\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 11, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/11\hfil On the essential spectra]
{On the essential spectra of matrix operators and applications}

\author[M. Damak, A. Jeribi\hfil EJDE-2007/11\hfilneg]
{Mondher Damak, Aref Jeribi}  % in alphabetical order

\address{Mondher Damak \newline
D\'epartement de Math\'ematiques \\
 Universit\'e de Sfax \\
Facult\'e des Sciences de Sfax \\
Route de Soukra, Km 3.5, B.P. 802 \\
 3018  Sfax, Tunisie}
\email{Mondher.Damak@fss.rnu.tn}

\address{Aref Jeribi \newline
D\'epartement de Math\'ematiques \\
 Universit\'e de Sfax \\
Facult\'e des Sciences de Sfax \\
Route de Soukra, Km 3.5, B.P. 802 \\
 3018  Sfax, Tunisie}
\email{Aref.Jeribi@fss.rnu.tn}

\thanks{Submitted October 3, 2006. Published January 8, 2007.}
\subjclass[2000]{34L40, 47A11, 47A53, 47B50}
\keywords{Matrix operators; essential spectra; Fredholm perturbation}

\begin{abstract}
 In this paper, we investigate the essential spectra of some
 matrix operators on Banach spaces. The  results obtained
 are used for describing the essential spectra of differential
 operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

 This paper is devoted to the study of the essential spectra of
$2\times 2$ matrix operators of the form
\[
L_0=\begin{pmatrix}
A&B\\
C&D \end{pmatrix}
\]
considered on the product Banach space $X:=X_1\times X_2$.
In general, the operators
occurring in the representation $L_0$ are unbounded.  The operator
$A$ acts on the Banach space $X_1$ and has the domain
$\mathcal{D}(A)$, $D$ is defined on $\mathcal{D}(D)$ and acts on the Banach
space $X_2$, and the intertwining operator $B$ (resp. $C$) is
defined on the domain $\mathcal{D}(B)$ (resp. $\mathcal{D}(C)$) and acts
from $X_2$ to $X_1$ (resp. from $X_1$ to $X_2$). Below, we shall
assume that $\mathcal{D}(A)\subset \mathcal{D}(C)$ and
$\mathcal{D}(B)\subset \mathcal{D}(D)$, and then the matrix operator $L_0$
defines a linear operator in $X$ with domain $\mathcal{D}(A)\times
\mathcal{D}(B)$.

One of the problems in the study
of such operators is that in general $L_0$ is not closed or even
closable, even if its entries are closed. Important results
concerning the spectral theory of this type of operators have been
obtained during the last years. One of these works, the paper by
Atkinson and all \cite{Atkinson} concerns the essential spectrum
of such an operator. First, they give some sufficient conditions
under which $L_0$ is closable and describe its closure which we
shall denote by $L$. Second, they study the Wolf essential
spectrum of a matrix differential operator in the case where the
operators are defined on a bounded domain
$\Omega\subset \mathbb{R}^n$.

 The first main purpose of this paper is
in a generalization of the results of \cite{Atkinson} on the
essential spectrum of the closure $L$ to the case where the
assumptions of \cite{Atkinson} concerning the resolvent of the
entry $A$ are weakened. In \cite{Atkinson} it is assumed that the
resolvent $(A-\lambda I)^{-{1}}$ for some (and hence for all)
$\lambda$ in the resolvent set $\rho(A)$ of $A$ is a compact
operator on $X_1$; whereas in our paper we assume that only
$(A-\lambda I)^{-{1}}$, $\lambda\in \rho(A)$ belongs to a
two-sided closed ideal $\mathcal{I}(X_1)\subset \mathcal{F}(X_1)$ of
$\mathcal{L}(X_1)$ where $\mathcal{F}(X_1)$ is the set of Fredholm
perturbation on $X_1$ (see Definition \ref{def2.1}) and $\mathcal{L}(X_1)$
denotes the Banach algebra of all bounded linear operators on
$X_1$. We shall show under certain additional assumptions that the
study of the essential spectrum of $L$ will be reduced to that of
the Schur complement.
\[
D-\overline{C(A-\mu
I)^{-1}B}
\]
for some $\lambda\in \rho(A)$, where
$\overline{C(A-\mu I)^{-1}B}$ denotes the closure
of the operator $C(A-\mu I)^{-1}B$. Note that the condition made
on the resolvent of $A$ in \cite{Atkinson} fails if $A$ is an
elliptic operator on a domain of infinite measure. Then, this
condition is more restrictive in the applications, and the class
of entries of the matrix $L_0$ that we consider here is
essentially more general than in \cite{Atkinson}.

There are several, and in general, non-equivalent definitions
of the essential spectrum of a closed operator on a Banach space.
Through all this paper we are concerned with six of them (see
Section 2). The second main purpose of this paper is to study and
characterize the essential spectrum of $L$ in all cases. In
particular, the Wolf essential spectrum studied in \cite{Atkinson}
is included. Then, the aim of this work is to pursue the analysis
of the Wolf essential spectrum started in \cite{Atkinson}. Indeed,
we extend the results obtained in \cite{Atkinson} to a large class
of operators and at the same time to the six essential spectra.
Therefore the results obtained in \cite{Atkinson} turn out to be a
particular case of the results proven in this paper.

Let us conclude this introduction with some historical comments and some
bibliographical references, which do not intend to be complete. The
problem of the essential spectrum of differential operators of mixed
order, which appears in mathematical physics was studied by many
authors. Among such works we can quote for example
\cite{Kopachevskii, Malyshev, Lifschitz, Adam, Goedbloed}. The
particular case of symmetric block differential operators with
Dirichlet boundary conditions for $A$, the essential spectrum was
studied in \cite{Adam, Goedbloed}. An example from
magnetohydrodynamics can be found in Section 5 in
\cite{Atkinson}.

  For  Agmon,  Douglis and Nirenberg elliptic system \cite{Agmon},
the most general results
were obtained by  Grubb and Geymonat \cite{Grubb}. We recall,
the abstract model $L_0$ was introduced in \cite{Atkinson}. This
model clarifies the essence of the problem and allows us to uncover
details that have not been noticed before, even for concrete
problems, for example, the condition that the matrix operators admit
a closure and the description of the domain of this closure.
Recently, in  Kurasov and  Nabako \cite{Kurasov}, it has been
proven that the essential spectrum of self-adjoint operator
associated with matrix differential operator appearing in problems
of magnetic hydrodynamics, consists of two branches. The first one
is called regularity spectrum and the second branch is called
singularity spectrum which appears due to singularity of the
coefficients.



 Our paper is organised as follows: In Section 2, we
introduce the algebraic framework in which our investigation will be
done. The analysis is based on the concept of Fredholm
perturbations. In Section 3, we introduce a general hypotheses on
different entries of the operators matrix $L$. In Section 4, we
investigate some results concerning essential spectra of $L$. The
main results of this section are Theorems \ref{thm4.1} and \ref{thm4.2}
 which contain
a general description of different types of the essential spectra of
the operator $L$. In the end of Section 4 we give some sufficient
conditions to verify our hypothesis. Finally, in Section 5 we apply
the results obtained in Sections 4 to study the essential spectra of
an example where $A$, $B$ and $C$ are ordinary differential
operators on spaces of vector functions and $D$ is a multiplication
operator.

\section{Preliminary Results}

Let $X$ and $Y$ be Banach spaces and let $A$ be an operator from
$X$ into $Y$. We denote by $\mathcal{D}(A)\subset X$ its domain and
$R(A)\subset Y$ its range. We denote by $\mathcal{C}(X,Y)$ (resp.
$\mathcal{L}(X,Y)$) the set of all closed, densely defined linear
operators (resp. the Banach algebra of all bounded linear
operators) from $X$ into $Y$. For $A\in \mathcal{C}(X,Y)$, by
$\sigma(A)$, $\rho(A)$ and $N(A)$ we denote the spectrum, the
resolvent set and the null space of $A$ respectively. The nullity,
$\alpha(A)$, of $A$ is defined as the dimension of $N(A)$ and the
deficiency, $\beta(A)$, of $A$ is defined as the codimension of
$R(A)$ in $Y$. The set of upper semi-Fredholm operators from $X$
into $Y$ is defined by
\[
\Phi_{+}(X,Y)=\{A\in \mathcal{C}(X,Y) \hbox{s uch
that } \alpha(A)<\infty  \hbox{ and } R(A) \hbox{ is closed in } Y
\},
\]
the set of lower semi-Fredholm operators from $X$ into $Y$ is defined by
 \[
\Phi_{-}(X,Y)=\{A\in \mathcal{C}(X,Y): \beta(A)<\infty  \hbox{ and } R(A) \hbox{ is closed in } Y\},
\]
 the set of semi-Fredholm operators from $X$ into $Y$ is defined by
\[
\Phi_{\pm}(X,Y)=\Phi_{+}(X,Y)\cup \Phi_{-}(X,Y),
\]
 the set of Fredholm
operators from $X$ into $Y$ is defined by
 \[
\Phi(X,Y)=\Phi_{+}(X,Y)\cap \Phi_{-}(X,Y),
\]
the set of bounded Fredholm operators from
$X$ into $Y$ is defined by
\[
\Phi^{b}(X,Y)=\Phi(X,Y)\cap \mathcal{
L}(X,Y),
\]
 and the set $\Phi_A$ is defined by
 \[
\Phi_A=\{\lambda\in \mathbb{C}:
\lambda-A\in \Phi(X,Y)\}.
 \]
If $A\in\Phi(X,Y)$, the number $i(A)=\alpha(A)-\beta(A)$ is called
the index of $A$. The subset of all compact operators of
$\mathcal{L}(X,Y)$ is denoted by $\mathcal{K}(X,Y)$.
If $X=Y$ then $\mathcal{L}(X,Y)$, $\mathcal{K}(X,Y)$,
 $\mathcal{C}(X,Y)$, $\Phi_{+}(X,Y)$,
$\Phi_{-}(X,Y)$, $\Phi_{\pm}(X,Y)$, $\Phi(X,Y)$ and
$\Phi^{b}(X,Y)$ are replaced, respectively, by $\mathcal{L}(X)$,
$\mathcal{K}(X)$, $\mathcal{C}(X)$, $\Phi_{+}(X)$, $\Phi_{-}(X)$,
$\Phi_{\pm}(X)$, $\Phi(X)$ and $\Phi^{b}(X)$.

\begin{definition} \label{def2.1} \rm
Let $X$ and $Y$ be two Banach
spaces and let $F \in \mathcal{L}(X,Y)$. $F$ is called a Fredholm
perturbation if $U+F\in\Phi^b(X,Y)$ whenever $U\in
\Phi^b(X,Y)$.
\end{definition}

The set of Fredholm perturbations is denoted by
$\mathcal{F}^b(X,Y)$. This class of operators is introduced and investigated
in \cite{Gohberg1}. In particular, it is shown that
$\mathcal{F}^b(X,Y)$ is a closed subset of $\mathcal{L}(X,Y)$ and if $X=Y$,
then $\mathcal{F}^b(X):=\mathcal{F}^b(X,X)$ is a closed two-sided ideal
of $\mathcal{L}(X)$.

\begin{proposition}[{\cite[pp. 69-70]{Gohberg1}}] \label{prop2.1}
 Let $X$, $Y$, $Z$ be Banach spaces. If at least one
of the sets $\Phi^b(X,Y)$ or $\Phi^b(Y,Z)$ is not empty, then
\begin{itemize}
\item[(i)] $F\in \mathcal{F}^b(X,Y)$,
$A\in\mathcal{L}(Y,Z)$ imply $AF\in \mathcal{F}^b(X,Z)$.

\item[(ii)]
$F\in \mathcal{F}^b(Y,Z)$, $A\in\mathcal{L}(X,Y)$ imply
$FA\in \mathcal{F}^b(X,Z)$.
\end{itemize}
\end{proposition}

\begin{definition} \label{def2.2} \rm
Let $X$ be a Banach space and $R \in \mathcal{L}(X)$.
$R$ is said to be a Riesz operator if $\Phi_R=\mathbb{C}\setminus\{0\}$.
\end{definition}

 For further information on the family of Riesz operators
we refer to \cite{Caradus, Kaashoek} and the references
therein.


\begin{remark} \label{rmk2.1} \rm
(i) In \cite{Schechter4}, it is
proved that $\mathcal{F}^b(X)$ is the largest ideal of $\mathcal{L}(X)$
contained in the family of Riesz operators.

(ii) Let $X$ and $Y$ be two Banach
spaces. If in Definition \ref{def2.1} we replace $\Phi^b(X,Y)$ by
$\Phi(X,Y)$ we obtain the set $\mathcal{F}(X,Y)$.
\end{remark}

 \begin{definition} \label{def2.3} \rm
Let $X$ and $Y$ be two Banach spaces and let $F \in \mathcal{L}(X,Y)$.
Then $F$ is called an upper (resp. lower) Fredholm perturbation if
$U+F\in\Phi_+^b(X,Y):=\Phi_+(X,Y)\cap\mathcal{L}(X,Y)$ (resp.
$\Phi_-^b(X,Y):=\Phi_-(X,Y)\cap\mathcal{L}(X,Y)$) whenever $U\in
\Phi_+^b(X,Y)$ (resp. $\Phi_-^b(X,Y)$).
\end{definition}

 The sets of upper semi-Fredholm and lower semi-Fredholm
perturbations are denoted by $\mathcal{F}^b_{+}(X,Y)$ and
$\mathcal{F}^b_{-}(X,Y)$, respectively. In \cite{Gohberg}, it is shown that
$\mathcal{F}^b_{+}(X,Y)$ and $\mathcal{F}^b_{-}(X,Y)$ are closed subsets
of $\mathcal{L}(X,Y)$, and if $X=Y$, then
$\mathcal{F}^b_{+}(X):=\mathcal{F}^b_{+}(X,X)$ is a closed two-sided
ideal of $\mathcal{L}(X)$.

 \begin{remark} \label{rmk2.2} \rm
Let $X$ and $Y$ be two
Banach spaces. If in Definition \ref{def2.3} we replace $\Phi^b_+(X,Y)$
(resp. $\Phi^b_-(X,Y)$) by $\Phi_+(X,Y)$ (resp. $\Phi_-(X,Y)$) we
obtain the set $\mathcal{F}_+(X,Y)$ (resp. $\mathcal{F}_-(X,Y)$).
\end{remark}

 \begin{definition} \label{def2.4} \rm
 An operator   $A  \in  \mathcal{L}(X,Y)$ is said to be weakly
compact if $A(B)$ is relatively
weakly compact in $Y$ for every bounded subset $B\subset X$.
\end{definition}

The family of weakly compact operators from $X$ into $Y$ is denoted by
$\mathcal{W}(X,Y)$. If $X=Y$, the family of weakly compact operators
on $X$, $\mathcal{W}(X):=\mathcal{W}(X,X)$, is a closed two-sided ideal
of $\mathcal{L}(X)$ containing $\mathcal{K}(X)$
(cf. \cite{Dunford2,Goldberg}).


\begin{definition} \label{def2.5} \rm
 Let $X$ and $Y$ be two Banach spaces.    An operator
$A  \in \mathcal{L}(X,Y)$ is   called   strictly singular if, for every
infinite-dimensional subspace $M$, the restriction of $A$ to $M$ is
not a homeomorphism.
\end{definition}

 Let $\mathcal{S}(X,Y)$ denote the set of strictly
singular operators from $X$ into $Y$.
The concept of strictly singular  operators was introduced in  the
pioneering paper by  Kato \cite{Kato1} as a generalization
of the notion of compact operators. For a detailed study of the
properties of strictly singular operators we refer to
\cite{Goldberg, Kato1}. For our own use, let us recall the following
four facts. The set $\mathcal{S}(X,Y)$ is a closed subspace of $\mathcal{
L}(X,Y)$, if $X=Y$, $\mathcal{S}(X):=\mathcal{S}(X,X)$ is        a closed
two-sided ideal of $\mathcal{L}(X)$ containing         $\mathcal{
K}(X)$. If $X$ is a Hilbert space then  $\mathcal{K}(X)=\mathcal{S}(X)$.
The class of weakly compact operators on $L_{1}$-spaces (resp.
$C(K)$-spaces with $K$ a compact Haussdorff space) is nothing else
but the family of strictly singular operators on $L_{1}$-spaces
(resp. $C(K)$-spaces) (see \cite[Theorem 1]{Pelczynski}).

 Let $X$ be a Banach space. If $N$ is a closed
subspace of $X$, we denote by $\pi_{N}^{X}$ the quotient map $X\to
X/N$. The codimension of $N$, $\mathop{\rm codim}(N$), is defined as the
dimension of the vector space $X/N$.

  \begin{definition} \label{def2.6} \rm
 Let $X$ and $Y$ be two Banach spaces and
$S \in \mathcal{L}(X,Y)$. $S$ is said to be strictly cosingular
operator from $X$ into $Y$, if there exists no closed subspace $N$
of $Y$ with $\mathop{\rm codim}(N)=\infty$ such that
$\pi_N^{Y} S:X\to Y/N$ is surjective.
\end{definition}

Let $C\mathcal{S}(X,Y)$ denote the set of strictly cosingular operators
from $X$ into $Y$. This class of operators was introduced by
Pelczynski \cite{Pelczynski}. It forms a closed subspace of $\mathcal{
L}(X,Y)$ which is, $C\mathcal{S}(X):=C\mathcal{S}(X,X)$, a closed
two-sided ideal of $\mathcal{L}(X)$ if $X=Y$ (cf. \cite{Vladimirskii}).

 \begin{definition} \label{def2.7}\rm
 A Banach space $X$ is said to have the
Dunford-Pettis property (for short property DP) if for each Banach
space $Y$ every weakly compact operator $T:X\to Y$ takes
weakly compact sets in $X$ into norm compact sets of $Y$.
\end{definition}

It is well known that any $L_{1}$
space has the DP property \cite{Dunford1}. Also, if $\Omega$ is a
compact Hausdorff space, $C(\Omega)$ has the DP property
\cite{Grothendieck}. For further examples we refer to \cite{Diestel}
or \cite[p. 494, 497, 508, 511]{Dunford2}. Note that the DP
property is not preserved under conjugation. However, if $X$ is a
Banach space whose dual has the DP property then $X$ has the DP
property (see \cite{Grothendieck}). For more information we
refer to the paper by Diestel \cite{Diestel} which contains a survey
and exposition of the Dunford-Pettis property and related
topics.


The following identity was established in
\cite[Lemma 2.3 (ii)]{Latrach7}.

\begin{lemma}[\cite{Latrach7}] \label{lem2.1}
 Let $X$ be an arbitrary Banach space. Then
 \[
\mathcal{F}(X)=\mathcal{F}^b(X),
\]
where $\mathcal{F}(X):=\mathcal{F}(X,X)$.
\end{lemma}

 An immediate consequence of this result is that
$\mathcal{F}(X)$ is a closed two-sided ideal of $\mathcal{L}(X)$.

 \begin{remark} \label{rmk2.3} \rm
Let $X$ and $Y$ be two Banach
spaces. In contrast to the result of Lemma \ref{lem2.1}, the fact that
$\mathcal{F}(X,Y)$ is equal or not to $\mathcal{F}^b(X,Y)$ seems to be
unknown.
\end{remark}

 In general, we have the following inclusions:
\begin{gather*}
\mathcal{{K}}(X)\subset \mathcal{{S}}(X)\subset \mathcal{
F}^b_+(X)\subset\mathcal{F}(X), \\
\mathcal{{K}}(X)\subset C\mathcal{{S}}(X)
\subset\mathcal{F}^b_-(X)\subset\mathcal{F}(X)
\end{gather*}
where $\mathcal{F}^b_{-}(X)=\mathcal{F}^b_{-}(X,X)$.

We say that $X$  is weakly compactly generating (w.c.g.) if the
linear span of some weakly compact subset is dense in $X$. For
more details and results we refer to \cite{Diestel}. In
particular, all separable and all reflexive Banach spaces are
w.c.g. as well as $L_1(\Omega,d\mu)$ if $(\Omega,\mu)$ is
$\sigma$-finite.

 It is proved in \cite{Weis} that if $X$ is a w.c.g.
Banach space then
\[
\mathcal{F}_{+}(X)= \mathcal{{S}}(X)\quad \text{and}\quad
\mathcal{F}_{-}(X)= C\mathcal{{S}}(X).
\]
We say that $X$ is subprojective, if given any closed
infinite-dimensional subspace $M$ of $X$, there exists a closed
and finite dimensional subspace $N\subset M$ and a continuous
projection from $X$ onto $N$. Clearly any Hilbert space is
subprojective. The spaces $c_0$, $l_p$, ($1\leq p<\infty$), and
$L_p$ ($2\leq p<\infty$), are also subprojective (cf.
\cite{Whitley}).

We say that $X$ is superprojective if every subspace $V$ having
infinite codimension in $X$ is contained in a closed subspace $W$
having infinite codimension in $X$ and such that there is a
bounded projection from $X$ to $W$. The spaces $l_p$,
($1<p<\infty$), and $L_p$ ($1< p\leq 2$), are superprojective
(cf. \cite{Whitley}).

 Let $X$ be a w.c.g. Banach space. If $X$ is
superprojective (resp. subprojective) then $\mathcal{{S}}(X)\subset
C\mathcal{{S}}(X)$ (resp. $C\mathcal{{S}}(X)\subset \mathcal{{S}}(X)$). So,
$\mathcal{{S}}(X)\subset \mathcal{F}_+(X)\cap \mathcal{F}_-(X)$
(resp. $C\mathcal{{S}}(X)\subset \mathcal{F}_+(X)\cap \mathcal{F}_-(X)$),
where $\mathcal{F}_+(X):=\mathcal{F}_+(X,X)$ and
$\mathcal{F}_-(X):=\mathcal{F}_-(X,X)$.

 Let $(\Omega,\Sigma,\mu)$ be a positive measure space
and let $X_p$ denote the spaces $L_p(\Omega,d\mu)$ with $1\leq
p<\infty$. Since $p\in [1,\infty)$ the spaces $X_p$ are w.c.g. and
consequently we have $\mathcal{F}_+(X_p)=\mathcal{S}(X_p)$ and
$\mathcal{{F}}_-(X_p)=C\mathcal{S}(X_p)$. In $X_p$ and in $C(E)$
(the Banach
space of continuous scalar-valued function on $E$ with the
supremum norm) provided that $E$ is a compact Hausdorff space we
have $\mathcal{S}(X_p)=C\mathcal{S}(X_p)=\mathcal{F}(X_p)$
(cf. \cite{Weis}).


 For a self-adjoint operator $A$ on a Hilbert space
$X$, the essential spectrum of $A$ is the set of limit points of
the spectrum of $A$ (with eigenvalues counted according to their
multiplicities), i.e., all points of the spectrum except isolated
eigenvalues of finite multiplicity (see, for example,
\cite{Wolf1, Wolf2}).

 If $X$ is a Banach space and
$A\in \mathcal{C}(X)$, there are many ways to define the essential
spectrum. We recall six of them:
\begin{itemize}

 \item   $\sigma_{e1}(A):= \{\lambda \in \mathbb{C}
: \lambda -A\not\in \Phi_{+}(X)\}
=\mathbb{C}\setminus \Phi_{+A}$,

\item $\sigma_{e2}(A):= \{\lambda \in \mathbb{C} : \lambda -A\not\in \Phi_{-}(X)\}=\mathbb{C}\setminus
\Phi_{-A}$,

\item $\sigma_{e3}(A)= \{\lambda \in \mathbb{C} :
 \lambda -A\not\in \Phi_{\pm}(X)\}=\mathbb{C}\setminus \Phi_{\pm A}$,

\item $\sigma_{e4}(A):=\{\lambda \in \mathbb{C} : \lambda -A\not\in \Phi(X)\}=\mathbb{C}\setminus \Phi_{A}$,

\item $\sigma_{e5}(A):= \mathbb{C}\setminus \rho_{5}(A)$,

\item $\sigma_{e6}(A):= \mathbb{C}\setminus \rho_{6}(A)$.

\end{itemize}
 where
$
\rho_{5}(A):=\{\lambda\in \Phi_{A} : i(\lambda-A)=0\}$ and
\[
\rho_{6}(A):=\{\lambda\in \rho_{5}(A) \hbox{ such that scalars
near $\lambda$  are in }\rho(A)\}.
\]
The subsets $\sigma_{e1}(.)$
and $\sigma_{e2}(.)$ are the Gustafson and Weidmann essential
spectra \cite{Gustafson}.
$\sigma_{e3}(.)$ is the Kato essential spectrum \cite{Kato}.
$\sigma_{e4}(.)$ is the Wolf essential
spectrum \cite{Gramsch,Wolf2,Schechter2,Wolf1}. $\sigma_{e5}(.)$
is the Schechter essential spectrum
\cite{Gramsch,Gustafson,Schechter2,Schechter3}, and
$\sigma_{e6}(.)$ denotes the Browder
essential spectrum \cite{Gustafson,Kaashoek,Nussbaum,Schechter2}.
Note that all these sets are closed and in general satisfy
the following inclusions
 \[
\sigma_{e1}(A)\cap \sigma_{e2}(A)
=\sigma_{e3}(A)\subset \sigma_{e4}(A)\subset
\sigma_{e5}(A)\subset \sigma_{e6}(A).
 \]
Note that, if $A$ is a self-adjoint operator on a
Hilbert space, then
 \[
\sigma_{e1}(A)=\sigma_{e2}(A)=\sigma_{e3}(A)=
\sigma_{e4}(A)=\sigma_{e5}(A)= \sigma_{e6}(A).
\]
The essential spectra were studied by many authors. Now, the main
question is about the classes of stability of the essential spectra
by a perturbation. Motivated by a problem concerning the spectrum of
the transport operator posed in \cite{Latrach2},  Latrach and
Jeribi \cite{Jeribi12} proved that the Shechter essential spectrum
is stable with the perturbations by weakly compact operators on
$L_1$-spaces. Then,  Jeribi,  Latrach and  Dehici have
extended the results about the classes of stability to strictly
singular operators on $L_p$-spaces (see \cite{Jeribi1, Jeribi2,
Jeribi6, Jeribi5, Jeribi11}), to weakly compact operators on
Dunford-Pettis spaces (see \cite{Jeribi4, Latrach6}), and to
Fredholm perturbation operators on Banach spaces (see \cite{Jeribi9,
Jeribi10, Jeribi20, Latrach7}). The Wolf essential spectrum for the
$N$-body problem, was studied in \cite{Damak1}.


 \begin{remark} \label{rmk2.4} \rm
If $\lambda\in\sigma C(A)$ (the continuous spectrum of $A$)
then $R(\lambda-A)$ is not
closed (otherwise $\lambda\in\rho(A)$ see
\cite[Lemma 5.1 p. 179]{Schechter1}). Therefore $\lambda\in\sigma_{ei}(A)$,
$i=1,\dots,6$. Consequently we have
$\sigma C(A)\subset \cap_{i=1}^{6}\sigma_{ei}(A)$. If the spectrum
of $A$ is purely continuous then $\sigma (A)=\sigma C(A)=\sigma_{ei}(A)$
$i=1,\dots,6$.
\end{remark}

\section{General Hypotheses}

The aim of this section is to present some hypotheses which we
need in the sequel. We begin with hypotheses which assure the
closedness of the operator $L_0$. Let $X_1$ and $X_2$ be Banach
spaces. In the product space $X:=X_1\times X_2$ we consider an
operator formally defined by a matrix
 \begin{equation}
L_0:=\begin{pmatrix}
A&B\\
C&D\end{pmatrix}. \label{e3.1}
\end{equation}
Our assumptions are as
follows: in general, the operators occurring in the representation
$L_0$ are unbounded, $A$ acts on the space $X_1$ and has the
domain $\mathcal{D}(A)$, $D$ is defined on $\mathcal{D}(D)$ and acts on
$X_2$, and the intertwining operators $B$ and $C$ are defined on
the domains $\mathcal{D}(B)$ and $\mathcal{D}(C)$, respectively, and
acts between these spaces.

 In what follows, we assume that the following conditions hold.
\begin{itemize}
\item[(H1)] $A$ is a densely defined operator in $X_1$ with
nonempty resolvent set $\rho(A)$.

\item[(H2)]  $B$ and $C$ are densely defined closable operators from
$X_2$ into $X_1$ and from $X_1$ into $X_2$, respectively, and
$\mathcal{D}(C)\supset \mathcal{D}(A)$.

\item[(H3)] For some $\mu\in \rho(A)$
the operator $(A-\mu I)^{-1}B$ is bounded on its domain
$\mathcal{D}(B)$.

\item[(H4)] $\mathcal{D}(B)\subset \mathcal{D}(D)$.

\item[(H5)] For some $\mu\in\rho(A)$ the
operator $D-C(A-\mu I)^{-1}B$ is closable. We denote its closure
by $S(\mu)$.
\end{itemize}

For a better understanding, we give some comments to explain these
hypothesis.

\begin{remark} \label{rmk3.1} \rm
(i) From the fact that $\mathcal{D}(C)\supset \mathcal{D}(A)$ and the closed
graph theorem we infer that for each $\mu\in \rho(A)$ the operator
$G(\mu):=C(A-\mu I)^{-1}$ is defined on $X_1$ and is bounded.

(ii) If the hypothesis (H3) holds for
some $\mu\in \rho(A)$ then it holds for all $\mu\in \rho(A)$.
Indeed, let $\mu_0\in \rho(A)$ be such that the operator
$(A-\mu_0 I)^{-1}B$ is  bounded. Then, for arbitrary $\mu\in \rho(A)$ the
relation
\begin{equation}
(A-\mu I)^{-1}B=(A-\mu_0 I)^{-1}B+ (\mu-\mu_0)(A-\mu I)^{-1}(A-\mu_0
I)^{-1}B \label{e3.2}
\end{equation}
shows that $(A-\mu I)^{-1}B$ is also bounded.

(iii) We denote the closure of $(A-\mu I)^{-1}B$ by $F(\mu)$.
Then the relation \eqref{e3.2} implies
\[
F(\mu)=F(\mu_0)+(\mu-\mu_0)(A-\mu I)^{-1}F(\mu_0).
\]

 (iv) The fact that $\mu\in \rho(A)$
implies that the operator $C(A-\mu I)^{-1}$ is defined everywhere
on $X_1$ and hence the operator $C(A-\mu I)^{-1}B$ is defined on
$\mathcal{D}(B)$.

 (v) According to  assumption (H4), the operator $D-C(A-\mu I)^{-1}B$
is defined on $\mathcal{D}(B)$.


(vi) If  hypothesis (H5) holds for
some $\mu\in \rho(A)$ then it holds for all $\mu\in \rho(A)$.

(vii) With the matrix defined in \eqref{e3.1}, we
associate the operator
$L_0: \mathcal{D}(L_0)\subset X$, with
$\mathcal{D}(L_0)=\mathcal{D}(A)\times\mathcal{D}(B)$ and
\[
L_0\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}=\begin{pmatrix}
Ax_1+Bx_2\\
Cx_1+Dx_2
\end{pmatrix}
\]
 The operator $L_0$ can be factored in the Frobenius-Shur
sense:
 \[
L_0-\mu I=\begin{pmatrix}
        I & 0 \\
        G(\mu) &I \end{pmatrix}
\begin{pmatrix}
 A-\mu I & 0 \\
 0       &D-C(A-\mu I)^{-1}B-\mu I \end{pmatrix}
\begin{pmatrix}
    I & F(\mu)  \\
    0 & I \end{pmatrix}
\]
 We recall now a result from \cite{Atkinson} regarding
the closability of the operator $L_0$.
\end{remark}

\begin{theorem}[{\cite[Theorem 1.1]{Atkinson}}] \label{thm3.1}
 Let hypotheses (H1)--(H5) be satisfied and let $\mu\in \rho(A)$.
Then $L_0$ is closable and its closure $L$ is given by
 \[
L=\mu I+\begin{pmatrix}
  I & 0  \\
 G(\mu)& I \end{pmatrix}
\begin{pmatrix}
 A-\mu I &  0 \\
   0 & S(\mu)-\mu I \end{pmatrix}
\begin{pmatrix}
 I& F(\mu)  \\
0 & I \end{pmatrix},
\]
or equivalently,
$ L: \mathcal{D}(L)\subset X \to X$  with
$\mathcal{D}(L)=  \big\{\begin{pmatrix}
x_1\\
x_2\end{pmatrix}\in X:x_1+F(\mu)x_2\in
\mathcal{D}(A)\text{ and }x_2\in\mathcal{D}(S(\mu))\big\}$
 and
\[
 L\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}=\begin{pmatrix}
A(x_1+F(\mu)x_2)-\mu F(\mu)x_2\\
C(x_1+F(\mu)x_2)+S(\mu)x_2
\end{pmatrix}
\]
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
Using the hypotheses (H2) and (H3) we infer that, for
$\mu,\ \mu_0\in \rho(A)$, the difference
\begin{equation}
\big(D-C(A-\mu )^{-1}B\big)-\big(D-C(A-\mu_0)^{-1}B\big)
=(\mu-\mu_0)C(A-\mu )^{-1}(A-\mu_0 )^{-1}B
\label{e3.3}
\end{equation}
 is a bounded operator.
Therefore if the operator $D-C(A-\mu I)^{-1}B$ is closable for some
$\mu\in \rho(A)$ then it is closable for all $\mu\in \rho(A)$. Since
the operator $L$ is the closure of $L_0$, it does not depend on the
choice of the point $\mu\in \rho(A)$ in its description above.
\end{remark}

 For the rest of this article, $\mathcal{I}(X)$ will
denote an arbitrary nonzero two-sided closed ideal of
$\mathcal{L}(X)$ satisfying the condition
\[
\mathcal{I}(X)\subset \mathcal{F}(X).
\]
Using Lemma \ref{lem2.1} and \cite[Proposition 4, p. 70]{Gohberg}
we deduce that
\[
\mathcal{{K}}(X)\subset \mathcal{{I}}(X),
\]
 where $\mathcal{K}(X)$ stands for the ideal of compact
operators. Hence the ideal of compact operators is the minimal
subset of $\mathcal{{L}}(X)$ (in the sense of the inclusion) for
which the results of this paper are valid.


 We conclude this section with the
following hypotheses:
\begin{itemize}
 \item[(H6)] For some $\mu\in\rho(A)$ the resolvent
$(A-\mu I)^{-1}\in \mathcal{I}(X_1)$.

\item[(H7)] For some $\mu\in\rho(A)$ the
operator
$G(\mu)F(\mu):=\overline {C(A-\mu I)^{-2}B}\in \mathcal{ I}(X_2)$.
\end{itemize}

\begin{remark} \label{rmk3.3} \rm
Note that the assumptions (H1)--(H6) do not imply (H7) even if the
operator $C(A-\mu I)^{-1}B$ is bounded (see \cite[p. 8]{Atkinson}
where $\mathcal{I}(X)=\mathcal{K}(X)$ is the ideal of compact
operators).
\end{remark}


\section{Main Results}

In this section we present some results concerning the essential
spectra of the operator $L$. We begin with the following
preparatory result which is crucial for our purposes.
It describes a sufficient condition for assumption (H1)--(H6)
ti imply (H7).

 \begin{theorem} \label{thm4.1}
Under the assumptions (H1)--(H6), condition (H7) is satisfied if
and only if the operator $S(\mu)$ admits the representation
\begin{equation}
S(\mu)=S_0+M(\mu)\quad \mu\in\rho(A) \label{e4.1}
\end{equation}
 with a closed operator $S_0$, which is independent of $\mu$, and an
operator $M(\mu)\in \mathcal{I}(X_2)$. In this case $S_0$ can be
chosen to be $S(\mu_0)$ for any $\mu_0\in\rho(A)$, and $M(\mu)$
depends holomorphically on $\mu$ in $\rho(A)$.
\end{theorem}

\begin{proof} If (H7) is satisfied, by \eqref{e3.3}, we write
$S(\mu)$ in the form
\begin{align*}
  S(\mu)& =S(\mu_0)+ (\mu-\mu_0)G(\mu)F(\mu_0)  \\
       & =S(\mu_0)+ (\mu-\mu_0)G(\mu_0)F(\mu_0)+
        (\mu-\mu_0)^{2}G(\mu_0)(A-\mu I)^{-1}F(\mu_0).
\end{align*}
  From  assumptions (H6) and
(H7) and Proposition \ref{prop2.1} we can deduce that the representation
\eqref{e4.1} follows, with $S_0=S(\mu_0)$.

Conversely, equation \eqref{e4.1} implies that
\begin{equation}
S(\mu)-S_0=(M(\mu)-M(\mu_0))_{{\vert }\mathcal{D}(S_0)}.\label{e4.2}
\end{equation}
 On the other hand,
\begin{equation}
S(\mu)-S(\mu_0)= (\mu-\mu_0)G(\mu)F(\mu_0)_{{\vert }\mathcal{D}(S_0)}.
\label{e4.3}
\end{equation}
 So, using  \eqref{e4.2} and \eqref{e4.3}, we
deduce that
 \[
G(\mu)F(\mu_0)=(\mu-\mu_0)^{-1}(M(\mu)-M(\mu_0)).
\]
 If $\mu\to\mu_0$ the operator
$G(\mu)F(\mu_0)$ tends to $G(\mu_0)F(\mu_0)$ in the operator norm
topology which mean that $(\mu-\mu_0)^{-1}(M(\mu)-M(\mu_0))$ also
converges to $G(\mu)F(\mu_0)$ in the operator norm topology.
Furthermore, $(M(\mu)-M(\mu_0))\in \mathcal{I}(X_2)$ and
$\mathcal{I}(X_2)$ is a closed sided ideal of $\mathcal{L}(X_2)$. So,
$G(\mu_0)F(\mu_0)\in \mathcal{I}(X_2)$ and this completes the
proof.
\end{proof}

 For $\mu\in\rho(A)$, we introduce the
following matrix operators which we shall need in the sequel:
\begin{gather*}
\mathbf{G}(\mu):=\begin{pmatrix}
        I & 0   \\
      G(\mu) & I \end{pmatrix}, \\
\mathbf{D}(\mu):=\begin{pmatrix}
      A-\mu I & 0  \\
     0 & S_0+M(\mu)-\mu I \end{pmatrix}, \\
\mathbf{F}(\mu):=\begin{pmatrix}
        I & F(\mu)  \\
        0 &  I \end{pmatrix}\,.
\end{gather*}
The following remark
will be used for proving the next Theorem \ref{thm4.2}.

\begin{remark} \label{rmk4.1} \rm
(a) Using Theorems \ref{thm3.1} and \ref{thm4.1} we can write the operator $L$ in the form
\begin{equation}
L-\mu I=\mathbf{G}(\mu) \mathbf{D}(\mu) \mathbf{F}(\mu).\label{e4.4}
\end{equation}
(b) If $\mu\in \rho(A)$, then
\begin{itemize}
\item[(i)] $\alpha(A-\mu I)=\beta(A-\mu I)=0$.
 \item[(ii)] $\alpha(\mathbf{D}(\mu))=\alpha(S_0+M(\mu)-\mu I)$.
 \item[(iii)] $\beta(\mathbf{D}(\mu))=\beta(S_0+M(\mu)-\mu I)$.
\end{itemize}
\end{remark}

 We now state the main result of this paper.

\begin{theorem} \label{thm4.2}
Assume hypotheses (H1)--(H7). Then
\begin{itemize}
\item[(i)] $\sigma_{ei}(L)=\sigma_{ei}(S_0)$ for $i=4, 5$.
 Moreover, if $C\sigma_{e5}(L)$ [the
complement of $\sigma_{e5}(L)$] is connected and neither
$\rho(S_0)$ nor $\rho(S(\mu))$ is empty, then
\[
\sigma_{e6}(L)= \sigma_{e6}(S_0).
\]

\item[(ii)] If $\mathcal{I}(X_2)\subset \mathcal{F}_+(X_2)$ then
$\sigma_{e1}(L)= \sigma_{e1}(S_0)$.

\item[(iii)] If $\mathcal{I}(X_2)\subset \mathcal{F}_-(X_2)$
or $[\mathcal{I}(X_2)]^{*}\subset \mathcal{F}_+(X_2^{*})$,
then
$\sigma_{e2}(L)= \sigma_{e2}(S_0)$.

\item[(iii)] If
$\mathcal{I}(X_2)\subset \mathcal{F}_+(X_2)\cap\mathcal{F}_-(X_2)$ then
$\sigma_{e3}(L)= \sigma_{e3}(S_0)$.
\end{itemize}
\end{theorem}

\begin{proof} (i) First assume that
$\lambda\in\rho(A)$. It is clear that $\mathbf{F}(\lambda)$ is a
bijection from $\mathcal{D}(L)$ onto
$\mathcal{D}(\mathbf{D}(\lambda))=\mathcal{D}(A)\times \mathcal{D}(S_0)$
and $\mathbf{G}(\lambda)$ is a bijection from $X$ onto $X$. Therefore,
\begin{gather}
\alpha(L-\lambda I)=\alpha(\mathbf{D}(\lambda)),\label{e4.5} \\
\beta(L-\lambda I)=\beta(\mathbf{D}(\lambda)).\label{e4.6}
\end{gather}
By Remark \ref{rmk4.1} (b) (ii)-(iii) taking into account \eqref{e4.5}, \eqref{e4.6},
we obtain
\begin{gather}
\alpha(L-\lambda I)=\alpha(S_0+M(\lambda)-\lambda I), \label{e4.7} \\
\beta(L-\lambda I)=\beta(S_0+M(\lambda)-\lambda I).\label{e4.8}
\end{gather}
 Since $M(\lambda)\in\mathcal{I}(X_2)$, the numbers
$\alpha(L-\lambda I)$ and $\beta(L-\lambda I)$ are finite if and
only if $\alpha(S_0-\lambda I)$ and $\beta(S_0-\lambda I)$ are
finite. Consequently, $L-\lambda I$ is a Fredholm operator if and
only if $S_0-\lambda I$ is a Fredholm operator and, if this is the
case, $i(L-\lambda I)=i(S_0-\lambda I)$.

 Let $\lambda\not\in\rho(A)$.  By hypothesis (H6), the
spectrum of $A$ is discrete. Therefore, $\lambda$ is an isolated
eigenvalue of $A$. Let
 \[
P_\lambda=-{\frac{1}{2\pi i}}\int_{\vert\lambda-\xi\vert
=\varepsilon}(A-\xi I)^{-1}d\xi
\]
be the Riesz projection with $\varepsilon$ sufficiently small.
Then $\lambda\in\rho(A_\lambda)$ where $A_\lambda$ is the
finite-dimensional perturbation of $A$ given by
 \[
A_\lambda:=A(I-P_\lambda)+\delta P_\lambda,\quad
\delta\neq\lambda.
\]
 Now, for $\mu\in\rho(A_\lambda)$, we have
\[
\overline{D-C(A_\lambda-\mu I)^{-1}B}=S_0+M_\lambda(\mu)
\]
where the operator $S_0$ is introduced in \eqref{e4.1} and operator
$M_\lambda(\mu)$ is an operator from $\mathcal{I}(X_2)$. Let
$L_\lambda$ be the closure of the operator
 \[
\begin{pmatrix}
      A_\lambda & B  \\
      C & D \end{pmatrix}.
\]
 Since $L_\lambda$ is a finite-dimensional perturbation of $L$,
$L-\lambda I$ is a Fredholm operator on $X$ if and only
if $L_\lambda-\lambda I$ is a Fredholm operator on $X$.
Now, with the first part of the present
proof, we conclude that $\lambda\in\sigma_{ei}(L_\lambda)$ if and
only if $\lambda\in\sigma_{ei}(S_0)$ $i=4, 5$.

 The proof of  statement (i) for $i=6$ is
essentially the same as that of the last assertion of
\cite[Theorem 3.1]{Latrach7}.
This completes the proof of part (i).

(ii) Let $\lambda\in \rho(A)$. Using the fact that
$\mathcal{I}(X_2)\subset \mathcal{F}_+(X_2)$ and
\cite[Lemma Lemma 2.2 (i)]{Latrach7} we deduce that
$\alpha(S_0+M(\lambda)-\lambda I)$ is finite if and only if
$\alpha(S_0-\lambda I)$ is finite. Hence, by  \eqref{e4.7}
$\alpha(L-\lambda I)$ is finite if and only if $\alpha(S_0-\lambda
I)$ is finite.

 Let $\lambda\not\in \rho(A)$. Since
$\mathcal{F}_+(X_1)\subset \mathcal{F}(X_1)$, then by
hypothesis (H6) we deduce that $\lambda$ is an isolated
eigenvalue of $A$. Now, the rest of the proof of (ii) may be
done in a way similar to that in (i). This completes the proof
of (ii).

 (iii) If $\mathcal{I}(X_2)\subset \mathcal{F}_-(X_2)$ then the proof
is the same as in (ii). It suffices to use  \eqref{e4.8} and
\cite[Lemma 2.2 (ii)]{Latrach7}.

 If $[\mathcal{I}(X_2)]^{*}\subset \mathcal{F}_+(X_2^{*})$, the
 result follows from the fact that
$\alpha(S_0^{*}+M(\lambda)^{*}-\bar\lambda
I)=\beta(S_0+M(\lambda)-\lambda I)$ (see \cite{Goldberg} or
\cite{Kato}).

(iv)  Assertion (iv) follows from \cite[Proposition 3.1 (iv)]{Latrach7}.
The proof is complete.
\end{proof}

 \begin{remark} \label{rmk4.2} \rm
 Let $X$ be a Banach space having the Dunford-Pettis property.
It is proved in \cite{Latrach6} that the ideal of weakly compact
 operators, $\mathcal{W}(X)$, leaves invariant the sets
$\Phi_+(X)$, $\Phi_-(X)$,
$\Phi_\pm(X)$ and $\Phi(X)$ under additive perturbations, i.e.,
$\mathcal{W}(X)\subset \mathcal{F}_+(X)\cap \mathcal{F}_-(X)$. Hence for
$\mathcal{I}(X)=\mathcal{W}(X)$, the assertions of the Theorem \ref{thm4.2} are
valid.
\end{remark}

 We deduce the following result.

\begin{corollary} \label{coro4.1}
If the operator $D$ is everywhere defined and
bounded and $C(A-\mu_0 I)^{-1}B$ (for some and hence for all
$\mu_0\in\rho(A)$) is bounded, then $S_0$ can be chosen to be
\[
S_0=D-\overline{C(A-\mu_0 I)^{-1}B}.
\]
In particular, under these assumptions,
$\sigma_{ei}(L)\neq \emptyset$, $i:=1\dots 6$,
if $\dim(X_2)=\infty$.
\end{corollary}

In applications (see Section 5),  hypotheses (H3), (H6) and
the boundedness of the operator $C(A-\mu I)^{-1}B$ are not easy to
verify. Now, we give some sufficient conditions which imply the
above assumptions that are easier to check. To this end, we
introduce the following concept.

 \begin{definition} \label{def4.1} \rm
The resolvent of the operator $A$ is
said to have a ray of minimal growth if there exists some
$\theta\in[0,2\pi)$ such that
 \[
\gamma_\theta:=\{\lambda\in\mathbb{C}:
\lambda=te^{i\theta},\ t\in\mathbb{R}^{+}\}\subset\rho(A)
\]
 and there is a positive constant $M$ such that
\begin{equation}
\Vert (A-\lambda I)^{-1}\Vert\leq
{\frac{M}{1+\vert \lambda\vert}} \hbox{ holds for
all } \lambda\in \gamma_\theta.
\label{e4.9}
\end{equation}
\end{definition}

 The domain $\mathcal{D}(A)$ of $A$ is
equipped with the graph norm topology i.e.,
$\Vert x\Vert_1= \Vert x\Vert+\Vert Ax\Vert$; hence $\mathcal{D}(A)$
 is a Banach space. Let $X_{1,1}$ denote $(\mathcal{D}(A),\Vert.\Vert_1)$.
If $A$ is an operator whose resolvent has a ray of minimal growth, then the
intermediate spaces
 \[
X_{1,\theta}=\mathcal{D}(A^\theta),\ \ 0\leq \theta\leq 1
\]
 between $X_1$ and $X_{1,1}=\mathcal{D}(A)$ with the norm
$\Vert x\Vert_\theta= \Vert x\Vert+\Vert A^\theta x\Vert$ are well
defined and the same holds for the intermediate spaces
$X_{1,\theta}^{*}$ between $X_{1,1}^{*}=\mathcal{D}(A^{*})$ and
$X_1^{*}$.

\begin{proposition}[{\cite[Proposition 3.1]{Atkinson}}] \label{prop4.1}
If the operators $A$ and $B$ satisfy properties (H1), (H2),
then the assumption (H3) holds if and
only if $\mathcal{D}(B^{*})\supset \mathcal{D}(A^{*})$.
\end{proposition}

\begin{theorem} \label{thm4.3}
Let $X$ be a Banach space and let $\mathcal{I}(X)$ denote an arbitrary nonzero
two-sided ideal of $\mathcal{{L}}(X)$ contained in $\mathcal{F}(X)$. Let
$T\in\mathcal{C}(X)$ be such that $\rho(T)\neq\emptyset$. Then
$(\lambda-T)^{-1}\in\mathcal{I}(X)$ for some $\lambda\in\rho(T)$ if
and only if the embedding of $\mathcal{D}(T)$ into $X$ is in
$\mathcal{I}(X)$.
\end{theorem}

 \begin{remark} \label{rmk4.3} \rm
Note that for $\mathcal{I}(X)=\mathcal{K}(X)$, Theorem \ref{thm4.3} is
nothing but the classical spectral theorem for compact operators.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm4.3}]
 Let $\lambda\in\rho(T)$ such that
$(\lambda-T)^{-1}\in\mathcal{ I}(X)$. The operator
$\lambda-T:\mathcal{D}(T)\rightarrow X$ is an
isomorphism when the domain, $\mathcal{D}(T)$, of the operator $T$ is
equipped with the graph norm. By using the fact that
$(\lambda-T)^{-1}\in\mathcal{I}(X)$ and writing the embedding $j$ of
$\mathcal{D}(T)$ into $X$ as $j:=(\lambda-T)^{-1}(\lambda-T)$ with
$\mathcal{D}(j):=\mathcal{D}(T)$ we deduce that $j\in \mathcal{I}(X)$.

 Inversely, let $\lambda\in\rho(T)$. Then we can
write $(\lambda-T)^{-1}=j\circ(\lambda-T)^{-1}$ where
$j:\mathcal{D}(T)\rightarrow X$ is in $\mathcal{I}(X)$ then
 $(\lambda-T)^{-1}$ is in $\mathcal{I}(X)$ as the compose of
a continuous map $(\lambda-T)^{-1}$ and a map $j$ in
$\mathcal{I}(X)$.
\end{proof}


\begin{lemma} \label{lem4.1}
Let  conditions (H1), (H2), (H6) be satisfied and assume that the
resolvent of $A$ has a ray of minimal growth. Furthermore, assume
that for some $\theta\in (0,1)$ the following inclusions hold:
\begin{gather}
\mathcal{D}(B^{*})\supset \mathcal{
D}((A^{*})^\theta):=\mathcal{D}((A^{\theta})^*), \label{e4.10}\\
\mathcal{D}(\overline{C})\supset \mathcal{D}(A^{1-\theta}).\label{e4.11}
\end{gather}
Then also conditions (H3) and (H7) are fulfilled and the
operator $C(A-\mu I)^{-1}B$ is bounded for
$\mu\in\rho(A)$.
\end{lemma}

\begin{proof}
It follows from the properties of fractional powers
\cite[Chapter 4]{Krasnoselskii} that $X_{1,\theta}^*\supset
X_{1,1}^*=\mathcal{D}(A^{*})$ for any $\theta\in [0,1)$. Moreover,
the embedding of $X_{1,1}^*$ into $X_{1,\theta}^*$ is a Fredholm
perturbation since $0\in\rho(A)$ by \eqref{e4.9} and
$A^{-1}\in \mathcal{I}(X_1)$ (see Theorem \ref{thm4.3}).
Using Proposition \ref{prop4.1} we conclude that
(H3) holds if and only if
$\mathcal{D}(B^{*})\supset \mathcal{D}(A^{*})$.
Thus, (H3) derives from \eqref{e4.10}.

 Now, write the operator $CA^{-1}B$ in the form
\begin{equation}
CA^{-1}B=\overline{C}A^{-(1-\theta)}A^{-\theta}B.
\label{e4.12}
\end{equation}
 The operator $\overline{C}A^{-(1-\theta)}$
is bounded on $X_1$ by \eqref{e4.11}. The operator $A^{-\theta}B$ is
bounded on $\mathcal{D}(B)$ by \eqref{e4.10} and Proposition \ref{prop4.1}. Hence,
the boundedness of the operator $CA^{-1}B$ follows from \eqref{e4.12}.
Finally, by writing
 \[
CA^{-2}B=\overline{C}A^{-(1-\theta)}A^{-1}(A^{-\theta}B)
\]
 the hypothesis (H7) follows from the
fact that $A^{-1}\in \mathcal{I}(X_1)$ and Proposition \ref{prop2.1}.
\end{proof}

 Let $X$ be a Banach space and let $T$ be
a closed operator on $X$. By $\Delta^{0}(T)$ we denote the maximal
open subset of $\mathbb{C}$ where the resolvent $(T-\lambda I)^{-1}$
is finitely meromorphic, i.e., it is meromorphic on
$\Delta^{0}(T)$ and all the coefficients in the principal parts of
the Laurent expansions at the poles are of finite rank.


 \begin{remark} \label{rmk4.4} \rm
The set $\Delta^{0}(T)$ is the union of all components $w$ of $\Phi_T$ for which
$w\cap\rho(T)\neq\emptyset$  \cite[Lemma 2.1]{Gohberg}.
\end{remark}

 Using representation \eqref{e4.4} we can prove the following result.


\begin{corollary} \label{coro4.2}
 Under assumptions (H1)--(H6), the set $\Delta^{0}(L)$ is the union of
all components $w$ of $\Phi_{S_0}$ such that for some $\mu\in w$
the operator $S(\mu)-\mu I$ maps $X_2$ bijectively onto
itself.
\end{corollary}

 We give now, a sufficient condition for
the fact that $\Delta^{0}(L)$ contains the unbounded component of
$\Delta(S_0)$, denoted by $\Delta^{0}_{ext}(S_0)$.

 \begin{corollary} \label{coro4.3}
 Let  conditions (H1), (H2, (H6) hold. Assume that the resolvent of $A$ has
a ray of minimal growth. Assume, in addition, that for some
$\theta\in (0,1)$ the inclusions \eqref{e4.10} and \eqref{e4.11} hold and the
operator $D$ is bounded. Then the inclusion
 \[
\Delta^{0}(L)\supset \Delta^{0}_{ext}(S_0)
\]
holds. In particular, if $\Delta(S_0)$ is simply connected, then the equality
$\Delta^{0}(L)=\Delta^{0}(S_0)$ holds.
\end{corollary}

\begin{proof} Let $\lambda\in \gamma_\theta$,
$0<\theta<1$, and consider the  identity:
 \[
S(\lambda)-\lambda I=-\lambda I+S(0)+[\overline{C}A^{-(1-\theta)}]
[\lambda(A-\lambda)^{-1}]\big[\overline{A^{-\theta}B}\big].
\]
By Lemma \ref{lem4.1} the operators $S(0)$,
$\overline{C}A^{-(1-\theta)}$ and $\overline{A^{-\theta}B}$ are
everywhere defined and bounded. Hence the operator
$S(\lambda)-\lambda I$ has a bounded inverse for all
$\lambda\in \gamma_\theta$ with $\vert \lambda\vert$ sufficiently large.
By Corollary \ref{coro4.2} the unbounded component $\Delta^{0}_{ext}(S_0)$ of
$\Delta(S_0)$ is a component of $\Delta^{0}(L)$.
\end{proof}

\section{An Example}

The aim of this section is in the analysis of the essential
spectra of the operator defined in \eqref{e3.1} where $A$, $B$, $C$ are
ordinary differential operators in spaces of vector functions and
$D$ is a multiplication operator defined as follows:
The operator $A: \mathcal{D}(A)\subset X_1 \to X_1$ is defined on
$ \mathcal{D}(A)=\{\varphi \in (H_{p}^l)^n :U\varphi=0\}$, as
\[
\varphi  \mapsto A\varphi(x)=  \sum_{k=0}^l a_k(x)\varphi^{(l-k)}(x)
\]
where $X_1:=(L_p(0,1))^n$, $p>1$,
$n\in \mathbb{N}$, $l>0$, and $a_i$, $0\leq i\leq l$ are $n\times n$ matrix
functions with sufficiently smooth entries and
$\det a_0(x)\neq 0$; $(H_p^l)^n:=(H_p^l(0,1))^n$ is a Sobolev space of
$n$-vector functions. The domain $\mathcal{D}(A)$ is supposed to be
given by general boundary conditions
\begin{equation}
U(\varphi):=U_0
\begin{pmatrix}
\varphi(0)\\
\varphi'(0)\\
\vdots\\
\varphi^{(l-1)}(0) \end{pmatrix}
+U_1\begin{pmatrix}
\varphi(1)\\
\varphi'(1)\\
\vdots\\
\varphi^{(l-1)}(1) \end{pmatrix}=0
\label{e5.1}
\end{equation}
where $U_0$ and $U_1$ are $nl\times nl$ matrices.

The operator $B: \mathcal{D}(B)\subset X_2\to X_1$ is defined on
$\mathcal{D}(B)=\{\varphi \in (H_{p}^s)^m :
    \hat{U}\varphi=0\}$, as
\[
\varphi \mapsto B\varphi(x)= \sum_{k=0}^s b_k(x)\varphi^{(s-k)}(x).
\]
where $X_2:=(L_p(0,1))^m$, $m\in \mathbb{N}$,
$0\leq s\leq l$; $b_i$, $0\leq i\leq s$ are $n\times m$ matrix
functions with sufficiently smooth entries. The system of boundary
conditions $U^{*}(v)=0$ ($v\in H_{q}^s$ with $q={\frac{p}{p-1}}$)
is the adjoint of the system \eqref{e5.1}. We take all boundary
conditions of order $\leq s-1$ in the system of boundary
conditions $U^{*}(v)=0$ and denote the corresponding subsystem of
linear forms by $\hat U^{*}(\cdot)$.
 The domain $\mathcal{D}(B)$ is chosen, on the one hand, to satisfy the
condition $\mathcal{D}(B^{*})\supset \mathcal{D}(A^{*})$ and, on the
other hand, to be as large as possible in order to cover examples
which are interesting in applications.

The operator $C: \mathcal{D}(C)\subset X_1 \to X_2$ is defined on
$\mathcal{D}(C)=\{\varphi \in (H_{p}^h)^n :
        U\varphi=0\}$, as
\[
\varphi  \mapsto C\varphi(x)=  \sum_{k=0}^h c_k(x)\varphi^{(h-k)}(x),
\]
 where $0\leq h\leq l$, $s+h=l$ and $c_i$,
$0\leq i\leq h$ are $m\times n$ matrix functions with sufficiently
smooth entries.

The operator $D: X_2 \to X_2$ is defined as
\[
\varphi  \mapsto D\varphi(x)=  d(x)\varphi(x),
\]
 where $d$ is an $m\times m$ matrix
function which is assumed to be measurable and essentially bounded
(hence $D$ is a bounded operator on $X_2$).

For more details we refer the reader to \cite{Atkinson}.
The next proposition contains  conditions that
we need in the sequel.


\begin{proposition} \label{prop5.1}
With the above notation, we have the following:
\begin{itemize}
\item[(i)] \cite[Proposition 4.1]{Atkinson}
$B$ is closable and the inclusion
$\mathcal{D}(B^{*})\supset \mathcal{D}((A^{*})^{\frac{s}{l}})$ holds.

\item[(ii)] \cite[Proposition 4.2]{Atkinson}
$C$ is closable and $\mathcal{D}(\bar C)\supset \mathcal{
D}((A^{*})^{\frac{h}{l}})$

\item[(iii)] \cite[Theorem 4.3]{Atkinson}
For $\mu\in\rho(A)$ the operator $D-C(A-\mu I)^{-1}B$ which is defined
on $\mathcal{D}(B)$, admits a
bounded closure $S(\mu)=S_0+K(\mu)$, where $S_0$ is the operator of
multiplication by the function
$d-c_0a_0^{-1}b_0$ and $K(\mu)$ a
compact operator in $X_2$.
\end{itemize}
\end{proposition}

  \begin{theorem} \label{thm5.1}
Let $L_0$ the operator defined as above and let $L$ denote the closure
of $L_0$. Then
\begin{equation}
\sigma_{ei}(L)=\{ \lambda\in \mathbb{C}:
\mathop{\rm ess\,inf}|
\det [d(x)-c_0(x)a_0^{-1}(x)b_0(x)-\lambda I]|=0\}\quad
i=1,\dots,6.
\label{e5.2}
\end{equation}
 Moreover, if the complement of this set is connected,
then this complement coincides with the domain of finite meromorphy
of the operator function $(L-\lambda I)^{-1}$.
\end{theorem}

\begin{proof}
Using Theorem \ref{thm4.2} and Proposition \ref{prop5.1} (iii) we deduce
that $\sigma_{ei}(L)=\sigma_{ei}(S_0)$,
$i=1,\dots,6$ where $S_0$ is the operator of multiplication by the
matrix function $d-c_0a_0^{-1}b_0$. On the other hand, it is shown
in \cite{Hardt} that the spectrum of $S_0$ is purely continuous and
is given by the expression on the right-hand side of \eqref{e5.2}. Now
the result follows from Remark \ref{rmk2.4}.
\end{proof}

\begin{thebibliography}{00}

\bibitem{Adam} J. A. Adam,
  \emph{Physics reports (Review section of Physics Letters)},
142, 5, 263-356 (1986).

\bibitem{Agmon} S. Agmon, A. Douglis, R. Mennicken and L.
Nirenberg, \emph{Estimates near the boundary for solutions of
elliptic partial differential equations satisfying general boundary
conditions. II}, Comm. Pure Appl. Math., 17, 35-92 (1964).

\bibitem{Atkinson} F. V. Atkinson, H. Langer, R. Mennicken and A. A.
Shkalikov, \emph{The essential spectrum of some matrix operators},
   Math. Nachr.,
167, 5-20 (1994).

\bibitem{Caradus} S. R. Caradus,
\emph{Operators of Riesz type},
   Pacific J. Math.,
18, 61-71 (1966).

\bibitem{Damak1} M. Damak,
\emph{On the spectral theory of dispersive N-Body Hamiltonians},
   J. Math. Phys.,
40, 35-48 (1999).

\bibitem{Diestel} J. Diestel, Geometry of Banach spaces-Selected topics
  Lecture Notes in Mathematics,
485, Springer, New-York (1975).

\bibitem{Dunford1} N. Dunford and Pettis,
\emph{Linear operations on summable functions},
  Tran. Amer. Math. Soc.,
47, 323-392 (1940).

\bibitem{Dunford2} N. Dunford and J. T. Schwartz,
\emph{Linears operators.}
  Interscience Publishers Inc. New-York,
Part 1 (1958).

\bibitem{Goedbloed} J. P. Goedbloed, \emph{Lecture note on ideal
magnetohydrodynamics}, in: Rijnhiuzen Report, From-Instituut voor
Plasmafysica, Nieuwegein 83-145 (1983).


\bibitem{Gohberg1} I. C. Gohberg, A. S. Markus, and I. A. Feldman,
\emph{Normally solvable operators and ideals associated with them}, Amer. Math.
Soc. Transl. ser. 2, 61, 63-84 (1967).

\bibitem{Gohberg} I. C. Gohberg and E. Sigal,
\emph{An operator Generalization of the logarithmic residue theorem and
the theorem of RouchŽ},
   Mat. Sbornik., 84, 126 (1971). Engl. Transl. in  Math. USSR
Sbornik, 13, 603-625 (1971).

\bibitem{Goldberg} S. Goldberg,
\emph{Unbounded Linear Operators.}
    McGraw-Hill, New-York,(1966).

\bibitem{Gramsch} B. Gramsch and D. Lay, \emph{Spectral mapping
theorems for essential spectra}, Math. Ann. 192, 17-32 (1971).

\bibitem{Grothendieck} A. Grothendieck,
\emph{Sur les applications linŽaires
 faiblement compactes d'espaces du type $C(K)$},
   Canad. J. Math., 5, 129-173 (1953).


\bibitem{Grubb} G. Grubb and G. Geymonat,
\emph{The Essential Spectrum of Elliptic Systems of Mixed Order},
 Math. Anal. Ann. 227, 245-276 (1977).

\bibitem{Gustafson} K. Gustafson and J. Weidmann,
\emph{On the essential spectrum.},
J. Math. Anal. Appl. 25, 6, 121-127 (1969).

\bibitem{Hardt} V. Hardt,
\emph{Uber ein im eigenwertparameter rationales
randeigenwertproblem bei differentialgleichungssystemen zweiter
ordnung.}
  Dissertation, Regensburg, (1992).

\bibitem{Jeribi1} A. Jeribi, \emph{Quelques  remarques  sur les op\'erateurs
de Fr\'edholm et application \`a l'\'equation de transport},
   C. R. Acad. Sci. Paris, t. 325, SŽrie I, pp. 43-48 (1997).

\bibitem{Jeribi2} A. Jeribi, \emph{Quelques remarques sur le spectre de Weyl et
applications},
   C. R. Acad. Sci. Paris, t. 327, SŽrie I, pp. 485-490 (1998).

\bibitem{Jeribi6} A. Jeribi, \emph{Une nouvelle caractŽrisation du spectre essentiel et
application},
   C. R. Acad. Sci. Paris, t. 331, 2000, SŽrie I, pp. 525-530 (2000).

\bibitem{Jeribi4} A. Jeribi, \emph{A characterization of the essential
 spectrum and applications},
  J. Boll. dell. Unio. Mate. Itali., (8) 5-B, 805-825 (2002).

\bibitem{Jeribi10} A. Jeribi,
\emph{A characterization of the Schechter essential spectrum on Banach
spaces and applications},
  J. Math. Anal. Appl., 271, 343-358 (2002).

\bibitem{Jeribi13} A. Jeribi,
\emph{Some remarks on the Schechter essential spectrum and applications
to transport equations.}
  J. Math. Anal. Appl., 275, 222-237 (2002).


\bibitem{Jeribi20} A. Jeribi,
\emph{On the Schechter essential spectrum on Banach spaces and
application},
  Facta Univ. Ser. Math. Inform., 17, 35-55 (2002).

\bibitem{Jeribi9} A. Jeribi,
\emph{Fredholm operators and essential spectra},
  Arch. Inequal. Appl.,  2, no. 2-3, 123-140 (2004).

\bibitem{Jeribi5} A. Jeribi, K. Latrach, \emph{Quelques  remarques  sur le spectre
essentiel et application ˆ l'Žquation de transport},
  C. R. Acad. Sci. Paris, t. 323, SŽrie I, 469-474 (1996).


\bibitem{Kaashoek} M.A. Kaashoek and D.C. Lay,
\emph{Ascent, descent, and commuting perturbations},
  Trans. Amer. Math. Soc., 169, 35-47 (1972).

\bibitem{Kato1} T. Kato, \emph{Perturbation theory for nullity, deficiency and other
quantities of linear operators},
  J. Anal. Math., 6, 261-322 (1958).

\bibitem{Kato} T. Kato, \emph{Perturbation theory for linear operators},
   Springer-Verlag, New York, (1966).

\bibitem{Kopachevskii} N. D. Kopachevskii, S. G. Krein and Ngo Zui Kan,
\emph{Operator methods in linear hydrodynamics. Evolution and spectral
problems [in Russian]},
   Nauka, Moscow, (1989).

\bibitem{Krasnoselskii} M. A. Krasnosel'skii, P. P. Zabreiko, E. I.
Pustylnik, P. E. Sobolevskii, \emph{Integral operators in space of
summable functions},
   Leyden : Noordhoff International Publishing, (1976).

\bibitem{Kurasov} P. Kurasov and S. Nabako,
\emph{Essential spectrum due to singularity},
   J. Nonl. Math. Phys., 10, 93-106 (2003).

\bibitem{Latrach2} K. Latrach, \emph{Some remarks on the essential spectrum of
transport operators with abstract boundary conditions},
   J. Math. Phys.,  35, 11, 6199-6212 (1994).

\bibitem{Latrach6} K. Latrach,
\emph{Essential spectra on spaces with the Dunford-Pettis property},
   J. Math. Anal. Appl., 233, 607-622 (1999).

\bibitem{Latrach7} K. Latrach, A. Dehici, \emph{Fredholm, Semi-Fredholm
Perturbations and Essential spectra},
   J. Math. Anal. Appl., 259, 277-301 (2001).

\bibitem{Jeribi12} K. Latrach, A. Jeribi, \emph{On the essential spectrum of
transport operators on $L_1$-spaces},
  J. Math. Phys., 37, 12, 6486-6494 (1996).

\bibitem{Jeribi11} K. Latrach, A. Jeribi, \emph{Some results on Fredholm
operators, essential spectra, and application},
   J. Math. Anal. Appl., 225, 461-485 (1998).

\bibitem{Lifschitz} A. E. Lifschitz, \emph{Magnetohydrodynamics and
spectral theory},
   Kluwer Acad. Publishers, Dordrecht, (1989).

\bibitem{Malyshev} V. A. Malyshev and R. A. Minlos,
\emph{Linear operators in infinite- particle system [in Russian]},
   Nauka, Moscow, (1994).

\bibitem{Nussbaum} R. D. Nussbaum,
\emph{Spectral mapping theorems and perturbation theorem for Browder's
essential spectrum},
  Trans. Amer. Math. Soc., 150, 445-455 (1970).

\bibitem{Pelczynski} A. Pelczynski,
\emph{On strictly singular and strictly cosingular operators. I.
Strictly singular and strictly cosingular operators in
$C(X)$-spaces. II. Strictly singular and strictly cosingular
operators in $L(\mu)$-spaces},
  Bull. Acad. Polon. Sci., 13, 13-36 and 37-41 (1965).


\bibitem{Schechter2} M. Schechter,
\emph{On the essential spectrum of an arbitrary operator},
  J. Math. Anal. Appl., 13, 205-215 (1966).

\bibitem{Schechter3} M. Schechter,
\emph{Invariance of essential spectrum},
  Bull. Amer. Math. Soc., 71, 365-367 (1971).

\bibitem{Schechter4} M. Schechter,
\emph{Riesz operators and Fredholm perturbations.}
  Bull. Amer. Math. Soc., 74, 1139-1144 (1968).

\bibitem{Schechter1} M. Schechter,
 \emph{Principles of  functional analysis},
Academic Press, 1971.


\bibitem{Vladimirskii} Ju. I. Vladimirskii,
\emph{Stricty cosingular operators},
  Soviet. Math. Dokl., 8, 739-740 (1967).

\bibitem{Weis} L. Weis,
\emph{Perturbation classes of semi-Fredholm operators},
  Math. Z., 178, 429-442 (1981).

\bibitem{Whitley} R. J. Whitley,
\emph{Strictly singular operators and their conjugates},
  Trans. Amer. Math. Soc., 18, 252-261 (1964).

\bibitem{Wolf1} F. Wolf,
\emph{On the essential spectrum of partial differential boundary
problems},   Comm. Pure Appl. Math., 12, 211-228 (1959).

\bibitem{Wolf2} F. Wolf,
\emph{On the invariance of the essential spectrum under a change of the
boundary conditions of partial differential operators},
  Indag. Math., 21, 142-147 (1959).

\end{thebibliography}

\end{document}