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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 175, pp. 1--10.\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-2007/175\hfil Boundary-value problems]
{Self-adjoint boundary-value problems on time-scales}

\author[F. A. Davidson, B. P. Rynne\hfil EJDE-2007/175\hfilneg]
{Fordyce A. Davidson, Bryan P. Rynne}  % in alphabetical order

\address{Fordyce A. Davidson \newline
Division of Mathematics, Dundee University, Dundee DD1 4HN,
Scotland}
\email{fdavidso@maths.dundee.ac.uk}

\address{Bryan  P. Rynne \newline
Department of Mathematics and the Maxwell Institute for Mathematical
Sciences, Heriot-Watt University, Edinburgh EH14 4AS,
Scotland}
\email{bryan@ma.hw.ac.uk}

\thanks{Submitted June 6, 2007. Published December 12, 2007.}
\subjclass[2000]{34B05, 34L05, 39A05}
\keywords{Time-scales; boundary-value problem; self-adjoint linear operators;
\hfill\break\indent  Sobolev spaces}

\begin{abstract}
 In this paper we consider a second order, Sturm-Liouville-type
 boundary-value operator of the form
 $$
 L u := -[p u^{\nabla}]^{\Delta} + qu,
 $$
 on an arbitrary, bounded time-scale $\mathbb{T}$, for suitable
 functions $p,q$, together with suitable boundary conditions.
 We show that, with a suitable choice of domain, this operator can
 be formulated in the Hilbert space $L^2(\mathbb{T}_\kappa)$,
 in such a way that the resulting operator is  self-adjoint,
 with compact resolvent (here, `self-adjoint' means in the standard
 functional analytic meaning of this term).
 Previous discussions of operators of this, and similar, form have
 described them as `self-adjoint', but have not demonstrated
 self-adjointness in the standard functional analytic sense.
\end{abstract}

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

\section{Introduction}  \label{intro.sec}

Over the past decade a large number of papers  on
second order, Sturm-Liouville-type boundary value problems on
bounded  time-scales $\mathbb{T}$ have appeared.
Most of these deal with a $\Delta\Delta$
formulation of the corresponding differential operator, viz.
\begin{equation}  \label{dede_op.eq}
L u := -(p u^{\Delta})^{\Delta} + q u^\sigma,  \quad u \in D(L),
\end{equation}
for suitable functions $p,\,q$, on a suitable domain $D(L)$ (the
specification of the domain $D(L)$ includes suitable boundary
conditions on $u$; in this introductory section we omit details of
spaces and domains). Much of the basic theory of such operators is
described in, for example, \cite[Chapter 4]{BP1}. Such operators
have often been termed `self-adjoint'. However, it was shown in
\cite{DR1} that expressions of this form do not, in general, yield
self-adjoint operators, in the standard functional-analytic
meaning of the term `self-adjoint'. Indeed, it is shown in
\cite{DR1} that a fundamental property of self-adjoint operators
can fail for operators of the form \eqref{dede_op.eq}, so that the
standard theory of self-adjoint operators cannot readily be
applied to such operators.

More recently, in an attempt to obtain  self-adjointness,
differential operators in the following $\nabla\Delta$ form
\begin{equation}  \label{nade_op.eq}
L u := -(p u^{\nabla})^{\Delta} + qu, \quad u \in D(L)
\end{equation}
have been considered,
see for example, \cite{AGH,GUS2} and the references therein.
Such `mixed' operators result in a symmetric Green's function,
which is taken to indicate that the corresponding operators possess some
of the features of self-adjoint operators.
However, the operators constructed in these papers map between (different)
Banach spaces of continuously differentiable functions on $\mathbb{T}$,
whereas, in the standard functional-analytic definition,
a self-adjoint operator is defined on a subspace of a Hilbert space $H$,
and maps this subspace of $H$ into $H$ itself.
This Hilbert space formulation is necessary to obtain
many of the desirable properties of such operators.


In this paper our goal is to formulate the $\nabla\Delta$ operator
in \eqref{nade_op.eq} in the setting of
the Hilbert space $L^2(\mathbb{T}_\kappa)$ defined in \cite{RYN}.
This formulation is based on the Sobolev-type spaces defined in \cite{RYN}
consisting of functions on $\mathbb{T}$ having $L^2$-type generalised derivatives.
We then show that the resulting operator $L$, in $L^2(\mathbb{T}_\kappa)$,
is an unbounded, self-adjoint operator, with compact resolvent
(in the standard functional-analytic sense).
The extensive functional-analytic theory of such operators
is then available for this operator, although, for brevity,
we will not discuss
any applications of this general theory to this operator.


\begin{remark} \label{Rm1} \rm
We consider the $\nabla\Delta$ operator in \eqref{nade_op.eq}, but
operators involving   $\Delta\nabla$ combinations (see e.g.
\cite{AGH, BP2, GUS2}) could be treated similarly, there is no
essential difference in these formulations.  Using the
$\nabla\Delta$ form allows us to apply the results in \cite{RYN}
(based on a Lebesgue-type `$\Delta$-integral') unaltered. A
corresponding Lebesgue-type `$\nabla$-integral' could be
constructed using the methods in \cite{RYN}, and this would then
allow $\Delta\nabla$ operators to be considered in a similar
manner.
\end{remark}

\section{Preliminaries}

Papers on time-scales usually go through a set of standard
definitions  of integration and differentiation on time-scales.
For brevity we will omit this and simply refer to
\cite[Section~2]{RYN} for this standard material (which is, of
course, also discussed in most other time-scales papers). In
particular, we will use the Lebesgue-type $\Delta$-integral
defined in \cite{RYN}. A similar Lebesgue-type $\nabla$-integral
could readily be defined, but will not be required here. However,
we will need to use spaces of $\nabla$-differentiable functions,
in addition to the spaces of $\Delta$--differentiable functions
discussed in \cite{RYN}. To distinguish between these spaces will
require some slight modifications to the notation used for various
spaces and norms in \cite{RYN}, so we briefly discuss time-scale
differentiation, and the notation we will use.

Recall that a function $u : \mathbb{T} \to \mathbb{R}$ is
\emph{$\nabla$-differentiable} on $\mathbb{T}$ if, at each $t \in
\mathbb{T}_\kappa$, there exists $u^\nabla(t)$  such that, for
any $\epsilon>0$ there exists $\delta > 0$ such that
$$
\text{$s \in \mathbb{T}$ and $|t-s| < \delta$} \implies | u(\rho(t)) - u(s) - u^\nabla(t) (\rho(t)-s) | \le \epsilon | \rho(t)-s| ,
$$
see, for example, \cite[Ch. 3]{BP2};
the $\Delta$-derivative is defined similarly, by replacing $\rho(t)$
with $\sigma(t)$ throughout.

We let $C^0(\mathbb{T})$ (respectively $C_{\rm rd}^0(\mathbb{T})$, $C_{\rm ld}^0(\mathbb{T})$)
denote the set of continuous
(respectively rd-continuous, ld-continuous)
functions on $\mathbb{T}$; with the norm
$$
|u|_{\mathbb{T}} := \sup_{t \in \mathbb{T}}|u(t)|,
\quad u \in C_{\rm rd}(\mathbb{T}) \cup C_{\rm ld}(\mathbb{T}) ,
$$
all these spaces are Banach spaces.
We now let
$C^1(\mathbb{T},\Delta)$ (respectively $C_{\rm rd}^1(\mathbb{T},\Delta)$) denote the set of
functions $u \in C^0(\mathbb{T})$ which are $\Delta$-differentiable and
for which   $u^\Delta \in C^0(\mathbb{T}^\kappa)$
(respectively $u^\Delta \in C_{\rm rd}^0(\mathbb{T}^\kappa)$);
with the norm
$$
|u|_{\mathbb{T},\Delta} := |u|_{\mathbb{T}} + |u^\Delta|_{\mathbb{T}^\kappa} ,
\quad u \in C_{\rm rd}^1(\mathbb{T},\Delta),
$$
these spaces are Banach spaces.
Similarly, we define the Banach spaces $C^1(\mathbb{T},\nabla)$ and $C_{\rm ld}^1(\mathbb{T},\nabla)$ with norm
$$
|u|_{\mathbb{T},\nabla}:= |u|_{\mathbb{T}} + |u^\nabla|_{\mathbb{T}_\kappa},
\quad u \in C_{\rm ld}^1(\mathbb{T},\nabla).
$$

The spaces $C^1(\mathbb{T},\nabla)$ and $C^1(\mathbb{T},\Delta)$
need not be equal. For example, let $\mathbb{T} = [-1,0] \cup
[1,2]$ and define the function $u \equiv 0$ on $[-1,0]$, $u(t) =
t$ on $[1,2]$. It can be verified that $u \in
C^1(\mathbb{T},\nabla)$, but $u \not\in C^1(\mathbb{T},\Delta)$.
However, the following result gives a simple relationship between
these spaces

\begin{lemma}[{\cite[Theorem~6]{GUS2}}] \label{na_de_si.lem}
$C^1(\mathbb{T},\nabla) \subset C_{\rm rd}^1(\mathbb{T},\Delta)$.
If $u \in C^1(\mathbb{T},\nabla)$ then $u^\Delta =
(u^\nabla)^\sigma$.
\end{lemma}

It will also be necessary to $\nabla$-differentiate indefinite
$\Delta$-integrals, for which we will require the following lemma.


\begin{lemma}[{\cite[Theorem~2.10]{AG}}]\label{na_deriv_int.lem}
If $u \in C^0(\mathbb{T})$, $t_0 \in \mathbb{T}$,  and
$$
U_{t_0}(t) := \int_{t_0}^t u \,\Delta, \quad t \in \mathbb{T},
$$
then $U_{t_0} \in C_{\rm ld}^1(\mathbb{T},\nabla)$ and
$U_{t_0}^\nabla = u^\rho$ on $\mathbb{T}_\kappa .$
\end{lemma}

We will also require the Sobolev-type space of functions with generalised
$\Delta$-derivatives defined in \cite{RYN}, which we will denote here
by $H^1(\mathbb{T},\Delta)$  with associated norm
$$
\|u\|_{\mathbb{T},\Delta}:= \|u\|_{\mathbb{T}} + \|u^\Delta\|_{\mathbb{T}},
\quad u \in H^1(\mathbb{T},\Delta),
$$
where
$$
\|u\|^2_{\mathbb{T}}: = \int_{\mathbb{T}} |u|^2 \Delta, \quad
u \in L^2(\mathbb{T}).
$$
Note that the integral used here is the Lebesgue-type $\Delta$-integral
constructed in \cite{RYN}.
We also note that \cite[Lemma~3.5]{RYN} shows that
$C_{\rm rd}^1(\mathbb{T},\Delta) \subset H^1(\mathbb{T},\Delta)$,
so Lemma~\ref{na_de_si.lem}
has the following simple corollary, which will be required below.

\begin{corollary} \label{na_de_si.cor}
$C^1(\mathbb{T},\nabla) \subset H^1(\mathbb{T},\Delta)$.
\end{corollary}


Finally, in this preliminary section, we recall some basic
functional-analytic definitions, see for example
\cite[Ch. 13]{RUD}. Let $T : D(T) \subset H \rightarrow H$ be a linear
operator in a Hilbert space $H$, with inner product $\langle
\cdot\,,\cdot \rangle$. Then $T$ is \emph{symmetric} if
$$
\langle Tx,y \rangle = \langle x, Ty\rangle , \quad \forall\, x,\,y \in D(T) ,
$$
and $T$ is \emph{self-adjoint} if $D(T)$ is dense in $H$ and
$$
\langle Tx,y \rangle = \langle x,z\rangle , \quad \forall\, x \in D(T)
\implies  \text{$y \in D(T)$ and $ z = Ty.$}
$$

\section{A boundary value linear operator}  \label{form.sec}

\subsection{Definition of $L$}  \label{defn_L.sec}

Let $a=\inf \mathbb{T}$, $b = \sup \mathbb{T}$.
We are interested in the class of functions defined on $\mathbb{T}$
which satisfy the boundary conditions
\begin{equation}  \label{bc.eq}
u^\nabla(\sigma(a)) = \gamma_a u(\sigma(a)),
\quad
u^\nabla(b) = - \gamma_b u(b) ,
\end{equation}
with arbitrary constants
$\gamma_a \in (-\infty,\infty]$, $\gamma_b \in (-\infty,\infty]$,
and we define the following set of functions
%which will form the domain of our operator $L$:
\begin{gather*}
\mathcal{D} := \{ u \in C^1(\mathbb{T},\nabla) :
\text{$u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$ and $u$
satisfies \eqref{bc.eq}} \},
\\
D(L) :=  \{ w \in L^2(\mathbb{T}_\kappa) :
\text{$w = u|_{\mathbb{T}_\kappa^\kappa}$ for some $u \in \mathcal{D}$} \}
\end{gather*}
(in the definition of $D(L)$, $w = u|_{\mathbb{T}_\kappa^\kappa}$ denotes the
restriction of $u$ to the set $\mathbb{T}_\kappa^\kappa$,
and we recall from \cite{RYN} that the point $b$ has $\mu_\mathbb{T}$-measure
zero, so in the setting of equivalence classes of $L^2(\mathbb{T}_\kappa)$
functions, the value $u(b)$ is not well-defined).

Throughout, we impose the following additional assumption on
$\gamma_a$, $\gamma_b$.

\begin{assumption}  \label{gamma.ass}
(i) If $a$ is right-scattered then $\gamma_a < \infty$.
(ii) If $b$ is left-scattered then $1+\gamma_b ( b- \rho(b)) \neq0$.
\end{assumption}

These constructions require some further explanation and remarks.

\begin{enumerate}
\item[(a)]
In the above notation the cases
$\gamma_a=\infty$ or $\gamma_b=\infty$
are taken to mean the conditions
$u(\sigma(a)) = 0$ or $u(b) = 0$,
and in these cases it is the latter form that would be used in the calculations below.
Furthermore, if $a$ is right-dense, these cases correspond to the
Dirichlet-type conditions
$u(a) = 0$ or $u(b) = 0$.
It will be seen in Remark~\ref{D_bc_a.rem}
below that when $a$ is right-scattered the Dirichlet-type
condition at $a$ arises from a different value of $\gamma_a$.

\item[(b)]
Assumption~\ref{gamma.ass}
precludes the boundary conditions
$u(\sigma(a))=0$ (when $a$ is right-scattered)
or
$u(\rho(b))=0$ (when $b$ is left-scattered).
Either of these conditions lead to certain pathological properties of
the operator $L$ which we wish to avoid.

\item[(c)]
If $a$ is right-scattered it is natural,
in view of the definition of the $\nabla$-derivative,
to formulate the first boundary condition in \eqref{bc.eq}
in terms of $u^\nabla(\sigma(a))$, but the use of $u(\sigma(a))$,
rather than $u(a)$, may seem slightly strange.
This formulation is chosen, primarily,
to simplify certain formulae arising from
various integrations by parts below.
The following remark shows that $u(a)$ could be
used in \eqref{bc.eq} simply by changing the value of $\gamma_a$.

\item[(d)] \label{alt_bc.rem}
If $a$ is right-scattered or $b$ is left-scattered, the corresponding boundary conditions in \eqref{bc.eq}
can be rewritten in the alternative forms
\begin{equation}  \label{alt_BC.eq}
\begin{split}
 u(a) - \bigl(1 +  (\sigma(a)-a)\gamma_a \bigr) u(\sigma(a)) & = 0 ,
\\
 u(\rho(b)) - \bigl(1 +  (b-\rho(b))\gamma_b \bigr) u(b) & = 0 .
\end{split}
\end{equation}
Hence, by \eqref{alt_BC.eq} and Assumption~\ref{gamma.ass},
if $u \in \mathcal{D}$ then $u(a)$ and $u(b)$ are determined by
$u(\sigma(a))$ and $u(\rho(b))$,
that is,
$u$ is determined entirely by its
restriction $w = u|_{\mathbb{T}_\kappa^\kappa} \in D(L)$.
Conversely, by using \eqref{alt_BC.eq}, any function $w \in D(L)$ can be
extended to $\mathbb{T}$ to yield a function $u \in \mathcal{D}$.
Thus, the sets $\mathcal{D}$ and $D(L)$ are (algebraically) isomorphic,
and can be naturally identified with each other.
In our discussion of $L$ below we will make use of this identification,
and we will generally use the symbol $u$ interchangeably for an element
of either $\mathcal{D}$ or $D(L)$.

\item[(e)] \label{D_bc_a.rem}
If $a$ is right-scattered then it follows from \eqref{alt_BC.eq}
that we obtain the Dirichlet-type
condition $u(a) = 0$ by choosing $\gamma_a$ such that
$1 +  (\sigma(a)-a)\gamma_a = 0$.
\end{enumerate}

Having dealt with the boundary conditions, we now define the
desired differential operator $L$.
Suppose that
$p \in H^1(\mathbb{T}_\kappa,\Delta)$, $q \in L^2(\mathbb{T}_\kappa)$,
with
$$
p_{\min} := \min \{ p(t) : t \in \mathbb{T}_\kappa \} > 0 ,
$$
and define the linear operator
$L : D(L) \subset L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$
by
\begin{align*}
%D(L) &:= \{ u \in C^1(\mathbb{T}_\kappa,\nabla) : \text{$u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$ and $u$ satisfies \eqref{bc.eq}} \},
%\\
L u &:= -[p u^{\nabla}]^{\Delta_g} + qu, \quad u \in D(L),
\end{align*}
where $\Delta_g$ is the generalised $\Delta-$derivative constructed  in \cite{RYN}. The  definition $L$ also requires some further explanation and remarks.

\begin{enumerate}
\item[(a)]
The set $\mathcal{D}$ would be a natural domain for the operator $L$.
However, to obtain a self-adjoint operator it is necessary that the domain
and range of $L$ lie in the same Hilbert space
(which we take to be $L^2(\mathbb{T}_\kappa)$).
For this reason we introduce the domain $D(L) \subset L^2(\mathbb{T}_\kappa)$,
isomorphic to $\mathcal{D}$.

\item[(b)]
In light of the identification of $\mathcal{D}$ and $D(L)$ described in
Remark~\ref{alt_bc.rem} above,
we regard the calculation of $Lu \in L^2(\mathbb{T}_\kappa)$
from $u \in D(L)$ as proceeding in the following manner:
use \eqref{alt_BC.eq} to extend $u$ from the set $\mathbb{T}_\kappa^\kappa$
to $\mathbb{T}$
(yielding an element of $\mathcal{D}$, which we still write as $u$),
and then construct $u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$
and $(u^\nabla)^{\Delta_g} \in L^2(\mathbb{T}_\kappa)$
in the usual manner
(by the definition of $\mathcal{D}$, these are well-defined for $u \in \mathcal{D}$).

\item[(c)]
The operator $Lu = -[p u^{\Delta}]^{\Delta_g} + qu^\sigma$,
on a similar domain, was considered in \cite{RYN}.
However, we will see that the above operator is self-adjoint,
(in the functional-analytic sense),
whereas the operator in \cite{RYN} is not.
Despite this difference, the comments in \cite[Remarks~5.1 and~5.2]{RYN}
regarding the definition of $L$ there apply equally
well to the above operator.
\end{enumerate}

\subsection{Properties of $L$}  \label{props_L.sec}

We now obtain various basic properties of $L$.

\begin{lemma}  \label{L_symm.lem}
The operator $L$ is symmetric with respect to the inner product
$\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on $L^2(\mathbb{T}_\kappa)$,
that is,
%For any $u,\,v \in D(L)$,
\begin{equation}  \label{L_symm.eq}
\langle Lu , v \rangle_{\mathbb{T}_\kappa} = \langle u , Lv \rangle_{\mathbb{T}_\kappa} ,
\quad u,\,v \in D(L) .
\end{equation}
\end{lemma}

\begin{proof}
By definition, we can regard  $u,\,v$ as belonging to $\mathcal{D}$,
that is $u,\,v \in C^1(\mathbb{T},\nabla)$.
Thus, by Lemma~\ref{na_de_si.lem},
$u,\,v \in C_{\rm rd}^1(\mathbb{T},\Delta)$,
and hence, by  \cite[Corollary~4.6~(f)]{RYN},
\begin{equation}  \label{Luv_ip.eq}
\begin{split}
\langle L u , v \rangle_{\mathbb{T}_\kappa}
&=
\int_{\sigma(a)}^b  (p u^\nabla)^\sigma v^\Delta \,\Delta  - [p u^\nabla v ]_{\sigma(a)}^b
+ \int_{\sigma(a)}^b q u v \, \Delta
\\[1 ex]&=
\int_{\sigma(a)}^b \bigl( p^\sigma u^\Delta v^\Delta + q u v \bigr)\,\Delta  + B(u,v)
\end{split}
\end{equation}
where, by \eqref{bc.eq},
$$
B(u,v) = \gamma_a p(\sigma(a)) u(\sigma(a)) v(\sigma(a)) + \gamma_b p(b) u(b) v(b)
$$
(if $\gamma_a=\infty$ or $\gamma_b=\infty$ then we omit the corresponding
term in this formula).
The result now follows from the symmetry in $u$ and $v$ of the
right hand side of \eqref{Luv_ip.eq}.
\end{proof}


\begin{lemma} \label{L_pos_def.lem}
There exists a constant $C_L$ such that
\begin{equation}  \label{L_pos_def.eq}
\langle L u ,  u \rangle_{\mathbb{T}_\kappa}
\ge  \tfrac12 p_{\min}  \|u\|_{\mathbb{T}_\kappa,\Delta}^2 + C_L \|u\|_{\mathbb{T}_\kappa}^2 ,
\quad   u \in D(L).
\end{equation}
\end{lemma}

\begin{remark} \label{rmk3.4} \rm
The constant $C_L$ in Lemma~\ref{L_pos_def.lem} need not be positive.
\end{remark}


\begin{proof}
Suppose that $u \in D(L)$. Then we can regard $u$ as belonging to $\mathcal{D}$,
and it follows from  \eqref{Luv_ip.eq}
and the Cauchy-Schwarz inequality
that
\begin{equation}  \label{int_parts.eq}
\langle L u , u \rangle_{\mathbb{T}_\kappa}
\ge
p_{\min} \|u^\Delta\|_{\mathbb{T}_\kappa}^2 - C_1 |u|_{\mathbb{T}_\kappa}^2 ,
\end{equation}
for some constant $C_1 \ge 0$ (independent of $u$),
and by \eqref{alt_BC.eq} and Assumption~\ref{gamma.ass},
\begin{equation}\label{1}
|u|_{\mathbb{T}_\kappa}^2  \leq C_2 |u|_{\mathbb{T}^\kappa_\kappa}^2,
\end{equation}
for some constant $C_2>0$.
Also, a straightforward  modification of the proof of
\cite[Theorem~4.16]{RYN} shows that
for any $\epsilon>0$ there exists a constant $C_3(\epsilon)>0$ such that
\begin{equation} \label{4}
|w|_{\mathbb{T}_\kappa^\kappa}  \leq \epsilon \|w^\Delta\|_{\mathbb{T}_\kappa} + C_3(\epsilon) \|w\|_{\mathbb{T}_\kappa} ,
\quad w \in H^1(\mathbb{T}_\kappa,\Delta) .
\end{equation}
By Corollary~\ref{na_de_si.cor} and the definition of $\mathcal{D}$,
$u \in H^1(\mathbb{T}_\kappa,\Delta)$, so
putting $\epsilon$ sufficiently small  and $w = u$ in \eqref{4}
and combining this with
\eqref{int_parts.eq} and \eqref{1} yields \eqref{L_pos_def.eq}.
\end{proof}


Invertibility of $L$ will be important below,
and it will be seen that invertibility follows from injectivity of $L$,
so we now consider this.
In general, $L$ need not be injective, but the following result shows
that we can obtain injectivity by adding to $L$ a sufficiently large
scalar multiple of the identity operator $I : L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$.
In many situations, if $L$ itself is not injective
then it is possible, with no loss of generality,
to replace $L$ with the injective operator $L_c$
given by the following result.

\begin{theorem} \label{L_injective.thm}
If $c + C_L > 0$ then the operator
$L_c := L + c I$ is injective.
\end{theorem}

\begin{proof}
It follows from \eqref{L_pos_def.eq} that
$$
\langle L_c u ,  u \rangle_{\mathbb{T}_\kappa}
\ge  \tfrac12 p_{\min}  \|u\|_{\mathbb{T}_\kappa,\Delta}^2 + (c+C_L) \|u\|_{\mathbb{T}_\kappa}^2
>0,
\quad  0 \ne u \in D(L),
$$
which proves that $L_c$ is injective.
\end{proof}


The following result gives simple criteria under which $L$ itself is injective.


\begin{theorem} \label{L_injective_pos.thm}
Suppose that $q \ge 0$ on $\mathbb{T}_\kappa$ and $\gamma_a,\,\gamma_b \ge 0$.
Then $L$ is injective under  either of the hypotheses$:$
\begin{itemize}
\item[(i)]
$\gamma_a + \gamma_b > 0;$
\item[(ii)]
$\|q\|_{\mathbb{T}_\kappa} > 0.$
\end{itemize}
\end{theorem}


\begin{proof}
We consider hypothesis (i), a similar proof holds for hypothesis (ii).
Suppose that $0 \ne u \in D(L)$ and $Lu=0$.
It follows from this and \eqref{Luv_ip.eq} that
$$
0 = \langle Lu,u \rangle_{\mathbb{T}_\kappa} \ge p_{\min} \|u^\Delta\|_{\mathbb{T}_\kappa}^2 + B(u,u),
$$
and hence, by \cite[Corollary~4.6]{RYN}, $u \equiv 0$ on $\mathbb{T}_\kappa$.
\end{proof}

We will also need the following result regarding
solutions of the corresponding initial value problem.
This result can be proved in a similar manner to that of
\cite[Theorem~5.8]{RYN}.


\begin{theorem} \label{ivp.thm}
For any $h \in L^2(\mathbb{T}_\kappa)$ and $\tau \in \mathbb{T}_\kappa$, $\eta_1,\,\eta_2 \in \mathbb{R}$,
the initial value problem
\begin{equation} \label{ivp.eq}
\begin{gathered}
-(pu^\nabla)^{\Delta_g} + q u = h,
\\
u(\tau) = \eta_1, \quad u^\nabla(\tau) = \eta_2,
\end{gathered}
\end{equation}
has a unique solution $u \in C^1(\mathbb{T},\nabla)$,
with $u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$.
\end{theorem}


Let $\phi,\,\psi$ be the solutions of \eqref{ivp.eq} given by Theorem~\ref{ivp.thm},
with $h=0$ and  the `initial' conditions
\begin{equation} \label{ivpbc.eq}
\begin{array}{rr@{\ }l}
\phi(\sigma(a)) = 1, & \phi^{\nabla}(\sigma(a)) &= \gamma_a,\\
\psi(b) = 1, & \psi^{\nabla}(b) &= -\gamma_b ,\,
\end{array}
\end{equation}
with the obvious modification, here and below, when $\gamma_a=\infty$ or
$\gamma_b=\infty$.
Also, let
$$
W := p \bigl(\phi^{\nabla}\psi - \psi^{\nabla}\phi\bigr)
\in H^1(\mathbb{T}_\kappa,\Delta)
$$
(it follows from the properties of $\phi,\,\psi$ given by
Theorem~\ref{ivp.thm},
together with Corollary~\ref{na_de_si.cor} and \cite[Corollary~4.6]{RYN},
that $W \in H^1(\mathbb{T}_\kappa,\Delta)$).


\begin{lemma} \label{dn0.lem}
$W$ is constant on $\mathbb{T}_\kappa$.
The operator $L$ is injective if and only if  $W \neq 0$.
\end{lemma}

\begin{proof}
 From the definitions of $\phi$, $\psi$, Corollary~\ref{na_de_si.cor}
and \cite[Corollary~4.6]{RYN},
\begin{align*}
W^{\Delta_g} &= (p\phi^\nabla)^{\Delta_g} \psi + (p\phi^{\nabla})^\sigma \psi^{\Delta}
-(p\psi^\nabla)^{\Delta_g} \phi - (p\psi^{\nabla})^\sigma \phi^{\Delta}
\\&=
q \phi \psi  + (p\phi^{\nabla}\psi^{\nabla})^\sigma - q \psi \phi - (p\psi^{\nabla}\phi^{\nabla})^\sigma
= 0,
\end{align*}
so by \cite[Corollary~4.6]{RYN}, $W \equiv \text{const}$.
Moreover, by \eqref{ivpbc.eq},
$$
W = p(b) \bigl( \phi^\nabla(b) + \gamma_b  \phi(b)) ,
$$
so $W=0$ if and only if $\phi$ satisfies the boundary conditions
\eqref{bc.eq}.
Clearly, if $\phi$ satisfies \eqref{bc.eq}
then $L$ is not injective, and the converse follows immediately from
linearity and the uniqueness of the solution of the initial value
problem for $\phi$.
\end{proof}

We can now begin the construction of the inverse of $L$
(when $L$ is injective).
Equivalently, we construct a solution of the boundary value problem
\begin{equation}  \label{bvp.eq}
Lu = h, \quad h \in L^2(\mathbb{T}_\kappa) ,  \quad  u \in D(L),
\end{equation}
for any $h \in L^2(\mathbb{T}_\kappa)$.

\begin{definition} \label{xst.def} \rm
Suppose that $L$ is injective. For $(t,s) \in \mathbb{T} \times \mathbb{T}$ let
$$
g(t,s) :=
\begin{cases}
W^{-1} \psi(t) \phi(s), \quad \text{if $t \geq s$,} \\
W^{-1} \phi(t) \psi(s), \quad \text{if $t \leq s$.}
\end{cases}
$$
Clearly, $g$ is continuous on $\mathbb{T} \times \mathbb{T}$.
For any $h \in L^2(\mathbb{T}_\kappa)$, let
\begin{equation}  \label{Gh.eq}
Gh(t) := \int_{\sigma(a)}^b g(t,\cdot) h \,\Delta ,
\quad t \in \mathbb{T}_\kappa^\kappa.
\end{equation}
\end{definition}

\begin{theorem} \label{LGinverse.thm}
Suppose that $L$ is injective.
Then$:$
\begin{itemize}
\item[(i)]
for any $h \in L^2(\mathbb{T}_\kappa)$ the function $u = Gh \in D(L),$
and $u$ is the unique solution of \eqref{bvp.eq}$;$

\item [(ii)]
the operators
$L :  D(L) \subset L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$,
$G : L^2(\mathbb{T}_\kappa) \to D(L) \subset L^2(\mathbb{T}_\kappa)$,
are  invertible, linear operators and
$L^{-1} = G$, $G^{-1} = L$.
The operator $G$ is compact, while if $\dim L^2(\mathbb{T}_\kappa) = \infty$
then $L$ is unbounded.
\end{itemize}
\end{theorem}

\begin{remark}  \label{epresults.rem} \rm
We call $g$ the \emph{Green's function} and $G$ the \emph{Green's
operator} for the operator $L$.
\end{remark}


\begin{proof}
The uniqueness follows immediately from the
injectivity of $L$.
Now suppose that $h \in C^0(\mathbb{T}_\kappa)$.
To simplify the following calculations we will suppose that $p \equiv 1$;
the general proof is similar.

It is clear that the formula for $u = Gh$ in \eqref{Gh.eq} can be extended to define
a function on the whole of $\mathbb{T}$, which we continue to denote by $u$.
Now suppose that $a$ is right-dense.
Then by direct calculation
(using Lemma~\ref{na_deriv_int.lem} above, the product rule for nabla derivatives, see \cite{BP1},
and for generalised derivatives, see Corollary~4.6 in \cite{RYN},
and (3.3) in \cite{RYN}),
\begin{equation*}
\begin{split}
W u(t) &=
\psi(t) \int_{\sigma(a)}^t \phi h \,\Delta + \phi(t) \int_t^{b} \psi h \,\Delta ,
\quad t \in \mathbb{T},\\
W u^{\nabla}(t)
&= \psi^\rho(t) \phi^\rho(t) h^\rho(t)
 + \psi^\nabla(t) \int_{\sigma(a)}^t \phi h \,\Delta\\
&\quad - \phi^\rho(t) \psi^\rho(t) h^\rho(t)
 +\phi^\nabla(t) \int_t^{b} \psi h \,\Delta \\
&=\psi^\nabla(t) \int_{\sigma(a)}^t \phi h \,\Delta
 +\phi^\nabla(t) \int_t^{b} \psi h \,\Delta,
\quad t \in \mathbb{T},
\\
W (u^\nabla)^{\Delta_g}(t)
&=-W h(t) + (\psi^\nabla)^{\Delta_g}(t) \int_{\sigma(a)}^{\sigma(t)}
  \phi h \,\Delta
 + (\phi^\nabla)^{\Delta_g}(t) \int_{\sigma(t)}^{b} \psi h \Delta \\
&= -W h(t) + q(t) \biggl\{ \psi(t) \int_{\sigma(a)}^{\sigma(t)} \phi h \,\Delta
 + \phi(t) \int_{\sigma(t)}^{b} \psi h \,\Delta \biggr\} \\
&= -W h(t) + q(t) \biggl\{ \psi(t) \int_{\sigma(a)}^t \phi h \,\Delta
 + \phi(t) \int_t^{b} \psi h \,\Delta \biggr\} \\
&= -W h(t) + q(t) W u(t),
\quad \text{$\mu_\mathbb{T}$-a.e. $ t \in \mathbb{T}$.}
\end{split}
\end{equation*}
It follows directly from these formulae and \eqref{ivpbc.eq} that
$u = Gh \in \mathcal{D}$, with
\begin{equation}  \label{Gh_bnd.eq}
\|Gh\|_{\mathcal{D}} = \|u\|_{\mathcal{D}}:=  |u|_{\nabla,\mathbb{T}} + \|u^\nabla\|_{\mathbb{T}_\kappa,\Delta}
\le C \|h\|{_{\mathbb{T}_\kappa}},
\quad
h \in C^0(\mathbb{T}_\kappa),
\end{equation}
for some constant $C$.
On the other hand, if $a$ is right-scattered
the calculation of $Wu(a)$ and $Wu^\nabla(\sigma(a))$ needs to be amended
slightly to avoid reference to the (undefined) value $h(a)$,
in the following manner.
Directly from \eqref{Gh.eq},
\begin{gather*}
W u(a) = \phi(a) \int_{\sigma(a)}^b \psi h \,\Delta ,
\quad
W u(\sigma(a)) = \int_{\sigma(a)}^b \psi h \,\Delta ,
\\
W u^\nabla(\sigma(a)) = W \frac{u(\sigma(a))-u(a)}{\sigma(a)-a}
=
\phi^\nabla(\sigma(a)) \int_{\sigma(a)}^b \psi h \,\Delta
=
W \gamma_a u(\sigma(a)).
\end{gather*}
Thus the boundary condition at $\sigma(a)$ also holds in this case,
and it follows from the preceding formulae that
if $\sigma(a)$ is right-dense then
$\lim_{t\to\sigma(a)^+} u(t) = u(\sigma(a))$,
$\lim_{t\to\sigma(a)^+} u^\nabla(t) = u^\nabla(\sigma(a))$,
(here, $t\to\sigma(a)^+$ through points in $\mathbb{T}_\kappa$),
so that $u \in C^1(\mathbb{T},\nabla)$, that is we again have
$u \in \mathcal{D}$.
Clearly, the inequality \eqref{Gh_bnd.eq} also holds in this case.

Now, since $C^0(\mathbb{T}_\kappa)$ is dense in $L^2(\mathbb{T}_\kappa)$
(see Lemma~3.5 in \cite{RYN}),
and $\mathcal{D}$ is a Banach space (with respect to the above norm
$\|\cdot\|_{\mathcal{D}}$),
the inequality \eqref{Gh_bnd.eq}  extends from the set
$C^0(\mathbb{T}_\kappa)$ to the whole
of $L^2(\mathbb{T}_\kappa)$ by continuity.
Hence, in particular, $G$ is bounded as an operator from
$L^2(\mathbb{T}_\kappa)$
into $\mathcal{D}$.
The assertions about the operators $L$ and $G$ now follow immediately
(the compactness of $G$ follows from \eqref{Gh_bnd.eq},
Lemma~\ref{na_de_si.lem} and the compactness of the embedding
$C_{\rm rd}^1(\mathbb{T},\Delta) \to C^0(\mathbb{T})$,
see \cite[Lemma~2.2]{DR1}).
\end{proof}


We can now prove that $L$ is self-adjoint
(irrespective of injectivity of $L$).


\begin{theorem}  \label{L_sa.thm}
The domain $D(L)$ is dense in $L^2(\mathbb{T}_\kappa)$,
and the operator $L$ is self-adjoint with respect to the inner product
$\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on
$L^2(\mathbb{T}_\kappa)$.
\end{theorem}

\begin{proof}
It suffices to prove the result for $L_c$, for arbitrary
$c \in \mathbb{R}$, so without loss of generality we suppose that
$c=0$ and $L$ is injective.
If $D(L)$ is not dense in $L^2(\mathbb{T}_\kappa)$
then there exists $0 \ne w \in L^2(\mathbb{T}_\kappa)$ such that
$$
\langle u , w \rangle_{\mathbb{T}_\kappa} = 0,  \quad  \forall u \in D(L).
$$
Since $R(L) = L^2(\mathbb{T}_\kappa)$, we have $w = Lz$ for some $z \in D(L)$,
so by Lemma~\ref{L_symm.lem},
$$
0 = \langle u , Lz \rangle_{\mathbb{T}_\kappa}
= \langle Lu , z \rangle_{\mathbb{T}_\kappa},  \quad  \forall u \in D(L),
$$
and hence $z=0$ (again, since $R(L) = L^2(\mathbb{T}_\kappa)$).
However, this implies that $w=0$, which contradicts the choice of $w$,
and so proves that $D(L)$ is dense in $L^2(\mathbb{T}_\kappa)$.

Now suppose that
\begin{equation}  \label{sa_test.eq}
\langle Lu , v \rangle_{\mathbb{T}_\kappa}
= \langle u , w \rangle_{\mathbb{T}_\kappa},  \quad  \forall u \in D(L),
\end{equation}
for some $v ,\,w \in L^2(\mathbb{T}_\kappa)$.
We again have $w = Lz$, for some $z \in D(L)$,
and so from \eqref{sa_test.eq},
$$
\langle Lu , v-z \rangle_{\mathbb{T}_\kappa} = 0,  \quad  \forall u \in D(L),
$$
and hence $v=z$.
That is, $v \in D(L)$ and $w = Lv$, which proves that
$L$ is self-adjoint.
\end{proof}

\begin{corollary}  \label{G_sa.cor}
Suppose that $L$ is injective. Then the operator $G$ is self-adjoint with respect to the inner product
$\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on $L^2(\mathbb{T}_\kappa)$.
\end{corollary}

\begin{proof}
Let $u,\,v \in L^2(\mathbb{T}_\kappa)$ be arbitrary.
Then by Theorem~\ref{LGinverse.thm}, $u=Lx$, $v=Ly$, for some
$x,\,y \in D(L)$.
Hence, by Lemma~\ref{L_symm.lem} and Theorem~\ref{LGinverse.thm},
$$
\langle Gu , v \rangle_{\mathbb{T}_\kappa}
= \langle x , Ly \rangle_{\mathbb{T}_\kappa}
= \langle Lx , y \rangle_{\mathbb{T}_\kappa}
= \langle u , Gv \rangle_{\mathbb{T}_\kappa}  ,
$$
which proves that $G$ is self-adjoint.
\end{proof}


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\end{document}