho\in\left(0,\frac{1}{\mu}\right]$, the symbol
class, $M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ consists
of all functions $\sigma:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$
which are smooth in $x$ for all $k\in\mathbb{Z}$, and for
$\gamma\in\\{0,1\\},$

$k^{\gamma}\Delta_{k}^{\gamma}\sigma(x,k)\in
S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right).$

In the same manner, we write
$\operatorname{Op}(M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z}))$ for the
class of pseudo-differential operators associated with the symbol class
$M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.

######  Remark 2.6.

For every $m\in\mathbb{R}$ and $\rho\in\left(0,\frac{1}{\mu}\right]$, we have

$M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)\subset
S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right).$

######  Lemma 2.7.

For every $m\in\mathbb{R}$ and $0<\rho\leq\frac{1}{\mu}$, there exists a
positive integer $N_{0}$ such that

$\displaystyle
S_{\rho,\Lambda}^{m-N_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)\subset
M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)\subset
S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right).$ (2.4)

More precisely, $N_{0}:=\left(\frac{1}{\mu_{0}}-\rho\right)$. Moreover,

$\displaystyle\bigcap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)=\bigcap_{m\in\mathbb{R}}S_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)=S_{\rho,\Lambda}^{-\infty}\left(\mathbb{T}\times\mathbb{Z}\right).$
(2.5)

Proof  The proof of the above lemma is similar to the proof of Lemma 3.10 in
[23]. $\Box$

##  3\. Symbolic calculus and parametrix for
$M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$

In this section we study the symbolic calculus for pseudo-differential
operators associated with symbol in
$M_{\rho,\Lambda}^{m}(\mathbb{Z}^{n}\times\mathbb{T}^{n})$ by deriving
formulae for composition, adjoint, transpose of the operators. We also
construct the parametrix of $M$-elliptic pseudo-differential operators. We
start this section with the following result related to the asymptotic sums of
symbols.

######  Theorem 3.1.

Let $\left\\{m_{j}\right\\}_{j\in\mathbb{N}_{0}}$ be a strictly decreasing
sequence of real numbers such that $m_{j}\rightarrow-\infty$ as
$j\rightarrow\infty$. Suppose $\sigma_{j}\in
M_{\rho,\Lambda}^{m_{j}}\left(\mathbb{T}\times\mathbb{Z}\right),j\in\mathbb{N}_{0}$.
Then there exists a symbol $\sigma\in
M_{\rho,\Lambda}^{m_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)$ such that

$\sigma(x,k)\sim\sum_{j=0}^{\infty}\sigma_{j}(x,k),$

i.e.,

$\sigma(x,k)-\sum_{j=0}^{N-1}\sigma_{j}(x,k)\in
M_{\rho,\Lambda}^{m_{N}}\left(\mathbb{T}\times\mathbb{Z}\right),$

for every positive integer $N$.

Proof  Let $\sigma_{j}\in
M_{\rho,\Lambda}^{m_{j}}\left(\mathbb{T}\times\mathbb{Z}\right)$. Then from
Remark 2.6, we have $\sigma_{j}\in
S_{\rho,\Lambda}^{m_{j}}\left(\mathbb{T}\times\mathbb{Z}\right)$. Consider
$\psi\in C^{\infty}\left(\mathbb{R}\right)$ such that $0\leq\psi\leq 1$ and

$\psi(k)=\begin{cases}1,&\text{ if }|k|\geq 1\\\ 0,&\text{ if
}|k|\leq\frac{1}{2}\end{cases}.$

Let $\left(\epsilon_{j}\right)_{j=0}^{\infty}$ be a sequence of positive real
numbers such that $\epsilon_{j}>\epsilon_{j+1}\rightarrow 0$. Define
$\psi_{j}\in C^{\infty}\left(\mathbb{R}\right)$, by
$\psi_{j}(k):=\psi\left(\epsilon_{j}k\right)$. It is clear that if $\alpha\geq
1$, then the support of $\Delta_{k}^{\alpha}\phi_{j}$ is bounded. Since
$\sigma_{j}\in
S_{\rho,\Lambda}^{m_{j}}\left(\mathbb{T}\times\mathbb{Z}\right)$, so using
discrete Leibniz formula, we have

$\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left(\psi_{j}(k)\sigma_{j}(x,k)\right)\right|\leq
C_{j\alpha\beta}\Lambda(k)^{m_{j}-\rho\alpha}\text{, }$

where $C_{j\alpha\beta}$ is a positive constant. This means that,
$\psi_{j}(k)\sigma_{j}(x,k)\in
S_{\rho,\Lambda}^{m_{j}}\left(\mathbb{T}\times\mathbb{Z}\right).$ Note that,
when $j$ is large enough,
$\Delta_{k}^{\alpha}\left(\psi_{j}(k)\sigma_{j}(x,k)\right)$, (where
$\alpha\in\mathbb{N}_{0}$), vanishes for any fixed $k\in\mathbb{Z}$. This
justifies the definition

$\sigma(x,k):=\sum_{j=0}^{\infty}\psi_{j}(k)\sigma_{j}(x,k),\quad(x,k)\in\mathbb{T}\times\mathbb{Z}.$

Clearly, $\sigma\in
S_{\rho,\Lambda}^{m_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)$. Further, we
have

$\displaystyle\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left(\sigma(x,k)-\sum_{j=0}^{N-1}\sigma_{j}(x,k)\right)\right|$
$\displaystyle\leq\sum_{j=0}^{N-1}\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left\\{\left(\psi_{j}(k)-1\right)\sigma_{j}(x,k)\right\\}\right|+\sum_{j=N}^{\infty}\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left(\psi_{j}(k)\sigma_{j}(x,k)\right)\right|.$

Since $\epsilon_{j}>\epsilon_{j+1}$ and $\epsilon_{j}\rightarrow 0$ as
$j\rightarrow\infty$, so
$\sum_{j=0}^{N-1}\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left\\{(\psi_{j}(k)-1)\sigma_{j}(x,k)\right\\}\right|$
vanishes, whenever $|k|$ is large. Thus, there exists a positive constant
$C_{rN\alpha\beta}$ such that

$\sum_{j=0}^{N-1}\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left\\{\left(\psi_{j}(k)-1\right)\sigma_{j}(x,k)\right\\}\right|\leq
C_{rN\alpha\beta}\Lambda(k)^{-r}$

for any $r\in\mathbb{R}$. On the other hand, one can easily show that

$\sum_{j=N}^{\infty}\left|\Delta_{k}^{\alpha}\partial_{x}^{\beta}\left(\psi_{j}(k)\sigma_{j}(x,k)\right)\right|\leq
C_{N\alpha\beta}^{\prime}\Lambda(k)^{m_{N}-\rho\alpha},$

where $C_{N\alpha\beta}^{\prime}$ is a positive constant. This shows that for
every $N\in\mathbb{N}$, we have

$\sigma(x,k)-\sum_{j=0}^{N-1}\sigma_{j}(x,k)\in
S_{\rho,\Lambda}^{m_{N}}\left(\mathbb{T}\times\mathbb{Z}\right).$

Since $m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$, using left
inclusions in (2.4), we have $\sigma-\sum_{j=0}^{N-1}\sigma_{j}\in$
$S_{\rho,\Lambda}^{m_{N}}\left(\mathbb{T}\times\mathbb{Z}\right)\subset
M_{\rho,\Lambda}^{m_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)$ for a
sufficiently large $N$. Hence $\sigma(x,k)\in
M_{\rho,\Lambda}^{m_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)$. Furthermore,
for all $N\geq 2$ and $N^{\prime}>N$

$\sigma-\sum_{j=0}^{N-1}\sigma_{j}=\sum_{j=N}^{N^{\prime}-1}\sigma_{j}+r_{N^{\prime}}$

with $r_{N^{\prime}}\in
S_{\rho,\Lambda}^{m_{N^{\prime}}}\left(\mathbb{T}\times\mathbb{Z}\right)$. By
choosing a sufficiently large $N^{\prime}$ so that $m_{N^{\prime}}<m_{N}-$
$N_{0}$, we have $r_{N^{\prime}}\in
S_{\rho,\Lambda}^{m_{N}-N_{0}}\left(\mathbb{T}\times\mathbb{Z}\right)\subset
M_{\rho,\Lambda}^{m_{N}}\left(\mathbb{T}\times\mathbb{Z}\right)$ and therefore
$\sigma-\sum_{j=0}^{N-1}\sigma_{j}\in$
$M_{\rho,\Lambda}^{m_{N}}\left(\mathbb{T}\times\mathbb{Z}\right)$. This
completes the proof of the theorem. $\Box$

The following results on the basic symbolic calculus of pseudo-differential
operators with weighted $M$-symbols on $\mathbb{T}\times\mathbb{Z}$ are
analogs of results for pseudo-differential operators with symbols in
$S^{m}\left(\mathbb{T}^{n}\times\mathbb{Z}^{n}\right)$ given in [28] and
symbols in
$M_{\rho,\Lambda}^{m}\left(\mathbb{Z}^{n}\times\mathbb{T}^{n}\right)$ given in
[23].

######  Theorem 3.2.

Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$
and $\tau\in M_{\rho,\Lambda}^{\mu}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then $T_{\sigma}T_{\tau}=T_{\lambda}$, where $\lambda\in
M_{\rho,\Lambda}^{m+\mu}\left(\mathbb{T}\times\mathbb{Z}\right)$ and

$\lambda\sim\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}(\Delta_{k}^{\alpha}\sigma)(\partial_{x}^{\alpha}\tau).$

Here the asymptotic expansion means that

$\lambda-\sum_{|\alpha|<N}\frac{(-i)^{|\alpha|}}{\alpha!}(\Delta_{k}^{\alpha}\sigma)(\partial_{x}^{\alpha}\tau)\in
M_{\rho,\Lambda}^{m+\mu-\rho N}\left(\mathbb{T}\times\mathbb{Z}\right),$

for every positive integer $N$.

######  Theorem 3.3.

Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then the formal adjoint $T_{\sigma}^{\ast}$ of $T_{\sigma}$ is the pseudo-
differential operator $T_{\tau}$, where $\tau\in M_{\rho,\Lambda}^{m}$ and

$\tau\sim\sum_{\alpha}\frac{(-i)^{|\alpha|}}{\alpha!}\Delta_{k}^{\alpha}\partial_{x}^{\alpha}\overline{\sigma}.$

Here the asymptotic expansion means that

$\tau-\sum_{|\alpha|<N}\frac{(-i)^{|\alpha|}}{\alpha!}\Delta_{k}^{\alpha}\partial_{x}^{\alpha}\overline{\sigma}\in
M_{\rho,\Lambda}^{m-\rho N}\left(\mathbb{T}\times\mathbb{Z}\right),$

for every positive integer $N$.

Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$,
where $m\in\mathbb{R}$. Then $\sigma$ is said to be $M$-elliptic if there
exist positive constants $C$ and $R$ such that

$|\sigma(x,k)|\geq C\Lambda(k)^{m},\quad\forall
x\in\mathbb{T},k\in\mathbb{Z},$

with $|k|\geq R$. Naturally, a pseudo-differential operator $T_{\sigma}$
corresponding to such $\sigma$, is said to be $M$-elliptic.
The following lemma is analogous to the Lemma 2 in [15], which can be proved
in the similar way. So we skip the proof here.

######  Lemma 3.4.

Let $\sigma(x,k)\in M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$ and
$\psi(x,k)\in C^{\infty}\left(\mathbb{T}\times\mathbb{R}\right)$, be such that
there exist two sufficiently large positive constants
$R^{\prime\prime}>R^{\prime}$ so that $\psi(x,k)=0$, for all $x\in\mathbb{T}$
and $|k|\leq R^{\prime}$, and $\psi(x,k)=1$, for all $x\in\mathbb{T}$ and
$|k|\geq R^{\prime\prime}$. Then $q(x,k)=\frac{\psi(x,k)}{p(x,k)}\in
M_{\rho,\Lambda}^{-m}(\mathbb{T}\times\mathbb{Z})$.

Using the above lemma, we obtain the parametrix of the elliptic operators.

######  Theorem 3.5.

A symbol $\sigma$ is elliptic in
$M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ if and only if
there exists a symbol $\tau$ in
$M_{\rho,\Lambda}^{-m}\left(\mathbb{T}\times\mathbb{Z}\right)$ such that

$\displaystyle T_{\tau}T_{\sigma}=I+R$ (3.1)

and

$\displaystyle T_{\sigma}T_{\tau}=I+S,$ (3.2)

where $R$ and $S$ are pseudo-differential operators with symbols in
$\bigcap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$,
and $I$ is the identity operator.

Proof  First suppose that there exists a symbol $\tau$ in
$M_{\rho,\Lambda}^{-m}\left(\mathbb{T}\times\mathbb{Z}\right)$ such that (3.1)
and (3.2) are true. From (3.2), we can conclude that

$I-T_{\sigma}T_{\tau}\in\operatorname{Op}(M_{\rho,\Lambda}^{-\infty}(\mathbb{T}\times\mathbb{Z})).$

Hence, by Theorem 3.2, we have

$1-\sigma(x,k)\tau(x,k)\in
M_{\rho,\Lambda}^{-\rho}\left(\mathbb{T}\times\mathbb{Z}\right),$

So, we can find two positive constants $C$ and $C_{0}$ such that

$\left|1-\sigma(x,k)\tau(x,k)\right|\leq C_{0}\Lambda(k)^{-\rho}\leq
C(1+|k|)^{-\rho\mu_{0}},\quad(x,k)\in\mathbb{T}\times\mathbb{Z}.$

Let $R\in\mathbb{N}$ such that $C(1+R)^{-\rho\mu_{0}}<\frac{1}{2}$. Then it
follows that

$\left|\sigma(x,k)\tau(x,k)\right|\geq\frac{1}{2},\quad\forall\,\,|k|\geq R,$

and hence

$\left|\sigma(x,k)\right|\geq\frac{1}{2\left|\tau(x,k)\right|}\geq\frac{1}{2C^{\prime}}(\Lambda(k))^{m},\quad\forall\,\,|k|\geq
R,$

since

$\left|\tau(x,k)\right|\leq
C^{\prime}(\Lambda(k))^{-m},\quad(x,k)\in\mathbb{T}\times\mathbb{Z}.$

Hence $\sigma$ is an $M$-elliptic symbol of order $m$.

Conversely, let us assume that $\sigma$ is an $M$-elliptic symbol of order
$m$. Then there exist positive constants $C$ and $R$ such that

$|\sigma(x,k)|\geq C\Lambda(k)^{m},$

for all $x\in\mathbb{T}$ and for all $k\in\mathbb{Z}$ with $|k|\geq R$. The
idea is to find a sequence of symbols $\tau_{j}\in M_{\rho,\Lambda}^{-m-\rho
j}(\mathbb{T}\times\mathbb{Z}),j=0,1,2,\ldots.$ Let us assume that this can
been done. Then, by Theorem 3.1, there exists a symbol $\tau\in
M_{\rho,\Lambda}^{-m}(\mathbb{T}\times\mathbb{Z})$ such that
$\tau\sim\sum_{j=0}^{\infty}\tau_{j}$, and, by Theorem 3.2, the symbol
$\lambda$ of the product $T_{\tau}T_{\sigma}$ is in
$M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ such that

$\displaystyle\lambda-\sum_{|\gamma|<N}\frac{(-i)^{|\gamma|}}{\gamma!}\left(\partial_{x}^{\gamma}\sigma\right)(\Delta_{k}^{\gamma}\tau)\in
M_{\rho,\Lambda}^{-\rho N}(\mathbb{T}\times\mathbb{Z}),$ (3.3)

for every positive integer $N$. Also $\tau\sim\sum_{j=0}^{\infty}\tau_{j}$
implies that

$\displaystyle\tau-\sum_{j=0}^{N-1}\tau_{j}\in M_{\rho,\Lambda}^{-m-\rho
N}(\mathbb{T}\times\mathbb{Z}),$ (3.4)

for every positive integer $N$. Hence, by (3.3) and (3.4),

$\displaystyle\lambda-\sum_{|\gamma|<N}\frac{(-i)^{|\gamma|}}{\gamma!}\left(\partial_{x}^{\gamma}\sigma\right)\sum_{j=0}^{N-1}(\Delta_{k}^{\gamma}\tau_{j})\in
M_{\rho,\Lambda}^{-\rho N}(\mathbb{T}\times\mathbb{Z}),$ (3.5)

for every positive integer $N$. But we can write

$\displaystyle\sum_{|\gamma|<N}\frac{(-i)^{|\gamma|}}{\gamma!}\sum_{j=0}^{N-1}(\Delta_{k}^{\gamma}\tau_{j})\left(\partial_{x}^{\gamma}\sigma\right)$
(3.6) $\displaystyle=$
$\displaystyle\,\tau_{0}\sigma+\sum_{l=1}^{N-1}\left\\{\tau_{l}\sigma+\sum_{\begin{subarray}{c}|\gamma|+j=l\\\
j<l\end{subarray}}\frac{(-i)^{|\gamma|}}{\gamma!}(\Delta_{k}^{\gamma}\tau_{j})\left(\partial_{x}^{\gamma}\sigma\right)\right\\}$
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}|\gamma|+j\geq N\\\
|\gamma|<N,\,j<N\end{subarray}}\frac{(-i)^{|\gamma|}}{\gamma!}(\Delta_{k}^{\gamma}\tau_{j})\left(\partial_{x}^{\gamma}\sigma\right).$

To find a sequence of symbols $\tau_{j}\in M_{\rho,\Lambda}^{-m-\rho
j}(\mathbb{T}\times\mathbb{Z}),j=0,1,2,\ldots$, we choose $\psi$ to be any
function in $C^{\infty}\left(\mathbb{R}^{n}\right)$ such that $\psi(k)=1$, for
$|k|\geq 2R$ and $\psi(k)=0$, for $|k|\leq R$. Define

$\displaystyle\tau_{0}(x,k)=\begin{cases}\frac{\psi(k)}{\sigma(x,k)},&|k|>R,\\\
0,&|k|\leq R,\end{cases},\quad(x,k)\in\mathbb{T}\times\mathbb{Z}\text{.}$
(3.7)

From Lemma 3.4, it is clear that $\tau_{0}\in
M_{\rho,\Lambda}^{-m}(\mathbb{T}\times\mathbb{Z})$. Now define, $\tau_{l}$,
for $l\geq 1$, inductively by

$\displaystyle\tau_{l}=-\left\\{\sum_{\begin{subarray}{c}|\gamma|+j=l\\\
j<l\end{subarray}}\frac{(-i)^{|\gamma|}}{\gamma!}\left(\partial_{x}^{\gamma}\sigma\right)(\Delta_{k}^{\gamma}\tau_{j})\right\\}\tau_{0}.$
(3.8)

Then it can be shown that $\tau_{j}\in M_{\rho,\Lambda}^{-m-\rho
j}(\mathbb{T}\times\mathbb{Z}),j=0,1,2,\ldots$. Now, by (3.7),
$\tau_{0}\sigma=1$, for $|k|\geq 2R$. The second term on the right hand side
of (3.6) vanishes for $|k|\geq 2R$ by (3.7) and (3.8). Also the third term
there,

$(\Delta_{k}^{\gamma}\tau_{j})\left(\partial_{x}^{\gamma}\sigma\right)\in
M_{\rho,\Lambda}^{-\rho N}(\mathbb{T}\times\mathbb{Z}),$

whenever $|\gamma|+j\geq N$. Hence, by (3.6),

$\displaystyle\sum_{|\gamma|<N}\frac{(-i)^{|\gamma|}}{\gamma!}\sum_{j=0}^{N-1}(\Delta_{k}^{\gamma}\tau_{j})\left(\partial_{x}^{\gamma}\sigma\right)-1\in
M_{\rho,\Lambda}^{-\rho N}(\mathbb{T}\times\mathbb{Z}),$ (3.9)

for every positive integer $N$. Thus, by (3.5) and (3.9),

$\lambda-1\in M_{\rho,\Lambda}^{-\rho N}(\mathbb{T}\times\mathbb{Z}),$

for every positive integer $N$. Hence, if we pick $R$ to be the pseudo-
differential operator with symbol $\lambda-1$, then the proof of (3.1) is
complete.

By a similar argument, we can find another symbol, $\kappa\in
M_{\rho,\Lambda}^{-m}(\mathbb{T}\times\mathbb{Z})$, such that

$\displaystyle T_{\sigma}T_{\kappa}=I+R^{\prime},$ (3.10)

where $R^{\prime}$ is a pseudo-differential operator with symbol in
$\bigcap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$. By
(3.1) and (3.10),

$T_{\kappa}+RT_{\kappa}=T_{\tau}+T_{\tau}R^{\prime}.$

Since $RT_{\kappa}$ and $T_{\tau}R^{\prime}$ are pseudo-differential operators
with symbols in
$\cap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$, it
follows that

$\displaystyle T_{\kappa}=T_{\tau}+R^{\prime\prime},$ (3.11)

where

$R^{\prime\prime}=T_{\tau}R^{\prime}-RT_{\kappa},$

is another pseudo-differential operator with symbol in
$\bigcap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$.
Hence, by (3.10) and (3.11),

$T_{\sigma}T_{\tau}=I+S,$

where

$S=R^{\prime}-T_{\sigma}R^{\prime\prime}.$

Since $S$ is a pseudo-differential operator with symbol in
$\bigcap_{m\in\mathbb{R}}M_{\rho,\Lambda}^{m}(\mathbb{T}\times\mathbb{Z})$, it
follows that (3.2) is proved. $\Box$

##  4\. Compact $M$-elliptic pseudo-differential operators

This section is devoted to study the compact $M$-elliptic pseudo-differential
operators on $\mathbb{T}$ and $\mathbb{Z}$. First, we prove Gohberg’s lemma
for pseudo-differential operators on $\mathbb{T}$ and $\mathbb{Z}$ with symbol
in the weighted symbol class
$M_{\rho,\Lambda}^{0}(\mathbb{T}\times\mathbb{Z})$ and
$M_{\rho,\Lambda}^{0}(\mathbb{Z}\times\mathbb{T})$, respectively. Using
Gohberg’s lemma, we provide a necessary and sufficient condition to ensure the
compactness of a pseudo-differential operator on $L^{2}(\mathbb{T})$ and
$\ell^{2}(\mathbb{Z})$. We start this section by recalling the definition of
Fredholm operators.

Let $X$ and $Y$ be two complex Banach spaces. A closed linear operator
$A:X\rightarrow Y$ with dense domain $\mathcal{D}(A)$ is said to be Fredholm
if the null space $N(A)$ of $A$, and the null space $N\left(A^{t}\right)$ of
the adjoint $A^{t}$ of $A$ are finite-dimensional, and the range space $R(A)$
of $A$ is a closed subspace of $Y.$ For a Fredholm operator $A$, the index
$i(A)$ of $A$ is defined by

$i(A)=\operatorname{dim}N(A)-\operatorname{dim}N(A^{t}).$

Let $X$ be a complex Banach space and $A$: $X\rightarrow X$ be a closed linear
operator with dense domain $\mathcal{D}(A)$. Then the spectrum $\Sigma(A)$ of
$A$ is defined by

$\Sigma(A)=\mathbb{C}\backslash\rho(A),$

where $\rho(A)$ is the resolvent set of $A$ given by

$\rho(A)=\\{\lambda\in\mathbb{C}:A-\lambda I\text{ is bijective}\\},$

and $I$ is the identity operator on $X$. The essential spectrum
$\Sigma_{w}(A)$ of $A$ (defined in [35]) is given by

$\Sigma_{w}(A)=\mathbb{C}\backslash\Phi_{w}(A),$

where

$\Phi_{w}(A)=\\{\lambda\in\mathbb{C}:A-\lambda I\text{ is Fredholm}\\}.$

For a detailed study on essential spectrum of an operator, we refer to [32,
33].

For $-\infty<s<\infty$, let $J_{s}$ be the pseudo-differential operator with
symbol $\sigma_{s}$ given by

$\sigma_{s}(k)=\left(\Lambda(k)\right)^{-s},\quad k\in\mathbb{Z}.$

Clearly, $\sigma_{s}\in
M_{\rho,\Lambda}^{-s}\left(\mathbb{T}\times\mathbb{Z}\right)$. In literature,
$J_{s}$ is called the Bessel potential of order $s$. For $s\in\mathbb{R}$ and
$1\,<\,p\,<\,\infty$, we define the Sobolev space, $H_{\Lambda}^{s,p}$ by

$H_{\Lambda}^{s,p}=\\{u\in\mathcal{D}^{\prime}(\mathbb{T}):J_{-s}\,u\in\,L^{p}(\mathbb{T})\\}.$

Then $H_{\Lambda}^{s,p}$ is a Banach space in which the norm $\|\,\|_{s,p}$ is
given by

$\|u\|_{s,p,\Lambda}=\|J_{-s}\,u\|_{L^{p}(\mathbb{T})},\quad
u\,\in\,H_{\Lambda}^{s,p}.$

Note that $H_{\Lambda}^{0,p}=L^{p}(\mathbb{T}).$
The following well-known compact Sobolev embedding theorem is just the
weighted version of [24, Theorem 2.5].

######  Theorem 4.1.

Let $s<t$. Then the inclusion $i:H_{\Lambda}^{t,p}\hookrightarrow
H_{\Lambda}^{s,p}$ is compact for $1<p<\infty$.

The following boundedness result on weighted Sobolev space can be found in
[21].

######  Theorem 4.2.

Let $\sigma\in
M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right),-\infty<m<\infty$.
Then $T_{\sigma}:H_{\Lambda}^{s,p}\rightarrow H_{\Lambda}^{s-m,p}$ is a
bounded linear operator for $1<p<\infty$.

The following proposition is the weighted version of [26, Theorem 4.5] for
$M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$ class.

######  Proposition 4.3.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$ be
such that

$\lim_{|k|\rightarrow\infty}\left\\{\sup_{x\in[-\pi,\pi]}|\sigma(x,k)|\right\\}=0.$

Then

$\Sigma_{e}\left(T_{\sigma}\right)=\\{0\\}.$

A bounded linear operator $A$ on a complex separable and infinite-dimensional
Hilbert space $X$ is essentially normal if $AA^{t}-A^{t}A$ is compact.

The next result is about the essential normality of pseudo-differential
operators with wrighted symbol of order 0.

######  Proposition 4.4.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then the bounded linear operator $T_{\sigma}$ :
$L^{2}\left(\mathbb{T}\right)\rightarrow L^{2}\left(\mathbb{T}\right)$ is
essentially normal.

Proof  Let $\tau\in
M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$ be such that
$T_{\sigma}^{t}=T_{\tau}$. Then using Theorem 3.2, we have

$T_{\sigma}T_{\tau}=T_{\gamma}\quad\text{ and }\quad
T_{\tau}T_{\sigma}=T_{\tilde{\gamma}}\text{, }$

where $\gamma$ and $\tilde{\gamma}$ are symbols of order 0. Moreover,
$\gamma-\sigma\tau\in
M_{\rho,\Lambda}^{-\rho}\left(\mathbb{T}\times\mathbb{Z}\right)$ and
$\tilde{\gamma}-\sigma\tau\in
M_{\rho,\Lambda}^{-\rho}\left(\mathbb{T}\times\mathbb{Z}\right)$. Therefore,
$\gamma-\tilde{\gamma}\in
M_{\rho,\Lambda}^{-\rho}\left(\mathbb{T}\times\mathbb{Z}\right)$. Hence, by
Theorem 4.1 and 4.2, we get

$T_{\sigma}T_{\sigma}^{t}-T_{\sigma}^{t}T_{\sigma}=T_{\gamma-\tilde{\gamma}}:L^{2}\left(\mathbb{T}\right)\rightarrow
H_{\Lambda}^{\rho,2}\hookrightarrow L^{2}\left(\mathbb{T}\right)$

is compact, which completes the proof. $\Box$

The following theorem is known as Gohberg’s lemma in the literature.

######  Theorem 4.5.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then for all compact operators $K$ on $L^{2}\left(\mathbb{T}\right)$,

$\displaystyle\left\|T_{\sigma}-K\right\|_{*}\geq d,$ (4.1)

where

$d=\limsup_{|k|\rightarrow\infty}\left\\{\sup_{x\in[-\pi,\pi]}|\sigma(x,k)|\right\\}.$

Here $\|\cdot\|_{*}$ denotes the norm in the $C^{*}$-algebra of all bounded
linear operators on $L^{2}(\mathbb{T})$.

Proof  Let $u$ be a nonzero function in $C^{\infty}\left(\mathbb{T}\right)$.
Then

$\displaystyle\left(T_{\sigma}u\right)(x)=(2\pi)^{-1}\sum_{k\in\mathbb{Z}}\left\\{\int_{-\pi}^{\pi}e^{ik(x-y)}\sigma(x,k)u(y)dy\right\\}$
$\displaystyle=(2\pi)^{-1}\int_{-\pi}^{\pi}\left(\mathcal{F}_{\mathbb{Z}}\sigma)(x,x-y)u(y)dy\right.$
$\displaystyle=(2\pi)^{-1}\int_{-\pi}^{\pi}\left(\mathcal{F}_{\mathbb{Z}}\sigma\right)(x,y)u(x-y)dy,\quad
x\in[-\pi,\pi],$

where $\mathcal{F}_{\mathbb{Z}}\sigma$ is the Fourier transform of $\sigma$
with respect to the second variable in the sense of distribution. Since
$\sigma\in C^{\infty}\left(\mathbb{T}\times\mathbb{Z}\right)$, it follows that
for all $k\in\mathbb{Z}$, there exists $x_{k}\in[-\pi,\pi]$ such that

$\left|\sigma\left(x_{k},k\right)\right|=\sup_{x\in[-\pi,\pi]}|\sigma(x,k)|.$

By the definition of $d$, there exists a sequence
$\left\\{\left(x_{k_{m}},k_{m}\right)\right\\}_{m=1}^{\infty}$ such that

$\left|k_{m}\right|\rightarrow\infty,$

and

$\left|\sigma\left(x_{k_{m}},k_{m}\right)\right|\rightarrow d$

as $m\rightarrow\infty$. For $m=1,2,\ldots$, we define the function
$u_{k_{m}}$ on $\mathbb{T}$ by

$u_{k_{m}}(x)=e^{ik_{m}x}u\left(x-x_{k_{m}}\right),\quad x\in[-\pi,\pi].$

Then

$\left\|u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}=\|u\|_{L^{2}\left(\mathbb{T}\right)},\quad
m=1,2,\ldots,$

and moreover, using the Riemann-Lebesgue lemma, one can easily show that
$u_{k_{m}}\rightarrow 0$ weakly as $k\rightarrow\infty$. Let
$K:L^{2}(\mathbb{T})\rightarrow L^{2}(\mathbb{T})$ be a compact operator and
$\epsilon$ be an arbitrary positive number. Then

$\left\|Ku_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}\rightarrow 0$

as $m\rightarrow\infty$, and hence, for sufficiently large $m$,

$\displaystyle\left\|Ku_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}\leq\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)}.$
(4.2)

######  Lemma 4.6.

$\left\|\sigma\left(\cdot,k_{m}\right)u_{k_{m}}-T_{\sigma}u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}\rightarrow
0$ as $m\rightarrow\infty$.

We assume the lemma for a moment and continue with the proof of Theorem 4.5.
By Lemma 4.6, we have

$\displaystyle\left\|\sigma\left(\cdot,k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}-\left\|T_{\sigma}u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}\leq\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)},$
(4.3)

for sufficiently large $m$. Since $\sigma\in
C^{\infty}\left(\mathbb{T}\times\mathbb{Z}\right)$, so all derivatives of
$\sigma\left(\cdot,k_{m}\right)$ exist and are bounded, and hence, there
exists a positive number $\delta$ such that for all $x$ in
$\left[-\pi+x_{k_{m}},\pi+x_{k_{m}}\right]$ with
$\left|x-x_{k_{m}}\right|<\delta$, we have

$\displaystyle\left|\sigma\left(x,k_{m}\right)-\sigma\left(x_{k_{m}},k_{m}\right)\right|<\epsilon.$
(4.4)

Choose $u\in C^{\infty}\left(\mathbb{T}\right)$ be such that

$u(x)=0,\quad|x|\geq\delta.$

Then $u_{k_{m}}(x)=0$ for all $x$ in
$\left[-\pi+x_{k_{m}},\pi+x_{k_{m}}\right]$ with
$\left|x-x_{k_{m}}\right|\geq\delta.$ So,

$\displaystyle\left\|\sigma\left(x_{k_{m}},k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}-\left\|\sigma\left(\cdot,k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\leq\left\|\sigma\left(x_{k_{m}},k_{m}\right)u_{k_{m}}-\sigma\left(\cdot,k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle=\left\\{\int_{-\pi+x_{k_{m}}}^{\pi+x_{k_{m}}}\left|\sigma\left(x,k_{m}\right)-\sigma\left(x_{k_{m}},k_{m}\right)\right|^{2}\left|u_{k_{m}}(x)\right|^{2}dx\right\\}^{1/2}$
$\displaystyle=\left\\{\int_{\left\\{x\in\left[-\pi+x_{k_{m}},\pi+x_{k_{m}}\right]:\left|x-x_{k_{m}}\right|<\delta\right\\}}\left|\sigma\left(x,k_{m}\right)-\sigma\left(x_{k_{m}},k_{m}\right)\right|^{2}\left|u_{k_{m}}(x)\right|^{2}dx\right\\}^{1/2}.$

Hence, by (4.4),

$\displaystyle\left|\sigma\left(x_{k_{m}},k_{m}\right)\right|\|u\|_{L^{2}\left(\mathbb{T}\right)}-\left\|\sigma\left(\cdot,k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}\leq\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)}.$
(4.5)

Thus, by (4.2), (4.3), and (4.5), for sufficiently large $m$, we get

$\displaystyle\|u\|_{L^{2}\left(\mathbb{T}\right)}\left\|T_{\sigma}-K\right\|_{*}$
$\displaystyle\geq\left\|\left(T_{\sigma}-K\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq\left\|T_{\sigma}u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}-\left\|Ku_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq\left\|T_{\sigma}u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}-\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq\left\|\sigma\left(\cdot,k_{m}\right)u_{k_{m}}\right\|_{L^{2}\left(\mathbb{T}\right)}-2\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq\left|\sigma\left(x_{k_{m}},k_{m}\right)\right|\|u\|_{L^{2}\left(\mathbb{T}\right)}-3\epsilon\|u\|_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle=\left(\left|\sigma\left(x_{k_{m}},k_{m}\right)\right|-3\epsilon\right)\|u\|_{L^{2}\left(\mathbb{T}\right)}.$

Letting $m\rightarrow\infty$, we get

$\left\|T_{\sigma}-K\right\|_{*}\geq d-3\epsilon.$

Finally, using the fact that $\epsilon$ is an arbitrary positive number, we
have

$\left\|T_{\sigma}-K\right\|_{*}\geq d.$

$\Box$

###### Proof of Lemma 4.6.

Let

$K(x,y)=\left(\mathcal{F}_{\mathbb{Z}}\sigma\right)(x,y),\quad
x,y\in[-\pi,\pi].$

Then for $k=1,2,\ldots$,

$\displaystyle\left(T_{\sigma}u_{k_{m}}\right)(x)=(2\pi)^{-1}\int_{-\pi}^{\pi}e^{ik_{m}(x-y)}K(x,y)u_{x_{k_{m}}}(x-y)dy,\quad
x\in[-\pi,\pi],$ (4.6)

where $u_{x_{k_{m}}}=u(x-x_{k_{m}}).$ In the view of Example 2.4 in [30], the
function $q(x)=e^{-2\pi ix}-1$ gives rise to a strongly admissible difference
operator on $\mathbb{T}$ with the property that

$\Delta p(k)=\Delta_{q}p(k)=p(k+1)-p(k),\quad k\in\mathbb{Z}.$

Let $N$ be a positive integer. Then by the Taylor expansion formula (see [30])

$\displaystyle
u_{x_{k_{m}}}(x-y)=u_{x_{k_{m}}}(x)+\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}q^{\alpha}(y)\partial^{\alpha}u_{x_{k_{m}}}(x)+\mathcal{O}\left(h(y)^{N}\right),$
(4.7)

where $h(x)$ is the geodesic distance from $x$ and the identity element of
$\mathbb{T}$. Using (4.6) and (4.7), we can write

$\displaystyle T_{\sigma}u_{k_{m}}(x)$
$\displaystyle=(2\pi)^{-1}\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)u_{x_{k_{m}}}(x)dy$
$\displaystyle+(2\pi)^{-1}\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}q^{\alpha}(y)\partial^{\alpha}u_{x_{k_{m}}}(x)dy$
$\displaystyle+(2\pi)^{-1}\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)\mathcal{O}\left(h(y)^{N}\right)dy.$

Define

$\displaystyle
I_{1}:=\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)u_{x_{k_{m}}}(x)dy,$
$\displaystyle
I_{2}:=\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}q^{\alpha}(y)\partial^{\alpha}u_{x_{k_{m}}}(x)dy,$

and

$I_{3}:=\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)\mathcal{O}\left(h(y)^{N}\right)dy.$

So, we have

$\displaystyle I_{1}$
$\displaystyle=\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)u_{x_{k_{m}}}(x)dy$
$\displaystyle=u_{k_{m}}(x)\sigma\left(x,k_{m}\right),$ $\displaystyle I_{2}$
$\displaystyle=\int_{\mathbb{T}}e^{ik_{m}(x-y)}K(x,y)\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}q^{\alpha}(y)\partial^{\alpha}u_{x_{k_{m}}}(x)dy$
$\displaystyle=\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}e^{ik_{m}x}\partial^{\alpha}u_{x_{k_{m}}}(x)\int_{\mathbb{T}}e^{-ik_{m}y}K(x,y)q^{\alpha}(y)dy$
$\displaystyle=\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}e^{ik_{m}x}\partial^{\alpha}u_{x_{k_{m}}}(x)\Delta_{q}^{\alpha}\sigma\left(x,k_{m}\right)$
$\displaystyle=\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}e^{ik_{m}x}\partial^{\alpha}u_{x_{k_{m}}}(x)\Delta^{\alpha}\sigma\left(x,k_{m}\right),$
$\displaystyle I_{3}$
$\displaystyle=\int_{\mathbb{T}}K(x,y)\mathcal{O}\left(h(y)^{N}\right)e^{ik_{m}(x-y)}dy$
$\displaystyle=\int_{\mathbb{T}}K(x,y)q^{N}(y)e^{ik_{m}(x-y)}dy$
$\displaystyle=e^{ik_{m}x}\Delta_{q}^{N}\sigma\left(x,k_{m}\right)$
$\displaystyle=e^{ik_{m}x}\Delta^{N}\sigma\left(x,k_{m}\right),$

where $q^{N}(x):=\mathcal{O}\left(h(x)^{N}\right)$ vanishes at the identity
element of $\mathbb{T}$. Hence,

$T_{\sigma}u_{k_{m}}(x)-u_{k_{m}}(x)\sigma\left(x,k_{m}\right)=\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}e^{ik_{m}x}\
\partial^{\alpha}u_{x_{k_{m}}}(x)\Delta^{\alpha}\sigma\left(x,k_{m}\right)+e^{ik_{m}x}\Delta^{N}\sigma\left(x,k_{m}\right).$

Define

$T_{N}^{1}(x):=\sum_{\alpha=1}^{N-1}\frac{1}{\alpha!}e^{ik_{m}x}\partial^{\alpha}u_{x_{k_{m}}}(x)\Delta^{\alpha}\sigma\left(x,k_{m}\right),$

and

$T_{N}^{2}(x):=e^{ik_{m}x}\Delta^{N}\sigma\left(x,k_{m}\right).$

Now using the fact that $u_{x_{k_{m}}}\in C^{\infty}(\mathbb{T})$, we obtain

$\displaystyle|T_{N}^{1}(x)|$
$\displaystyle\leq\sum_{\alpha=1}^{N-1}C_{\alpha}|\Delta^{\alpha}\sigma\left(x,k_{m}\right)|$
$\displaystyle\leq\sum_{\alpha=1}^{N-1}C_{\alpha}\,\Lambda(k_{m})^{-\rho\alpha}$
$\displaystyle\leq C\,\Lambda(k_{m})^{-\rho},\quad x\in\mathbb{T},$

where $C=\sum_{\alpha=1}^{N-1}C_{\alpha}.$ Clearly $|T_{N}^{1}(x)|\rightarrow
0$ uniformly on $\mathbb{T}$ as $m\rightarrow\infty$, and hence,
$\|T_{N}^{1}\|_{L^{2}(\mathbb{T})}\rightarrow 0$ as $m\rightarrow\infty$.
Similarly, $|T_{N}^{2}(x)|\leq C_{0}\,\Lambda(k_{m})^{-\rho\,N}\leq
C_{0}\,\Lambda(k_{m})^{-\rho}\rightarrow 0$ uniformly on $\mathbb{T}$ as
$m\rightarrow\infty$, and hence, $\|T_{N}^{2}\|_{L^{2}(\mathbb{T})}\rightarrow
0$ as $m\rightarrow\infty$. This completes the proof of the theorem. ∎

Now we recall the definition of the Calkin algebra, which will be used to
prove the main result of this section. Let
$B\left(L^{2}\left(\mathbb{T}\right)\right)$ and
$K\left(L^{2}\left(\mathbb{T}\right)\right)$ denotes the $C^{*}$-algebra of
bounded linear operators on $L^{2}\left(\mathbb{T}\right)$ and the ideal of
compact operators on $L^{2}\left(\mathbb{T}\right)$, respectively. The Calkin
algebra
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$
is a *-algebra with respect to the product and the adjoint defined as follows:

$[A][B]=[AB]$

and

$[A]^{*}=\left[A^{*}\right]$

for all $A$ and $B$ in $B\left(L^{2}\left(\mathbb{T}\right)\right).$ Two
elements $[A]$ and $[B]$ be in the Calkin algebra
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$
are equal if and only if $A-B\in K\left(L^{2}\left(\mathbb{T}\right)\right)$.

The norm $\|\cdot\|_{C}$ in
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$
is given by

$\|[A]\|_{C}=\inf_{K\in
K\left(L^{2}\left(\mathbb{T}\right)\right)}\|A-K\|_{*},\quad[A]\in
B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right).$

It can be shown that
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$
is a $C^{*}$-algebra. Using the Calkin algebra, (4.1) in Gohberg’s lemma is
the same as

$\left\|\left[T_{\sigma}\right]\right\|_{C}\geq d.$

Now we are ready to prove our main theorem in this section.

######  Theorem 4.7.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then $T_{\sigma}$ is a compact operator on $L^{2}\left(\mathbb{T}\right)$ if
and only if $d=0$, where

$d=\limsup_{|k|\rightarrow\infty}\left\\{\sup_{x\in[-\pi,\pi]}|\sigma(x,k)|\right\\}.$

Proof  First, let us assume that $d=0$. Then $T_{\sigma}$ is compact if and
only if $\left[T_{\sigma}\right]=0$ in
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$.
By Proposition 4.4, $T_{\sigma}$ is essentially normal on
$L^{2}\left(\mathbb{T}\right)$, which implies that $\left[T_{\sigma}\right]$
is normal in the Calkin algebra
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$.
Hence,

$r\left(\left[T_{\sigma}\right]\right)=\left\|\left[T_{\sigma}\right]\right\|_{C},$

where $r\left(\left[T_{\sigma}\right]\right)$ is the spectral radius of
$\left[T_{\sigma}\right]$, and by Proposition 4.3, we get
$\Sigma_{e}\left(T_{\sigma}\right)=\\{0\\}$. Therefore, by Atkinson’s theorem
(see [1]), the spectrum of $\left[T_{\sigma}\right]$ in the Calkin algebra
$B\left(L^{2}\left(\mathbb{T}\right)\right)/K\left(L^{2}\left(\mathbb{T}\right)\right)$
is

$\Sigma\left(\left[T_{\sigma}\right]\right)=\\{0\\}.$

This implies that

$\left\|\left[T_{\sigma}\right]\right\|_{C}=r\left(\left[T_{\sigma}\right]\right)=0.$

Hence, it follows that

$\left[T_{\sigma}\right]=0.$

Therefore $T_{\sigma}$ is compact.
Now, to prove the converse part, assume that $d\neq 0$, then we need to show
that $T_{\sigma}$ is not compact on $L^{2}\left(\mathbb{T}\right)$. Suppose
that $T_{\sigma}$ is compact, then putting $K=T_{\sigma}$ in (4.1) will
contradict our assumption that $d\neq 0$. This completes the proof of the
theorem. $\Box$

Now our aim is to study the Gohberg’s lemma and characterization of compact
operators on $\ell^{2}(\mathbb{Z})$ with symbol in
$M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right)$ class (defined
in [23]). The main ingredient is the relation between the weighted periodic
and discrete symbols which can be found in [23].

######  Theorem 4.8.

Let $\sigma:\mathbb{Z}\times\mathbb{T}\rightarrow\mathbb{C}$ be a measurable
function such that the pseudo-differential operator
$T_{\sigma}:\ell^{2}\left(\mathbb{Z}\right)\rightarrow\ell^{2}\left(\mathbb{Z}\right)$
is a bounded linear operator. If we define
$\tau:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$ by

$\tau(x,k)=\overline{\sigma(-k,x)},\quad x\in\mathbb{T},k\in\mathbb{Z},$

then

$\displaystyle
T_{\sigma}=\mathcal{F}_{\mathbb{Z}}^{-1}T_{\tau}^{*}\mathcal{F}_{\mathbb{Z}},$
(4.8)

where $T_{\tau}^{*}$ is the adjoint of $T_{\tau}$. We also have

$\displaystyle
T_{\tau}=\mathcal{F}_{\mathbb{Z}}T_{\sigma}^{*}\mathcal{F}_{\mathbb{Z}}^{-1},$
(4.9)

where $T_{\sigma}^{*}$ is the adjoint of $T_{\sigma}$.

As a corollary of Theorem 4.5 and Theorem 4.8, we obtain the following
estimates for the distance between a given operator and the space of compact
operators on $\ell^{2}(\mathbb{Z}).$

######  Corollary 4.9.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right)$.
Then for all compact operators $K$ on $\ell^{2}\left(\mathbb{Z}\right)$,

$\displaystyle\left\|T_{\sigma}-K\right\|_{**}\geq d,$ (4.10)

where

$d=\limsup_{|k|\rightarrow\infty}\left\\{\sup_{x\in[-\pi,\pi]}|\sigma(k,x)|\right\\}.$

Here $\|\cdot\|_{**}$ denotes the norm in the $C^{*}$-algebra of all bounded
linear operators on $\ell^{2}(\mathbb{Z})$.

Proof  We define $\tau:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$ by

$\tau(x,k)=\overline{\sigma(-k,x)},\quad x\in\mathbb{T},k\in\mathbb{Z}.$

Since $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right)$,
we have $\tau\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Then, by Theorem 4.5, we have the following estimate for all compact operators
$K^{\prime}$ on $L^{2}\left(\mathbb{T}\right):$

$\displaystyle\left\|T_{\tau}-K^{\prime}\right\|_{*}\geq d.$ (4.11)

We need to show that

$\displaystyle\left\|T_{\sigma}-K\right\|_{**}\geq d,$ (4.12)

for all compact operators $K$ on $\ell^{2}\left(\mathbb{Z}\right).$ Let
$K:\ell^{2}\left(\mathbb{Z}\right)\rightarrow\ell^{2}\left(\mathbb{Z}\right)$
be any arbitrary compact operator. This implies that
$K^{*}:\ell^{2}\left(\mathbb{Z}\right)\rightarrow\ell^{2}\left(\mathbb{Z}\right)$
is also a compact operator. Then
$K_{1}=\mathcal{F}_{\mathbb{Z}}K^{*}\mathcal{F}_{\mathbb{Z}}^{-1}:L^{2}(\mathbb{T})\rightarrow
L^{2}(\mathbb{T})$ is a compact operator. Hence, by (4.9) and (4.11), we have

$\displaystyle\hskip
28.1pt\left\|T_{\tau}-K_{1}\right\|_{L^{2}(\mathbb{T})}\geq d,$
$\displaystyle\implies\left\|\mathcal{F}_{\mathbb{Z}}T_{\sigma}^{*}\mathcal{F}_{\mathbb{Z}}^{-1}-\mathcal{F}_{\mathbb{Z}}K^{*}\mathcal{F}_{\mathbb{Z}}^{-1}\right\|_{L^{2}(\mathbb{T})}\geq
d,$
$\displaystyle\implies\left\|\mathcal{F}_{\mathbb{Z}}(T_{\sigma}-K)^{*}\mathcal{F}_{\mathbb{Z}}^{-1}\right\|_{L^{2}(\mathbb{T})}\geq
d,$
$\displaystyle\implies\left\|(T_{\sigma}-K)^{*}\right\|_{\ell^{2}(\mathbb{Z})}\geq
d,$
$\displaystyle\implies\left\|T_{\sigma}-K\right\|_{\ell^{2}(\mathbb{Z})}\geq
d,$

and this completes the proof of the estimate (4.12). $\Box$

The following corollary gives us a necessary and sufficient condition for an
operator to be compact on $\ell^{2}(\mathbb{Z})$ for symbol class
$M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right).$

######  Corollary 4.10.

Let $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right)$.
Then $T_{\sigma}$ is a compact operator on $\ell^{2}\left(\mathbb{Z}\right)$
if and only if $d=0$, where

$d=\limsup_{|k|\rightarrow\infty}\left\\{\sup_{x\in[-\pi,\pi]}|\sigma(k,x)|\right\\}.$

Proof  First, let us assume that $d=0.$ Define

$\tau(x,k)=\overline{\sigma(-k,x)},\quad x\in\mathbb{T},k\in\mathbb{Z}.$

Since $\sigma\in M_{\rho,\Lambda}^{0}\left(\mathbb{Z}\times\mathbb{T}\right)$,
we have $\tau\in M_{\rho,\Lambda}^{0}\left(\mathbb{T}\times\mathbb{Z}\right)$.
Hence, by Theorem 4.7, $T_{\sigma}$ is compact on
$L^{2}\left(\mathbb{T}\right)$. This implies that $T_{\sigma}^{*}$ is a
compact operator on $L^{2}\left(\mathbb{T}\right)$. Hence, by (4.8),
$T_{\sigma}$ is a compact operator on $\ell^{2}\left(\mathbb{Z}\right)$.
Now, to prove the converse part, assume that $d\neq 0$, then we need to show
that $T_{\sigma}$ is not compact on $\ell^{2}\left(\mathbb{Z}\right)$. Suppose
that $T_{\sigma}$ is compact, then putting $K=T_{\sigma}$ in (4.10) will
contradict our assumption that $d\neq 0$. This completes the proof of the
theorem. $\Box$

##  5\. Gårding’s and sharp Gårding’s inequalities on $\mathbb{T}$ and
$\mathbb{Z}$

The main aim of this section is to prove Gårding’s and sharp Gårding’s
inequalities for $M$-elliptic operators on $\mathbb{T}$ and $\mathbb{Z}$,
respectively. First, we state the Gårding’s inequality for $M$-elliptic
operators on $\mathbb{T}$, which is analogous to the [3, Corollary 5.7]. The
proof can be done in similar lines, so we skip the proof here.

######  Theorem 5.1.

(Gårding’s inequality for $M$-elliptic operators on $\mathbb{T}$)
Let $m>0$. Let $\sigma\in
M_{\rho,\Lambda}^{2m}\left(\mathbb{T}\times\mathbb{Z}\right)$ be elliptic such
that $\sigma(x,k)\geq 0$ for all $x$ and co-finitely many $k$. Then there
exist $C_{0},C_{1}>0$ such that for all $f\in
H_{\Lambda}^{m,2}\left(\mathbb{T}\right)$, we have

$\displaystyle\operatorname{Re}(T_{\sigma}f,f)_{L^{2}\left(\mathbb{T}\right)}\geq
C_{0}\|f\|_{H_{\Lambda}^{m,2}\left(\mathbb{T}\right)}^{2}-C_{1}\|f\|_{L^{2}\left(\mathbb{T}\right)}^{2}.$
(5.1)

Now, we will show that Theorem 5.1 implies the corresponding Gårding
inequality for $M$-elliptic operators on $\mathbb{Z}$. As there is no
regularity concept on the lattice, the statement is given in terms of weighted
$\ell^{2}$-spaces. For this, we need the following definition:

######  Definition 5.2.

For $s\in\mathbb{R}$, let us define the weighted space
$\ell_{s,\Lambda}^{2}\left(\mathbb{Z}\right)$ as the space of all
$f:\mathbb{Z}\rightarrow\mathbb{C}$ such that

$\|f\|_{\ell_{s,\Lambda}^{2}\left(\mathbb{Z}\right)}:=\left(\sum_{k\in\mathbb{Z}}\Lambda(k)^{2s}|f(k)|^{2}\right)^{1/2}<\infty.$

We observe that the symbol $\sigma_{s}(k)=\Lambda(k)^{s}$ belongs to
$M_{\rho,\Lambda}^{s}\left(\mathbb{Z}\times\mathbb{T}\right)$, and
$f\in\ell_{s,\Lambda}^{2}\left(\mathbb{Z}\right)$ if and only if
$T_{\sigma_{s}}f\in\ell^{2}\left(\mathbb{Z}\right)$.

######  Theorem 5.3.

(Gårding’s inequality for $M$-elliptic operators on $\mathbb{Z}$)
Let $m>0$. Let $\sigma\in
M_{\rho,\Lambda}^{2m}\left(\mathbb{Z}\times\mathbb{T}\right)$ be elliptic such
that $\sigma(k,x)\geq 0$ for all $x$ and co-finitely many $k$. Then there
exist $C_{1},C_{2}>0$ such that for all
$f\in\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)$, we have

$\operatorname{Re}(T_{\sigma}f,f)_{\ell^{2}\left(\mathbb{Z}\right)}\geq
C_{0}\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}^{2}-C_{1}\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}^{2}.$

Proof  Let $\tau(x,k)=\overline{\sigma(-k,x)}$. Then by Theorem 4.8, we have

$T_{\sigma}=\mathcal{F}_{\mathbb{Z}}^{-1}T_{\tau}^{*}\mathcal{F}_{\mathbb{Z}},$

and if $\sigma$ is elliptic on $\mathbb{Z}\times\mathbb{T}$, then $\tau$ is
elliptic on $\mathbb{T}\times\mathbb{Z}$. Also, if $\sigma\geq 0$, then
$\tau\geq 0$. Then by Theorem 5.1, for all $g\in
H_{\Lambda}^{m,2}\left(\mathbb{T}\right)$, we have

$\displaystyle\operatorname{Re}\left(T_{\tau}^{*}g,g\right)_{L^{2}\left(\mathbb{T}\right)}=\operatorname{Re}\left(T_{\tau}g,g\right)_{L^{2}\left(\mathbb{T}\right)}\geq
C_{0}\|g\|_{H_{\Lambda}^{m,2}\left(\mathbb{T}\right)}^{2}-C_{1}\|g\|_{L^{2}\left(\mathbb{T}\right)}^{2}.$
(5.2)

Let $f\in\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)$ and
$g=\mathcal{F}_{\mathbb{Z}}f$. Then $g\in
H_{\Lambda}^{m,2}\left(\mathbb{T}\right)$, and

$\displaystyle\|g\|_{H_{\Lambda}^{m,2}\left(\mathbb{T}\right)}=\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}\quad\text{
and
}\quad\|g\|_{L^{2}\left(\mathbb{T}\right)}=\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}.$
(5.3)

Now by Theorem 4.8,

$T_{\sigma}f=\mathcal{F}_{\mathbb{Z}}^{-1}\circ
T_{\tau}^{*}\circ\mathcal{F}_{\mathbb{Z}}f=\mathcal{F}_{\mathbb{Z}}^{-1}\circ
T_{\tau}^{*}g,$

so that $\mathcal{F}_{\mathbb{Z}}T_{\sigma}f=T_{\tau}^{*}g.$ Substituting
(5.3) into (5.2), we get

$\displaystyle\operatorname{Re}\left(T_{\tau}^{*}g,g\right)_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq
C_{0}\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}^{2}-C_{1}\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}^{2},$
$\displaystyle\operatorname{Re}\left(\mathcal{F}_{\mathbb{Z}}T_{\sigma}f,\mathcal{F}_{\mathbb{Z}}f\right)_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq
C_{0}\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}^{2}-C_{1}\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}^{2},$
$\displaystyle\operatorname{Re}\left(\mathcal{F}_{\mathbb{Z}}^{*}\mathcal{F}_{\mathbb{Z}}T_{\sigma}f,f\right)_{\ell^{2}\left(\mathbb{Z}\right)}$
$\displaystyle\geq
C_{0}\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}^{2}-C_{1}\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}^{2},\quad\text{
since }\mathcal{F}_{\mathbb{Z}}^{*}\mathcal{F}_{\mathbb{Z}}=Id,$
$\displaystyle\operatorname{Re}(T_{\sigma}f,f)_{\ell^{2}\left(\mathbb{Z}\right)}$
$\displaystyle\geq
C_{0}\|f\|_{\ell_{m,\Lambda}^{2}\left(\mathbb{Z}\right)}^{2}-C_{1}\|f\|_{\ell^{2}\left(\mathbb{Z}\right)}^{2},$

and this completes the proof. $\Box$

We now proceed to prove the sharp Gårding inequality for $M$-elliptic
operators on $\mathbb{T}$ and $\mathbb{Z}$, respectively. For this, first we
will state the sharp Gårding’s inequality for $M$-elliptic operators on
$\mathbb{T}$ without proof as it follows similar lines to the proof of [3,
Corollary 5.9].

######  Theorem 5.4.

(Sharp Gårding’s inequality for $M$-elliptic operators on $\mathbb{T}$)
Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right)$ be
such that $\sigma(x,k)\geq 0$, for all $(x,k)\in\mathbb{T}\times\mathbb{Z}$.
Then there exists a positive constant $C$ such that for all $f\in
H_{\Lambda}^{\frac{m-1}{2},2}\left(\mathbb{T}\right)$, we have

$\operatorname{Re}(T_{\sigma}f,f)_{L^{2}\left(\mathbb{T}\right)}\geq-C\|f\|_{H_{\Lambda}^{\frac{m-1}{2},2}\left(\mathbb{T}\right)}^{2}.$

In the following result, we prove that Theorem 5.4 implies the corresponding
sharp Gårding’s inequality for $M$-elliptic operators on $\mathbb{Z}$.

######  Theorem 5.5.

(Sharp Gårding’s inequality for $M$-elliptic operators on $\mathbb{Z}$)
Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{Z}\times\mathbb{T}\right)$ be
such that $\sigma(k,x)\geq 0$ for all $(k,x)\in\mathbb{Z}\times\mathbb{T}$.
Then there exists a positive constant $C$ such that for all
$f\in\ell_{\frac{m-1}{2},\Lambda}^{2}\left(\mathbb{Z}\right)$, we have

$\operatorname{Re}(T_{\sigma}f,f)_{\ell^{2}\left(\mathbb{Z}\right)}\geq-C\|f\|_{\ell_{\frac{m-1}{2},\Lambda}^{2}\left(\mathbb{Z}\right)}.$

Proof  Let $\tau(x,k)=\overline{\sigma(-k,x)}$. Then by Theorem 4.8, we have

$T_{\sigma}=\mathcal{F}_{\mathbb{Z}}^{-1}T_{\tau}^{*}\mathcal{F}_{\mathbb{Z}}.$

Using the same argument and notation as in the proof of Theorem 5.3, and by
Theorem 5.4, we get

$\displaystyle\operatorname{Re}(T_{\sigma}f,f)_{\ell^{2}\left(\mathbb{Z}\right)}$
$\displaystyle=\operatorname{Re}\left(\mathcal{F}_{\mathbb{Z}}^{-1}T_{\tau}^{*}g,\mathcal{F}_{\mathbb{Z}}^{-1}g\right)_{\ell^{2}\left(\mathbb{Z}\right)}$
$\displaystyle=\operatorname{Re}\left(T_{\tau}^{*}g,g\right)_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle=\operatorname{Re}\left(T_{\tau}g,g\right)_{L^{2}\left(\mathbb{T}\right)}$
$\displaystyle\geq-C\|g\|_{H_{\Lambda}^{\frac{m-1}{2},2}\left(\mathbb{T}\right)}^{2}$
$\displaystyle=-C\|f\|_{\ell_{\frac{m-1}{2},\Lambda}^{2}\left(\mathbb{Z}\right)},$

and this completes the proof of the theorem. $\Box$

##  6\. Applications

In this section, we present an application of Gårding’s inequality for the
class $M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right).$ Let
$T_{\sigma,0}$ and $T_{\sigma,1}$ are the minimal and maximal pseudo
differential operator of $T_{\sigma}$ on $L^{2}(\mathbb{T})$ defined as in
[21]. First, we recall the following definitions about strongly elliptic
symbols and strong solutions.

######  Definition 6.1.

Let $\sigma\in M_{\rho,\Lambda}^{m}\left(\mathbb{T}\times\mathbb{Z}\right),$
$m\in\mathbb{R}$. Then $\sigma$ is said to be strongly $M$-elliptic if there
exist positive constants $C$ and $R$ for which

$\operatorname{Re}(\sigma(x,k))\geq C\,\Lambda(k)^{m},\quad|k|\geq R.$

######  Definition 6.2.

Let $\sigma\in
M_{\rho,\Lambda}^{2m}\left(\mathbb{T}\times\mathbb{Z}\right),m>0$, and let
$f\in L^{p}\left(\mathbb{T}\right),1<p<\infty$. A function $u\in
L^{p}\left(\mathbb{T}\right)$ is a strong solution of the equation
$T_{\sigma}u=f$ if $u\in\mathcal{D}\left(T_{\sigma,0}\right)$ and
$T_{\sigma,0}u=f$.

Proceeding similarly as in Theorem 18.2 of [37], we have the following result
related to the strong solution of the equation $T_{\sigma}u=f$ on
$\mathbb{T}$.

######  Lemma 6.3.

Let $\sigma\in
M_{\rho,\Lambda}^{2m}\left(\mathbb{T}\times\mathbb{Z}\right),m>0$, be an
$M$-elliptic symbol such that

$\displaystyle\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)\geq
C\|\varphi\|_{m,2,\Lambda}^{2},\quad\varphi\in H_{\Lambda}^{m,2},$ (6.1)

for some positive constant $C$. Then for every function $f$ in
$L^{2}\left(\mathbb{T}\right)$, the pseudo-differential equation
$T_{\sigma}u=f$ has a unique strong solution $u$ in
$L^{2}\left(\mathbb{T}\right)$.

In the next result, we provide sufficient conditions for the existence and
uniqueness of strong solutions in $L^{2}(\mathbb{T})$ for the pseudo-
differential operator $T_{\sigma}$ with strongly elliptic symbol.

######  Theorem 6.4.

Let $\sigma\in
M_{\rho,\Lambda}^{2m}\left(\mathbb{T}\times\mathbb{Z}\right),m>0$, be a
strongly elliptic symbol. Then for all $f$ in $L^{2}\left(\mathbb{T}\right)$
there exists a real number $\lambda_{0}$ such that for all
$\lambda\geq\lambda_{0}$, the pseudo-differential equation
$\left(T_{\sigma}+\lambda I\right)u=f$ on $\mathbb{T}$ has a unique strong
solution $u$ in $L^{2}\left(\mathbb{T}\right)$, where $I$ is the identity
operator on $L^{2}\left(\mathbb{T}\right)$.

Proof  From Gårding’s inequality (5.1), there exist constants $A>0$ and
$\lambda_{0}$ such that

$\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)\geq
A\|\varphi\|_{m,2,\Lambda}^{2}-\lambda_{0}\|\varphi\|_{0,2,\Lambda}^{2},\quad\varphi\in
H_{\Lambda}^{m,2}.$

Now for any $\lambda\geq\lambda_{0}$, we get

$\displaystyle\operatorname{Re}\left(\left(T_{\sigma}+\lambda
I\right)\varphi,\varphi\right)$
$\displaystyle=\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)+\left(\lambda-\lambda_{0}\right)\|\varphi\|_{2}^{2}-\lambda_{0}\|\varphi\|_{2}^{2}$
$\displaystyle\geq
A\|\varphi\|_{m,2,\Lambda}^{2}+\left(\lambda-\lambda_{0}\right)\|\varphi\|_{2}^{2}\geq
A\|\varphi\|_{m,2,\Lambda}^{2}.$

This shows that $T_{\sigma}$ satisfies the condition (6.1). Thus by Lemma 6.3,
the pseudo-differential equation $\left(T_{\sigma}+\lambda I\right)u=f$ has a
unique strong solution $u$ in $L^{2}\left(\mathbb{T}\right)$ and this
completes the proof. $\Box$

## References

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