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 \markboth{H.\,Tuna and A.\,Ery\i lmaz}{Completeness of the system of root functions of $q$-Sturm-Liouville operators}
\title[Completeness of the system of root functions of $q$-Sturm-Liouville operators]{Completeness of the system of root functions of $q$-Sturm-Liouville operators{}}

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%\author[D. Juki\'c]{Dragan Juki\'c\corrauth}
%\address{Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia}
%\email{{\tt jukicd@mathos.hr} (D. Juki\'c)}


% multiple authors:
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\author[H.\,Tuna, A.\,Ery\i lmaz]{H\"{u}seyin Tuna\affil{1} and Aytekin Ery\i lmaz\affil{2}\comma\corrauth}
\address{\affilnum{1}\ Department of Mathematics, Mehmet Akif Ersoy
University, 15\,100, Burdur, Turkey\\
\affilnum{2}\ Department of Mathematics, Nev\c{s}ehir University,
50\,300, Nev\c{s}ehir, Turkey}
\emails{{\tt hustuna@gmail.com}\,\,(H.\,Tuna), {\tt aeryilmaz@nevsehir.edu.tr} (A. Ery\i lmaz)}
%\author[AUTHOR1 and AUTHOR2]{AUTHOR1\affil{1}\comma\corrauth and AUTHOR2\affil{2}}
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%%%%% Begin Abstract %%%%%%%%%%%
\begin{abstract}
In this paper, we study $q$-Sturm-Liouville operators. We construct
a space of boundary values of the minimal operator and describe all
maximal dissipative, maximal accretive, self-adjoint and other
extensions of $q$-Sturm-Liouville operators in terms of boundary
conditions. Then we prove a theorem on completeness of the system
of eigenfunctions and associated functions of dissipative operators
by using the Lidskii's theorem.
\end{abstract}
%%%%% end %%%%%%%%%%%

%%%%% Keywords %%%%%%%%%%%
\keywords{$q$-Sturm-Liouville operator, dissipative operator,
completeness of the system of eigenvectors and associated vectors,
Lidskii's theorem}

%%%% AMS subject classifications %%%%
\ams{34L10, 39A13}

%%%% maketitle %%%%%
\maketitle

%%%% Start %%%%%%
\section{\small{Introduction}}


Quantum calculus was initiated at the beginning of the 19th century
and recently there has been great interest therein. The subject of
$q$-difference equations has evolved into a multidisciplinary
subject \cite{er}. There are several physical models involving
$q$-difference and their related problems (see
\cite{an,1,2,3,4,5}). Many
works have been devoted to some problems of a $q$-difference equation.
In particular, Ad\i var and Bohner \cite{ad} investigated
eigenvalues and  spectral singularities of non-self-adjoint
$q$-difference equations of second order with spectral
singularities. Huseynov and Bairamov \cite{hus} examined the
properties of eigenvalues and eigenvectors of a quadratic pencil of
$q$-difference equations. In \cite{an}, Annaby and Mansour studied a
$q$-analogue of Sturm-Liouville eigenvalue problems and formulated a
self-adjoint $q$-difference operator in a Hilbert space. They also
discussed properties of eigenvalues and eigenfunctions. One
can see the reference \cite{kac} for some definitions and theorems
on $q$-derivative, $q$-integration, $q$-exponential function,
$q$-trigonometric function, $q$-Taylor formula, $q$-Beta and Gamma
functions, Euler-Maclaurin formula, etc.

Many problems in mechanics, engineering, and mathematical physics
lead to the study of completeness and basic properties of all or
part of eigenvectors and associated vectors corresponding to
some operators. For instance, when we apply the method of separation of variables
to solve an equation like the reduced wave equation or Helmholtz equation,
we assume the solution expanded in a series of eigenfunctions of one of the variables.
The coefficients depend upon the other variable. We substitute the expansion into the equation,
thereby obtaining ordinary differential equations for the coefficients. The method relies upon completeness
of eigenfunctions corresponding to one of the variables \cite{mm}.
Dissipative operators are an important part of non-self-adjoint operators. In the spectral analysis of a dissipative
operator, we should answer the question whether all
eigenvectors and associated vectors of a dissipative operator span
the whole space or not. Many authors investigated the problem of
completeness of the system of eigenvectors and associated vectors of
boundary value problems for differential and difference operators
(see \cite{ba1,ba2,ae,gus,gus2,gus3,ek}).

The organization of this paper is as follows: In Section 2, some
preliminary concepts related to $q$-difference equation and
Lidskii's theorem essentials are presented for the convenience of
the readers. In Section 3, we construct a space of boundary values
of the minimal operator and describe all maximal dissipative,
maximal accretive, self-adjoint and other extensions of
$q$-Sturm-Liouville operators in terms of boundary conditions.
Finally, in Section 4, we proved a theorem on completeness of the
system of eigenvectors and associated vectors of dissipative
operators under consideration.

\section{{Preliminaries}}

In this section, we introduce some of the required $q$-notations and
Lidskii's theorem essentials.

Following the standard notations in \cite{ae} and \cite{kac}, let
$q\ $be a positive number with $0<$ $q<1,$ $A\subset \mathbb{R\
}$and $a\in \mathbb{C}.\ $A $q$-difference equation is an equation
that contains $q$-derivatives of a function defined on $A.$ Let
$y\left( x\right) $ be \ a complex-valued function on $x\in A.$ The
$q$-difference operator $D_{q}$ is defined by
\begin{equation}
D_{q}y\left( x\right) =\frac{y\left( qx\right) -y\left( x\right)
}{\mu \left( x\right) },\quad\text{ for all }x\in A,
\end{equation}
where $\mu \left( x\right) =\left( q-1\right) x$. The $q$-derivative
at zero is defined by
\begin{equation}
D_{q}y\left( 0\right) =\lim_{n\rightarrow \infty }\frac{y\left(
q^{n}x\right) -y\left( 0\right) }{q^{n}x},\quad \text{ }x\in A,
\end{equation}
if the limit exists and does not depend on $x.\ $A right inverse to
$D_{q}$, the Jackson $q$-integration is given by
\begin{equation}
\int_{0}^{x}f\left( t\right) d_{q}t=x\left( 1-q\right)
\sum_{n=0}^{\infty }q^{n}f\left( q^{n}x\right) ,\quad \text{ }x\in A,
\end{equation}
provided that the series converges, and
\begin{equation}
\int_{a}^{b}f\left( t\right) d_{q}t=\int_{0}^{b}f\left( t\right)
d_{q}t-\int_{0}^{a}f\left( t\right) d_{q}t,\quad a,b\in A.
\end{equation}
Let $L_{q}^{2}$ $(0,a)\ $be the space of all complex-valued
functions defined on $[0,a]$ such that
\begin{equation}
\left\Vert f\right\Vert :=\left( \int_{0}^{a}\left\vert f\left(
x\right) \right\vert d_{q}x\right) ^{1/2}<\infty .
\end{equation}
The space $L_{q}^{2}$ $(0,a)\ $is a separable Hilbert space with the
inner product
\begin{equation}
\left( f,g\right) :=\int_{0}^{a}f\left( x\right) \overline{g\left(
x\right) } d_{q}x,\quad f,g\in L_{q}^{2}(0,a).
\end{equation}
Let $A$ denote the linear non-self-adjoint operator in the Hilbert
space with domain $D\left( A\right) .$\ A complex number $\lambda
_{0}$ is called an eigenvalue of the operator $A$ if there exists a
non-zero element $y_{0}\in $ $D\left( A\right) \ $such that
$Ay_{0}=\lambda _{0}y_{0}$; in this case, $ y_{0}$ is called the
eigenvector of $A$ for $\lambda _{0}.$\ The\ eigenvectors for
$\lambda _{0}$ span a subspace of $D\left( A\right)$, called the
eigenspace for $\lambda _{0}.$

The element $y\in D\left( A\right) ,\ y\neq 0$ is called a root
vector of $A$ corresponding to the eigenvalue $\lambda _{0}$ if
$\left( T-\lambda _{0}I\right) ^{n}y=0$ for some $n\in \mathbb{N}.$
The root vectors for $ \lambda _{0}$ span a linear subspace of
$D\left( A\right) ,$ is called the root lineal for $\lambda _{0}.$\
The algebraic multiplicity of $\lambda _{0}$ is the dimension of its
root lineal. A root vector is called an associated vector if it is
not an eigenvector. The completeness of the system of all
eigenvectors and associated vectors of $A$ is equivalent to the
completeness of the system of all root vectors of this operator.

An operator $A$ is called dissipative if $Im\left(
Ax,x\right) \geq 0,\ \left( \forall x\in D\left( A\right) \right) .$
A bounded operator is dissipative if and only if
\begin{equation}
ImA=\frac{1}{2i}\left( A-A^{\ast }\right) \geq 0.
\end{equation}

\begin{theorem}[see \cite{w}]
Let $A$ be an invertible operator. Then $-A$ is dissipative if and
only if the inverse operator $A^{-1}$ of $A$ is dissipative.
\end{theorem}

A linear bounded operator $A\ $defined on the separable Hilbert
space $H$ is said to be of trace class (nuclear) if the series
\begin{equation}
\sum_{j}\left( Ae_{j},e_{j}\right)
\end{equation}
converges and has the same value in any orthonormal basis $\left\{
e_{j}\right\} $ of $H.$ The sum
\begin{equation}
TrA=\sum_{j}\left( Ae_{j},e_{j}\right)
\end{equation}
is called the trace of $A.$

The kernel $G(x,t),\ x,t\in \mathbb{R}$, of the integral operator
$K$ on $ L^{2}(\mathbb{R})$
\begin{equation}
Kf=\int_{\mathbb{R}}G(x,t)f\left( x\right) dx,\quad f\in
L^{2}(\mathbb{R}),
\end{equation}
is a Hilbert-Schmidt kernel if$\ |G(x,t)|^{2}$ is integrable on
$\mathbb{R} ^{2}$, i.e.,


\begin{equation}
\int_{\mathbb{R}^{2}}|G(x,t)|^{2}dxdt<\infty .
\end{equation}%
If $G(x,x)$ is measurable and summable, then it is called a
trace-class kernel (see \cite{e},\cite{f}). Recall that the integral
operator with a trace class-kernel is nuclear.

\begin{theorem}[see \protect\cite{gok,lid}]
If the dissipative operator $A$ is the nuclear operator, then the
system of root functions is complete in $H$.
\end{theorem}
\section{{Dissipative extensions of $q$-difference
operators}}

In this section, we describe all maximal dissipative, maximal
accretive, self-adjoint and other extensions of $q$-Sturm-Liouville
operators. We will consider a $q$-Sturm-Liouville operator
\begin{equation}\label{1}
l\left( y\right) :=-\frac{1}{q}D_{q^{-1}}D_{q}y\left( x\right)
+w\left( x\right) y\left( x\right) ,\quad 0\leq x\leq a<+\infty,
\end{equation}%
where $w\left( x\right) $ is defined on $[0,a]$ and continuous at
zero. The $q$-Wronskian of $y_{1}\left( x\right) ,\ y_{2}\left(
x\right) $ is defined
to be%
\begin{equation*}
W_{q}\left( y_{1},y_{2}\right) \left( x\right) :=y_{1}\left(
x\right) D_{q}y_{2}\left( x\right) -y_{2}\left( x\right)
D_{q}y_{1}\left( x\right) ,\quad x\in \lbrack 0,a].
\end{equation*}

Let $L_{0}\ $denote the closure of the minimal operator generated by (\ref{1})
and by $D_{0}$ its domain. Besides, by $D$ we denote the set of
all functions $y\left( x\right) $ from $L_{q}^{2}$ $(0,a)$ such that
$y\left(
x\right) $ and $D_{q}y\left( x\right) $ are continuous in $[0,a)$ and $%
l\left( y\right) \in $ $L_{q}^{2}$ $(0,a);\ D$ is the domain of the
maximal operator $L$. Furthermore, $L=L_{0}^{\ast }\ $\cite{na}$.\
$Suppose that the operator $L_{0}$ has a defect index $\left(
2,2\right) $, so the case of Weyl's $\ $limit-circle occurs for $l.$

For every $y,z\in D\ $\ we have $q$-Lagrange's identity (\cite{an})%
\begin{equation*}
\left( Ly,z\right) -\left( y,Lz\right) =[y,z]_{a}-[y,z]_{0}
\end{equation*}%
where $[y,z]_{x}:=y\left( x\right) \overline{D_{q^{-1}}z\left(
x\right) } -D_{q^{-1}}y\left( x\right) \overline{z\left( x\right)
}.$

By $u\left( x,\lambda \right) ,\ v\left( x,\lambda \right)
$ denote the solutions of the equation $l\left( y\right) =\lambda y$
satisfying the initial conditions
\begin{eqnarray}
u\left( 0,\lambda \right) &=&\cos \alpha ,\ \ D_{q^{-1}}u\left(
0,\lambda \right) =\sin \alpha,
\\
v\left( 0,\lambda \right) &=&-\sin \alpha, \ \ D_{q^{-1}}v\left(
0,\lambda \right) =\cos \alpha ,
\end{eqnarray}%
where $\alpha \in \mathbb{R}.\ $The solutions $u\left( x,\lambda \right) \ $%
and$\ v\left( x,\lambda \right) \ $\ form a fundamental system of
solutions of $l\left( y\right) =\lambda y$ and\ they are entire
functions of $\lambda \ ($see \cite{an}$).$ Let $u\left( x\right)
=u\left( x,0\right) \ $and$\
v\left( x\right) =v\left( x,0\right) \ $be the solutions of the equation $%
l\left( y\right) =0\ $satisfying the initial conditions
\begin{eqnarray}
u\left( 0\right) &=&\cos \alpha ,\ \ D_{q^{-1}}u\left( 0 \right)
=\sin \alpha,
\\
v\left( 0\right) &=&-\sin \alpha, \ \ D_{q^{-1}}v\left( 0\right)
=\cos \alpha .
\end{eqnarray} \\
\begin{lemma}
For arbitrary $y,z\in D,$ one has the equality
\begin{equation}
\lbrack y,z]_{x}[u,v]_{x}=[y,u]_{x}[z,v]_{x}-[y,v]_{x}[z,u]_{x},\ \ 0\leq x \leq +\infty
\label{w}
\end{equation}
\end{lemma}

\begin{proof}
Direct calculations verify equality (\ref{w}).
\end{proof}

Let us consider the functions $y\in D$\ satisfying the conditions%
\begin{eqnarray}
y\left( 0\right) \cos \alpha +D_{q^{-1}}y\left( 0\right) \sin \alpha
&=&0,
\\
\lbrack y,u]_{a}-h[y,v]_{a}&=&0,
\end{eqnarray}%
where $Imh>0,\ \alpha \in \mathbb{R}.\ $\\

We recall that a triple $\left( \mathbb{H},\Gamma _{1},\Gamma
_{2}\right) $ is called a space of boundary values of a closed
symmetric operator $A$ on a
Hilbert space $H$\ if $\Gamma _{1},\Gamma _{2}$ are linear maps from $%
D\left( A^{\ast }\right) $ to $H$ with equal deficiency numbers and
\begin{itemize}
\item[i)] for every $f,g\in D\left( A^{\ast }\right) $%
\begin{equation*}
\left( A^{\ast }f,g\right) _{H}-\left( f,A^{\ast }g\right)
_{H}=\left( \Gamma _{1}f,\Gamma _{2}g\right) _{\mathbb{H}}-\left(
\Gamma _{2}f,\Gamma _{1}g\right) _{\mathbb{H}},
\end{equation*}
\item[ii)] for any $F_{1},F_{2}\in H$\ there is a vector $f\in D\left( A^{\ast
}\right) $ such that$\ \Gamma _{1}f=F_{1},\ \Gamma _{2}f=F_{2}\
$(\cite{gor}).
\end{itemize}

Let us define by $\Gamma _{1},\Gamma _{2}$ linear maps from$\ D\ $to $%
\mathbb{C}^{2}$ by the formula%
\begin{equation}
\Gamma _{1}y=\left(
\begin{array}{c}
-y\left( 0\right) \\
\lbrack y,u]_{a}%
\end{array}%
\right) ,\ \Gamma _{2}y=\left(
\begin{array}{c}
D_{q^{-1}}y\left( 0\right) \\
\lbrack y,v]_{a}%
\end{array}%
\right) ,\quad y\in D.  \label{2}
\end{equation}%
For any $y,z\in D,$ using Lemma 1, we have
\begin{eqnarray}
\left( \Gamma _{1}y,\Gamma _{2}z\right) _{\mathbb{C}^{2}}-\left(
\Gamma
_{2}y,\Gamma _{1}z\right) _{\mathbb{C}^{2}} &=&-y\left( 0\right) \overline{%
D_{q^{-1}}z\left( 0\right) }+z\left( 0\right)
\overline{D_{q^{-1}}y\left(
0\right) }  \notag \\
&&+[y,u]_{a}[z,v]_{a}-[z,u]_{a}[y,v]_{a}  \notag \\
&=&[y,z]\left( a\right) -[y,z]\left( 0\right)  \notag \\
&=&\left( Ly,z\right) -\left( y,Lz\right).  \label{3}
\end{eqnarray}

\begin{theorem}
The triple $\left( \mathbb{C}^{2},\Gamma _{1},\Gamma _{2}\right) $
defined by (\ref{2}) is a boundary space of the operator $L_{0}.$
\end{theorem}

\begin{proof}
The proof is obtained from (\ref{3}) and the definition of the boundary
value space.
\end{proof}

\ The following result is obtained from Theorem 1.6 in Chapter 3 of \cite{gor}.

\begin{theorem}
For any contraction $K$ in $\mathbb{C}^{2}$ the restriction of the
operator $L$ to the set of functions $y\in D$ satisfying either
\begin{equation}
\left( K-I\right) \Gamma _{1}y+i\left( K+I\right) \Gamma _{2}y=0
\label{4}
\end{equation}
or
\begin{equation}
\left( K-I\right) \Gamma _{1}y-i\left( K+I\right) \Gamma _{2}y=0
\label{5}
\end{equation}
is a maximal dissipative and accretive extension of
the operator $L_{0}$, respectively. Conversely, every maximally dissipative
(accretive) extension of $L_{0}$ is the restriction of
$L$ to the set of functions $y\in D$ satisfying
(\ref{4})((\ref{5})), and the contraction $K$ is uniquely determined by the extension.
Conditions (\ref{4})((\ref{5}))) define self-adjoint extensions if $K$ is unitary.
\end{theorem}

In particular, boundary conditions%
\begin{eqnarray}
\cos \alpha y\left( 0\right) +\sin \alpha D_{q^{-1}}y\left( 0\right)
&=&0 \label{6}
\\
\lbrack y,u]_{a}-h[y,v]_{a}&=&0  \label{7}
\end{eqnarray}%
with $Imh>0,\ $describe the maximal dissipative
(self-adjoint) extensions of\ $L_{0}$ with separated boundary
conditions.

We know that all eigenvalues of a dissipative operator lie in the
closed upper half-plane. By virtue of Theorem 4, all the eigenvalues
of $L$ lie in the closed upper half-plane $Im\lambda \geq 0.$

\section{{Completeness theorem for the $q$-difference operators}}

\begin{theorem}
The operator $L$ has no any real eigenvalue.
\end{theorem}

\begin{proof}
Suppose that the operator $L$ has a real eigenvalue $\lambda _{0}.$ Let $%
\eta _{0}\left( x\right) =\eta \left( x,\lambda _{0}\right) $ be the
corresponding eigenfunction. Since
\begin{equation*}
Im\left( L\eta _{0},\eta _{0}\right) =Im\left( \lambda
_{0}\left\Vert \eta _{0}\right\Vert ^{2}\right) =Imh\left(
[\eta _{0},v]_{a}\right) ^{2},
\end{equation*}%
we get $[\eta _{0},v]_{a}=0.$ By boundary condition (\ref{7}),
we have $ [\eta _{0},u]_{a}=0.$ Thus
\begin{equation}
\lbrack \eta _{0}\left( t,\lambda _{0}\right) ,u]_{a}=[\eta
_{0}\left( t,\lambda _{0}\right) ,v]_{a}=0.  \label{15}
\end{equation}%
Let $\xi _{0}\left( t\right) =v\left( t,\lambda _{0}\right) .$ Then%
\begin{equation*}
1=[\eta _{0},\xi _{0}]_{a}=[\eta _{0},u]_{a}[\xi _{0},v]_{a}-[\eta
_{0},v]_{a}[\xi _{0},u]_{a}.
\end{equation*}%
By equality (\ref{15}), the right-hand side is equal to $0$. This contradiction proves Theorem 5.
\end{proof}

\begin{definition}
Let $f$ be an entire function. If for each $\varepsilon >0$ there
exists a finite constant $C_{\varepsilon }>0$, such that
\begin{equation}
|f\left( \lambda \right) |\leq C_{\varepsilon }e^{\varepsilon
\left\vert \lambda \right\vert },\quad \lambda \in \mathbb{C},
\end{equation}%
then $f$ is called an entire function of order $\leq 1$ of growth
and minimal type \cite{gok}.
\end{definition}

Let $\varphi \left( x,\lambda \right) $ be a single linearly
independent solution of the equation $l\left( y\right) =\lambda y,$
and
\begin{eqnarray*}
\tau _{1}\left( \lambda \right) &:&=[\varphi \left( x,\lambda
\right)
,u\left( x\right) ]_{a}, \\
\tau _{2}\left( \lambda \right) &:&=[\varphi \left( x,\lambda
\right)
,v\left( x\right) ]_{a}, \\
\tau \left( \lambda \right) &:&=\tau _{1}\left( \lambda \right)
-h\tau _{2}\left( \lambda \right) .
\end{eqnarray*}%
It is clear that%
\begin{equation*}
\sigma _{p}\left( L\right) =\left\{ \lambda :\lambda \in
\mathbb{C},\ \tau \left( \lambda \right) =0\right\},
\end{equation*}%
where $\sigma _{p}\left( L\right) $ denotes the set of all eigenvalues of $%
L.\ $Since $\varphi \left( a,\lambda \right) $ and
$D_{q^{-1}}\varphi \left(
a,\lambda \right) $ are entire functions of $\lambda $ of order $\leq \frac{1%
}{2}\ ($see \cite{an}$),$ consequently, $\tau \left( \lambda \right)
$ have the same property. Then $\tau \left( \lambda \right) $ is
entire functions of order $\leq 1\ $of\ growth, and of minimal
type. It is clear that all zeros of $\tau \left( \lambda \right) $
(all eigenvalues of $L$) are discrete and possible limit points of
these zeros (eigenvalues of $L$) can only occur at infinity.

For $y\in D\left( L\right) ,$ $Ly\left( x\right) =f\left( x\right) \
\left( x\in (0,a),\ f\left( x\right) \in L_{q}^{2}(0,a)\right) $ is
equivalent to the nonhomogeneous differential equation $l\left(
y\right) =f\left( x\right) \
\left( x\in (0,a)\right) ,$ subject to the conditions%
\begin{eqnarray*}
\cos \alpha y\left( 0\right) +\sin \alpha D_{q^{-1}}y\left( 0\right) &=&0, \\
\lbrack y,u]_{a}-h[y,v]_{a} &=&0,\quad Imh>0.
\end{eqnarray*}

By Theorem 5, there exists an inverse operator $L^{-1}.$ In order to
describe the operator $L^{-1}$, we use the Green's function method.
We consider the functions $v\left( x\right) $ and $\theta \left(
x\right)
=u\left( x\right) -hv\left( x\right) .$ These functions belong to the space $%
L_{q}^{2}$ $(0,a).$ Their Wronskian $W_{q}\left( v,\theta \right)
=-1.$

Let%
\begin{equation}
G\left( x,t\right) =\left\{
\begin{array}{cc}
v\left( x\right) \theta \left( t\right) , & 0\leq x\leq t\leq a \\[1ex]
v\left( t\right) \theta \left( x\right) , & 0\leq t\leq x\leq a%
\end{array}%
\right. .
\end{equation}%
Let us consider the integral operator $K$ defined by the formula
\begin{equation}
Kf=\int_{0}^{a}G\left( x,t\right) \overline{f\left( t\right)
}d_{q}t,\quad \left( f\in L_{q}^{2}(0,a)\right) .
\end{equation}%
It is evident that$\ K=L^{-1}$. Consequently,  the root lineals of
the
operators $L$ and $K$ coincide and, therefore, the completeness in $%
L_{q}^{2}(0,a)$ of the system of all eigenfunctions and associated functions of $%
L$ is equivalent to the completeness of those for $K$.

We obtain that $G\left( x,t\right) \ $is a Hilbert-Schmidth kernel since $%
v,\theta \in $ $L_{q}^{2}$ $(0,a).$ Furthermore, $G\left( x,t\right)
\ $is measurable and integrable on $(0,a).$ Hence $K$ is of trace
class. Since $L$ is a dissipative operator, $-K$ is a dissipative
operator by Theorem 1. Thus all conditions are satisfied for the
Lidskii's theorem. Hence we have;

\begin{theorem}
The system of all root functions of $-K$ (also $K$) is complete in
$L_{q}^{2}$ $(0,a)$.
\end{theorem}

From all above conclusions, we have;

\begin{theorem}
All eigenvalues of the operator $L$ lie in the open upper half-plane
and they are purely discrete. The limit points of these eigenvalues
can only occur at infinity. The system of all eigenfunctions and associated functions of the $%
L $ is complete in the space $L_{q}^{2}$ $(0,a)$.
\end{theorem}
%%%% Acknowledgment %%%%%%%%
%\section*{Acknowledgement}
%The author would like to thank the referees for the helpful
%suggestions.

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\end{thebibliography}
%\end{linenumbers}
\end{document}