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Abstract boundary value problem, symmetric operators, Green formula, Weyl 
function
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\begin{document}

\title{A General Boundary Value Problem and its Weyl Function}

\author{ Vladimir Ryzhov }



\maketitle


\begin{center}{ \it
Dedicated to Prof. Boris Pavlov on the occasion of his 70th birthday
}
\end{center}

\smallskip

%%% ----------------------------------------------------------------------


%\bigskip
\begin{abstract}
We study the abstract boundary value problem defined in terms of the
Green identity and introduce the concept of Weyl operator
function~$M(\cdot)$ that agrees with other definitions found in the
current literature.
%
In typical cases of problems arising from the multidimensional
partial equations of mathematical physics the function~$M(\cdot)$
takes values in the set of unbounded densely defined operators
acting on the auxiliary boundary space.
%
Exact formulae are obtained and essential properties of~$M(\cdot)$
are studied.
%
In particular, we consider boundary problems defined by various
boundary conditions and justify the well known procedure that
reduces such problems to the ``equation on the boundary'' involving
the Weyl function, prove an analogue of the Borg-Levinson theorem,
and link our results to the classical theory of extensions of
symmetric operators.
\end{abstract}




\keywords{Abstract boundary value problem, symmetric
     operators,  Green formula, Weyl function.}


\subjclass{Primary 47B25, 47F05; \  Secondary 35J25, 31B10}



%%% ----------------------------------------------------------------------
%%% ----------------------------------------------------------------------



\section*{Introduction and notation}


The abstract approach to various linear boundary value problems
arising in mathematical physics is commonly based on the extension
theory of symmetric operators that was developed in classic works of
J. von Neumann, H.~Weyl, D.~Hilbert, K.~Friedrichs, M.~Krein and
many others.
%
In fact, the extension theory as we know it today can be seen as a
cumulative result of intensive research on various problems of
physics carried out in the last century.
%
The basic assumption under which the fruitful theory was developed
is the validity of the so-called Green formula that furnish a
convenient setting for the study of various boundary conditions in
the form of linear dependencies between traces of the solution being
sought and its normal derivatives on the boundary.
%
The abstract framework allowed one to treat classical boundary value
problems such as problems of Dirichlet, Neumann, and Robin for
partial differential equations of mathematical physics as direct
applications of the extension theory.
%

%%==========================================================================


%
 An abstract approach based on the Green
formula that only requires the main operator to be symmetric and
densely defined and does not impose any constraints on its nature,
be it differential or otherwise, was developed later and summarized
in many works, see~\cite{Bru,DM1, DM2,GG,Ko, KK, LS} and references
therein.
%
The main object of the theory is the so called boundary triple that
consists of the main operator assumed to be the adjoint to a
symmetric densely defined one, and two boundary mappings defined on
its domain with values in an auxiliary Hilbert space.
%
An abstract Green formula then can explicitly written in terms of
these objects and subsequently studied.
%
The authors of~\cite{DM1, DM2} succeeded in advancing the theory to
the point where an abstract version of the Weyl-Titchmarsh function,
previously known only for the Sturm-Liouville type of problems, was
successfully introduced and utilized for the following spectral
analysis.
%
Conditions imposed in~\cite{DM1, DM2} are rather strict and don't
allow one to consider the case of symmetric operators with
non-surjective boundary mappings.
%
Unfortunately, this class includes many interesting partial
differential operators of mathematical physics with infinite defect
indices.
%
A few successful attempts have been made recently in order to
overcome these limitations and extend the theory of boundary triples
and respective Weyl functions to situations where requirements
of~\cite{DM1, DM2} are not met, see~\cite{AP, KK} for examples.
%


%%==========================================================================


The paper is dedicated to the systematic treatment of general
boundary value problems and associated Weyl functions by means of
operator theory.
%
The approach offered here is not based on the theory of symmetric
operators and their extensions; we engage another line of reasoning.
%
The starting point is the Green formula for the main operator and
certain assumptions regarding its restriction to the null set of one
of the boundary mappings.
%
The illustrative example supplied at the end of paper deals with the
Laplacian defined on its natural domain in the bounded region
in~$\Real^3$ as the main operator and two boundary operators that
map a smooth function defined inside this region to its trace and
trace of its normal derivative on the boundary.
%
The restriction mentioned above turns out to be the Dirichlet
Laplacian which is a selfadjoint operator with the bounded inverse.
These two properties of the Laplacian are in fact the main
conditions imposed on the abstract problem under consideration.
%
It is useful to keep this example in mind reading the paper.
%
Further, the Weyl function defined in the manner consistent with the
previous definitions is an analytic function whose values are
unbounded operators.
%
However, after substraction of its value at the origin it becomes an
analytic bounded operator function of Herglotz class, i.~e. the
operator function with the positive imaginary part in the upper half
plane.
%
In the example cited above the Weyl function coincides with the
Dirichlet-to-Neumann map for the Laplacian, see~\cite{SU}.
%
It has to be noted that all results of the paper are obtained under
minimal assumptions about participating objects.
%
In particular, the main operator is not assumed to be closed.


%%============================================================================

There exits a tight relationship of our results with the theory of
open systems as developed by M.~S.~Liv\^{s}i\c{c} in his seminal
book~\cite{Liv}.
%
Broadly speaking, the main operator of the boundary problem can be
understood as a main operator of a certain open system
 whose input and output transformations are
described in terms of boundary mappings.
%
Correspondingly, in light of this interpretation the Weyl function
becomes a transfer function of this system and as such describes the
system's properties.
%
More importantly, the approach based on the open system theory
offers an intuitive way to model various boundary value problems as
closed systems perturbed by certain ``channel vectors''. The latter
are naturally interpreted as external control and observation
channels  attached to the system.
%
The author hopes to expound these ideas and address other relevant
topics in a subsequent publication.

%%============================================================================

\smallskip

The paper is organized as follows.


%%============================================================================
In the Section~1 after definition of the Green formula two main
assumptions are formulated.
%
Then we study their immediate consequences and make a few
observations that will be used later in the paper.
%
The Section concludes with two equivalent descriptions of the
boundary value problem subject to the main assumptions.



%%============================================================================
Section~2 links the setting introduced in the previous section with
the extension theory of symmetric operators.
%
 It is possible to define the minimal operator
and show that it is symmetric, and if its domain is dense, the
standard extension theory can be applied.
%
Then we formulate the criteria for complete nonselfadjointness of
the minimal operator and distinguish two special cases which are
equivalent to the Friedrichs and Krein extensions in more
conventional setting of semibounded densely defined symmetric
operators.



%%============================================================================
Third Section is dedicated to the Weyl function and its properties.
%
Here we study the question of recovery of the boundary value problem
by its Weyl function and prove an analogue of the Borg-Levinson
theorem.
%
The last paragraph deal with general boundary value conditions and
offers a recipe to reduce the corresponding boundary value problem
to the equation ``on the boundary''.
%
Solvability criteria of such obtained boundary equation are
formulated in terms of the Weyl function and boundary conditions.

%%=============================================================================
Final Section~4 illustrates the theory by means of an example of the
Laplacian on the smooth bounded domain.
%
The Weyl function is expressed in terms of the surface potentials
and some related results regarding its properties are formulated.


\smallskip

%%=============================================================================



The author would like to thank Prof. S.~N.~Naboko for his interest
to the work and continual encouragement and to the anonymous referee
for the close attention and valuable remarks.




\smallskip



The symbol~$\BB(H_1,H_2)$ where $H_1$, $H_2$ are separable Hilbert
spaces is used for the Banach algebra of bounded operators, defined
everywhere in ~$H_1$ with values in $H_2$. The notation $A : H_1 \to
H_2 $ is equivalent to $A \in \BB(H_1,H_2)$. Also, $\BB(H) := \BB(H,
H)$. The real axis, complex plane are denoted as ~$\Real$,
$\Complex$, respectively. Further, $\Complex_\pm := \{ z \in
\Complex \, :\, \pm \IM z > 0\}$, $\Real_\pm := \{x \in \Real \, :\,
\pm\; x > 0\}$, where $\IM$ stands for the imaginary part of a
complex number. The domain, range and kernel of a linear
operator~$A$ are denoted as $\D(A)$, $\Ran(A)$, and $\ker (A)$; the
symbol~$\rho(A)$ is used for the resolvent set of ~$A$. For a
Hilbert space the term \textit{subspace} will denote a closed linear
set. The orthogonal complement to a linear set in a Hilbert space is
always closed, i.~e. is a subspace.




%%====================================================================
%%
%%====================================================================





\section{Green formula}


\paragraph{  1.}
Let  $H$ be a separable Hilbert space and $A$ be a linear operator on $H$
with dense domain~$\mathcal{D}(A)$.

%%===============================================================

\begin{defn}
We will call the \textbf{Green formula} for $A$ the equality
\begin{equation} \label{GreenFormula}
 (Au,v)_{H} - (u,Av)_{H} = (\Gn
u, \Gd v)_E -(\Gd u, \Gn v)_E , \qquad u,v \in \D(A),
\end{equation}
where linear operators $\Gd$, $\Gn$ map vectors from
$\D(\Gamma_{\!k}) \subset H$, $k = 0, 1$  into an auxiliary Hilbert
space~$E$.
%
It is assumed that $\D(\Gamma_{\!k}) \supset \D(A)$, $k = 0,1$.
%
In the sequel the space~$E$ is called~\textbf{boundary space}.
\end{defn}


%%===============================================================

\begin{assump}\label{AS0}
Green formula ~(\ref{GreenFormula}) for operator~$A$ holds with some
operators $\Gd$, $\Gn$ and space~$E$.
\end{assump}


%%===============================================================

\begin{rem}
Note that we do not impose any conditions on boundedness or
closability of the operators $A$, $\Gd$, $\Gn$. In particular,  $A$,
$\Gd$, $\Gn$ acting on the domain~$\mathcal{D}(A)$ are not assumed
to be closed or even closable. If the  domains~$\D(\Gamma_{\! k})$,
$k = 0,1$ are wider than~$\D(A)$, the restrictions  of mappings
$\Gd$, $\Gn$  to the sets $\D(\Gd) \backslash \D(A)$ and $\D(\Gn)
\backslash \D(A)$ respectively are irrelevant to the following
considerations. For that reason we will assume in the sequel
 that~$\D(\Gd)= \D(\Gn) = \D(A)$ without any loss of generality.
\end{rem}


%%===============================================================


\begin{rem}\label{RemarkOnSymmOp}
The well known situation when Green formula~(\ref{GreenFormula})
holds is the case when $A$ is the adjoint of a symmetric operator
with equal deficiency indices defined on the dense domain.
%
The validity of~(\ref{GreenFormula}) with some~$\Gd$, $\Gn$, $E$ is
readily verified on the ground of the Neumann formula for $\D(A)$.
(See, e.~g. \cite{BirS}.)
%
In a series of papers dedicated to this
case with an additional condition~$\Gd\D(A) = \Gn\D(A) = E$
the collection~$\{\Gd, \Gn, E\}$ is called a \textit{boundary triple} or
a \textit{boundary values space} of the operator~$A$,
see~\cite{Bru, DM1, DM2,GG,Ko} for more details.
\end{rem}



%%====================================================================

Let us introduce the null set~$\ker(A)$ of operator $A$ and denote
it $\HH$:
\begin{equation}\label{H}
\HH := \ker (A) =  \{ u \in \D(A) \:|\: Au = 0 \}
\end{equation}
%
The set~$\HH$ is closed in $H$ if~$A$ is closed, as in the case
described in Remark~\ref{RemarkOnSymmOp}.
%
For our purposes, however, the closedness of~$\HH$ is not relevant.
%
Denote $A_0$ the restriction of $A$ to the intersection of $\D(A)$
and $\ker (\Gd)$:
\begin{equation}\label{A0}
A_0 := A |_{\D(A_0)},\;\; \text{where} \;\; \D(A_0) := \{ u \in
\D(A) \:|\: \Gd u=0\}
\end{equation}
%
Following equalities are direct consequence of the definitions
above:
\begin{equation}\label{Ker}
\ker (A_0) = \ker (\Gd)\cap \HH
\end{equation}
\begin{equation}\label{AA0-1}
(A - zI)(A_0 - zI)^{-1} u = u, \;\; u \in H, \; z \in \rho(A_0)
\end{equation}
where the resolvent set in~(\ref{AA0-1}) is assumed to be non-empty.
%
According to ~(\ref{GreenFormula}), (\ref{A0}),
operator~$A_0$ is symmetric on the possibly non-dense
domain~$\D(A_0)$.



\smallskip


Now we can formulate our second assumption imposed
on the introduced objects:


%%==============================================


\begin{assump}\label{AS1}
In notation (\ref{GreenFormula})--(\ref{A0}) following conditions
are satisfied:
  \begin{enumerate}
    \item  Operator $A_0$ is selfadjoint on the domain
    $\D(A_0)$ and there exists its bounded inverse~$A_0^{-1}$:
\begin{equation}\label{A0Invertable}
    A_0 = A_0^*, \;\;\; \exists \; A_0^{-1} : H \rightarrow H
\end{equation}
    \item Set $\Gd \HH$ is dense in $E$:
\begin{equation}\label{GHDensity}
    \overline{\Gd \HH } = E
\end{equation}
  \end{enumerate}
\end{assump}


\noindent In the sequel we will always assume that both Assumptions
are valid and will use the following symbol
 \begin{equation}\label{BVPNotation}
   \{A, \Gd, \Gn, H, E\}
 \end{equation}
for the collection of three operators~$A$, $\Gd$, $\Gn$, and two
Hilbert spaces $H$, $E$ subjected to these Assumptions.


%%=============================================================

\begin{rem}
Since $A_0$ is selfadjoint, it is closed and its resolvent set is
not empty. It follows that (\ref{AA0-1}) holds at least for $\I z
\neq 0$  and $z\in\Complex$ in some neighborhood of the origin.
%
Furthermore, domains $\D(A_0)$ and $\D(A)$ are dense in $H$.
%
The inverse operator~$A_0^{-1}$  is defined on the whole
space $H$ and its range coincides with $\D(A_0)$. Now
formula~(\ref{Ker}) shows that
\begin{equation}\label{Ker0}
\D(A_0)\cap \HH =  \{0\}.
\end{equation}
%
It means that the set $\HH$ could not be ``too large'' in the
space~$H$.
%
Particularly, (\ref{Ker0}) may be not valid if the set~$\HH$
is replaced with its closure $\overline\HH$.
\end{rem}


%%==================================================================

\begin{rem}\label{NonUniquenessG}
 Operators~$\Gd$, $\Gn$ are not determined uniquely
 by the Green formula.
%
 In particular, the equality~(\ref{GreenFormula})
 holds true if the operator~$\Gn$ is replaced by the
 sum~$\Gn + N \Gd$, where $N$ is
 a symmetric operator on~$E$ defined on the domain~$\Gd \D(A)$.
%
 It is easily seen that Assumption~\ref{AS1}
 holds or not for
 the sets~$\{A,\Gd, \Gn + N \Gd, H,E\}$ and~(\ref{BVPNotation})
 at the same time.
%
 Indeed, neither of conditions~(\ref{A0Invertable})
 and~(\ref{GHDensity}) affects the operator~$\Gn$.
\end{rem}





\begin{rem}
For any $a\in\Real$, the Green formula (\ref{GreenFormula}) is valid
for both $A$ and $A - aI$ simultaneously with the same  $E$,
$\Gd$, $\Gn$.
%
It means that the requirement $0\in\rho(A_0)$ in
Assumption~\ref{AS1} always can be ensured provided that there
exists a real number $a\in\Real$ such that $a\in\rho(A_0)$.
%
To that end, it is sufficient to pass to the boundedly invertible
selfadjoint operator~$A - aI$ in place of $A$.
%
Note as well that the condition~(\ref{GHDensity}) will be satisfied
automatically if the space~$E$ is \textit{defined} as the closure
of~$\Gd \HH$.
\end{rem}


%%=================================================================


\paragraph{  2.}
Our next objective is to establish the relationships
among vectors from linear sets~$\ker(A-zI)$ for
different values of~$z\in\rho(A_0)$
and describe the domain of~$A$ in terms of
operator~$A_0$ and vectors from~$\HH$.


%%=================================================================

\begin{prop}\label{PropDirectSum}
\begin{enumerate}
 \item
 The mapping
\begin{equation}\label{HtoHZMap}
 h \longmapsto (I - zA_0^{-1})^{-1} h, \quad h\in \HH, \, z\in\rho(A_0)
\end{equation}
establishes a one-to-one correspondence between $\HH = \ker (A) $
and $\ker (A - zI)$. For any $h\in \HH$ and $h_z := (I -
zA_0^{-1})^{-1} h \in \ker(A-zI)$, $z\in\rho(A_0)$ the equality $\Gd
h = \Gd h_z$ holds.
  \item For any $z_1, z_2   \in \rho(A_0)$
    \[
      \ker(A - z_1 I)\cap \ker(A - z_2 I) = \{0\},
      \qquad z_1 \neq z_2
    \]
  \item
For any~$z\in\rho(A_0)$ the domain~$\D(A)$ allows the representation
in the form of direct sum
\begin{equation}\label{DirectSumGeneral}
 \D(A) = \D(A_0) \dot{+} \ker(A-zI) = \D(A_0) \dot{+} (I - zA_0^{-1})^{-1}\HH,
 \qquad z\in\rho(A_0).
\end{equation}
 \end{enumerate}
\end{prop}


%%=================================================================


\begin{proof}
Let $z\in\rho(A_0)$ be a complex number.

%%=================================================================

(1) \quad Due to (\ref{AA0-1})  we have for any vector~$h\in\HH$
\[
(A - zI)(I -z A_0^{-1})^{-1} h = (A - zI)[I + z ( A_0 - zI)^{-1}] h
= (A -zI) h + z h = 0. \]
 Conversely, for $h_z \in \ker (A - zI)$ and
 $h := (I - zA_0^{-1})h_z$ we obtain
\[
Ah = A h_z - z A A_0^{-1}h_z = (A - zI)h_z = 0.
\]
Finally, the equality $\Gd h = \Gd h_z$ follows directly from the
relation $h = (I - zA_0^{-1})h_z$ and definition of the
domain~$\D(A_0)$.

%%=================================================================



(2) \quad
 Let $z_1, z_2 \in \rho(A_0)$ be two complex numbers.
 If~$(A - z_1)u =0 $ and $(A - z_2)u =0$
 for some vector~$u\in \D(A)$, then $A u = z_1 u = z_2 u$.
 Therefore~$(z_1 - z_2)u =0 $, which implies the required equality.


%%=================================================================



(3) \quad
 For an arbitrary number $z\in\rho(A_0)$ the
sum in~(\ref{DirectSumGeneral}) is direct because
the equality~$(A - zI)f = 0 $ for a  vector~$f\in \D(A_0)$
implies $(A_0 - zI)f=0$.
%
Therefore $f=0$ because $\ker (A_0 - zI) = \{0\}$.
%
Further, any
vector~$f\in \D(A)$ can be represented in the form:
\begin{equation}\label{DirectSumOfZ}
f = (A_0 - zI)^{-1} (A - zI)f  + \left[  f - (A_0 - zI)^{-1} (A -
zI)f \right], \qquad z\in \rho(A_0)
\end{equation}
%
The first summand here belongs to $\D(A_0)$, and the second one lies
in $\ker(A - zI)$ according to the formula~(\ref{AA0-1}).
%
The rest follows from the already proven statement~(1).
%
In the particular
case of $z = 0 $ we arrive at the
decomposition
\begin{equation}\label{DirectSum}
f = A_0^{-1}Af + (f - A_0^{-1} Af), \quad f \in \D(A).
\end{equation}
%
Operator $A_0^{-1} A $ here can be interpreted as an oblique
projection of the sum~$\D(A) = \D(A_0) \dot {+} \HH$ onto the first
summand.

The proof is complete.
\end{proof}



%%=================================================================


\paragraph{  3. }
Now we can introduce one of the main objects of the theory
under development, operator $\Pi$.
Its properties and relationships with
operators~$A$ and $\Gd$ play a significant role in the sequel.
%%=================================================================


\begin{prop}\label{PropAboutOperatorPi}
\begin{enumerate}
\item Operator  $\Gn A_0^{-1}$ is defined everywhere on the  space $H$
and bo\-un\-ded. Let $\Pi$ be its
adjoint.
\begin{equation}\label{PiDefinition}
\Pi \equiv (\Gn A_0^{-1})^* :
E\rightarrow H
\end{equation}
\item The set $\HH$ permits the characterization:
\begin{equation} \label{PiGamma}
\HH = \{ u \in \D(A) \; | \; \Pi\Gd u = u \}
\end{equation}
\item The range of operator~$\Pi$ is dense in the subspace~$\overline\HH$:
\begin{equation} \label{RanPi}
 \overline{\R(\Pi)} = \overline{\HH}
\end{equation}
\end{enumerate}
\end{prop}



\begin{proof} (1)\quad
Operator~$\Gn A_0^{-1}$ is defined in the whole~$H$ because for any
element~$x\in H$ the vector~$A_0^{-1}x$ belongs to the
domain~$\D(A_0) =\D(\Gn)$.
%
Let us write the Green formula
(\ref{GreenFormula}) with $u \in \D(A)$ and $v = A_0^{-1} x$, where
$x \in H$ and use the equality~(\ref{AA0-1}) with $z = 0$:
\[
(Au, A_0^{-1}x) - (u, A A_0^{-1}x) = (\Gn u, \Gd A_0^{-1} x) -
(\Gd u, \Gn A_0^{-1} x),
\]
%
Since $A A_0^{-1} x = x$ and $\Gd A_0^{-1} =0$, we obtain
\begin{equation}\label{Pi0}
(Au, A_0^{-1}x) = (u, x) - (\Gd u , \Gn A_0^{-1} x), \qquad
u \in\D(A), \quad x\in H
\end{equation}
%
Further,  for $u  = h \in \HH$ the relation~(\ref{Pi0}) implies
\[
(h, x) = (\Gd h , \Gn A_0^{-1} x), \quad x\in H, \; h \in
\HH.
\]
This identity means that the vector~$\Gd h$ belongs to the domain
of operator $(\Gn A_0^{-1})^*$, acting as
\begin{equation} \label{PiMap}
(\Gn A_0^{-1})^* : \; \Gd h \longmapsto h, \quad h \in \HH
\end{equation}
%
The set $\Gd \HH$ is dense in $E$ according to Assumption~\ref{AS1},
so that the operator~$\Gn A_0^{-1}$ defined on~$H$ has the densely
defined adjoint~$(\Gn A_0^{-1})^*$. Consequently, $\Gn A_0^{-1}$ is
closable in $H$. (See \cite{BirS}, p.~70-71.) As the domain of
$\Gn A_0^{-1}$ is the whole space $H$, it implies that~$\Gn
A_0^{-1}$ is already closed. By the virtue of closed graph theorem
$\Gn A_0^{-1}$ is bounded, so is its adjoint~$\Pi = (\Gn
A_0^{-1})^*$.


\smallskip

(2) \quad Taking into account the definition~(\ref{PiDefinition}) of
operator~$\Pi$, the implication $u \in \HH \Rightarrow \Pi\Gd u = u$
follows directly from~(\ref{PiMap}).
%
In order to prove the inverse assertion note that for a vector~$u \in \D(A)$
for which~$\Pi\Gd u = u$,  the right hand side of~(\ref{Pi0}) is zero
for any $x\in H$.
%
The set~$\{A_0^{-1}x \, |\; x \in H\}$ is equal
to~$\D(A_0)$, hence is dense in $H$.
%
Now the inclusion $u \in\ker(A)$ is a consequence of the
equality to zero of the left hand side of~(\ref{Pi0}).



\smallskip


(3) \quad Equality (\ref{PiGamma}) means that $\HH \subset \R(\Pi)$.
Let us show that any vector from $\R(\Pi)$ can be approximated
by elements of~$\HH$.
%
It is sufficient for the proof
of~(\ref{RanPi}).
%
For a given  $\Pi e \in \R(\Pi)$, $e \in E$ and
$\eps > 0$ we choose the vector  $h \in \HH$ such that $\|\Gd h - e
\| \leq \eps \|\Pi\|^{-1}$.
%
It is always possible because $\Gd \HH $
is dense in the space~$E$ according to Assumption~\ref{AS1}.
%
Since
$h = \Pi\Gd h$ due to equality~(\ref{PiGamma}), we have $\|\Pi e - h
\| = \| \Pi e - \Pi \Gd h \| \leq \| \Pi \|\cdot \| e - \Gd h\| \leq
\eps $.

The proof is complete.
\end{proof}

%%===================================================================

From the description~(\ref{PiGamma}) we obtain following important equality
\begin{equation} \label{PiGamma0}
\Pi\Gd h = h, \quad h \in \HH.
\end{equation}
showing that the operator~$\Pi$ is the left inverse of the
restriction of~$\Gd$ to the set~$\HH$. Let us take notice of a few
consequences of~(\ref{PiGamma0}).


%%===================================================================

\begin{rem}
Since $\HH \subset \D(A)= \D(\Gd)$, operator $\Gd $ can be
applied to the both sides of~(\ref{PiGamma0}), which yields
\begin{equation}\label{GammaPi}
(\Gd \Pi - I) \Gd h = 0 , \quad h \in \HH
\end{equation}
This equality shows that the restriction~$\left.\Pi\right|_{\Gd\HH}$
is the right inverse of~$\Gd$ restricted to the set~${\HH}$.
\end{rem}


%%===================================================================

\begin{rem}
Operator~$\Pi\Gd$ defined on the domain~$\D(A) = \D(A_0) \dot{+}
\HH$ is an oblique projection to the second summand
\[
\Pi \Gd \; : \; f  +  h \longmapsto   h, \quad f \in \D(A_0), \, h
\in \HH.
\]
 In particular,
 $ \Pi \Gd \D(A)$ and $\HH$ coincide. This assertion follows
immediately  from the definition of domain~${\D(A_0)} = \D(A) \cap \ker
(\Gd)$ and~(\ref{PiGamma0}).
\end{rem}


%%===================================================================

\begin{rem}
The equality~(\ref{Ker0}) can be strengthened as follows
\begin{equation}\label{Ker0Strong}
 \D(A_0) \cap  \Ran (\Pi)   = \{0\}
\end{equation}
Indeed, let ~$e\in E$ be a vector from~$E$ such that
$\Pi e \in \D(A_0)$, in other words, $\Gd \Pi e = 0$. Owing to the
identity $\overline{\Gd\Pi} = I_E$ for the closure of operator~$\Gd
\Pi$ resulting from~(\ref{GammaPi}) and density of~$\Gd \HH$ in~$E$,
we conclude that~$\overline{\Gd\Pi} e = e$. Now the equality~$e =0$
follows.
\end{rem}

%%===================================================================

\begin{rem}
Since $f - A_0^{-1}A f \in \HH$, the relation~(\ref{PiGamma0}) and
 re\-la\-tion~$\Gd A_0^{-1} = 0$ imply that the
 sum~(\ref{DirectSum}) can be rewritten in a more elaborated form
\begin{equation}\label{DirectSumDetails}
 f = A_0^{-1} A f + \Pi \Gd f, \qquad  f \in \D (A).
\end{equation}
%
 Let us insert this re\-pre\-sen\-ta\-tion
 into the expression for the second summand in (\ref{DirectSumOfZ}):
 \begin{equation*}
  \begin{aligned}
  \lefteqn{
  f - (A_0 - zI)^{-1} (A - zI) f   =  [ I - (A_0 - zI)^{-1} (A - zI) ] f }
  \\
  & =  \left[I - (A_0 - zI)^{-1} (A - zI)\right] A_0^{-1} A f +
 \left[I - (A_0 - zI)^{-1} (A - zI)\right] \Pi \Gd f
  \\
  & = \left[ I  + z (A_0 - zI)^{-1}\right ]  \Pi \Gd f
  = A_0 (A_0 - zI)^{-1} \Pi \Gd f
  = (I - zA_0^{-1})^{-1}\Pi \Gd f.
  \end{aligned}
 \end{equation*}
%
Now (\ref{DirectSumOfZ}) can be rewritten as
 \begin{equation}\label{ExpansionDependentOnZ}
   f = (A_0 - zI)^{-1} (A - zI) f  + (I - zA_0^{-1})^{-1} \Pi \Gd f,
   \qquad f \in \D(A), \quad z\in\rho(A_0)
 \end{equation}
in complete accordance with the decomposition~(\ref{DirectSumGeneral}).
%
Obviously, this formula is a generalization
of~(\ref{DirectSumDetails}).
\end{rem}


%%===================================================================


\begin{rem}
If we represent $h\in\HH$  in~(\ref{PiGamma0}) as $h = (I -
zA_0^{-1})h_z$, where $h_z \in \ker (A - zI)$, $z\in\rho(A_0)$,
then~(\ref{PiGamma0}) takes the form
\begin{equation}\label{hzEquality}
(I - zA_0^{-1})^{-1} \Pi \Gd h_z = h_z, \quad h_z \in \ker (A - zI),
\quad z\in\rho(A_0).
\end{equation}
In particular, this equality shows that the intersection
$\ker(\Pi)\cap \Gd \ker (A- zI)$ is trivial:
\begin{equation}\label{1:KerPiHH}
\ker(\Pi)\cap \Gd \ker (A- zI) = \{0\}, \qquad z\in \rho(A_0),
\end{equation}
which is almost obvious if we take into consideration that $ \Gd
\ker(A - zI) = \Gd \HH$ due to the assertion~(1) of
Proposition~\ref{PropDirectSum} and recall that the operator~$\Pi$
is the left inverse of~$\left.\Gd\right|_\HH$.
\end{rem}

%%===================================================================
%%
%%===================================================================


\paragraph{  4.} The collection~(\ref{BVPNotation}) as defined above
naturally appears in the context of what could be called
 \textit{the theory of null extensions of selfadjoint operators}.

%%===================================================================

\begin{defn}\label{NullExtensionsDefinition}
Let~$T$ be an operator on the Hilbert space~$H$ with domain~$\D(T)$.
Linear operator~$S$  is called a \textit{null extension} of~$T$ if
$\D(T)\subset\D(S)$, $Sx = Tx$ for $x \in \D(T)$, and~$Sx =0$ for
$x\in\D(S)\setminus \D(T)$.
\end{defn}

%%===================================================================

Next two Propositions offer convenient ways to construct
collections~$\{A, \Gd, \Gn, H,E\}$ that automatically satisfy
both Assumptions~\ref{AS0} and~\ref{AS1}.

%%===================================================================

\begin{prop}\label{NullExtension}
Let $A_0$ be a selfadjoint boundedly invertible operator on the
space~$H$ with domain~$\D(A_0)$ and ~$\HH$ be a linear set in~$H$
with the property~$\D(A_0) \cap \HH = \{ 0\}$. Define operator~$A$
as a null extension of~$A_0$  to the set $\D(A) := \D(A_0) \dot{+}
\HH$ according to the formula~$A (A_0^{-1} f + h ) := f$, $f\in H$,
$h\in \HH$.
%
 Assume that there exists a Hilbert space~$E$ and a linear
mapping~$\Gd$ with domain~$\D(A_0) \dot{+}\HH$ and the
range~$\Ran(\Gd)$ dense in~$E$, such that $\ker(\Gd) = \D(A_0)$ and
there exists a bounded left inverse~$\Pi$ of the
restriction~$\left.\Gd\right|_{\HH}$ defined on the whole space~$E$.
Let~$N$ be a symmetric operator on~$E$ with the dense domain~$\Gd
\HH$ and $\Gn := \Pi^* A + N \Gd$, $\D(\Gn) := \D(A_0) \dot{+} \HH$
be a densely defined mapping from~$H$ to ~$E$. Then the
collection~$\{A, \Gd, \Gn, H, E\}$ of such defined objects satisfies
both Assumptions~\ref{AS0} and~\ref{AS1}.
\end{prop}


%%===================================================================

\begin{proof} Assumption~\ref{AS1} follows directly from the
premise of Proposition.
%
With regard to Assumption~\ref{AS0}, it is
sufficient to show that the Green formula~(\ref{GreenFormula}) is valid
for~$u = v \in\D(A)$.
%
For any vector~$u \in\D(A)$ of the form~$u = A_0^{-1} +h$
with some~$f\in H$ and $h\in\HH$ we have~$Au = f$.
%
Therefore, since~$A_0^{-1}$ is selfadjoint and $\Gd A_0^{-1} = 0$,
$ \Pi\Gd h = h$,
\[
 \begin{aligned}
(Au, u) - (u , Au)  & = (f, A_0^{-1} f + h) - (A_0^{-1} f + h, f) =
(f,h) - (h,f)
 \\
 & = (f,\Pi\Gd h) - (\Pi\Gd h , f) = (\Pi^*f,\Gd h) - (\Gd h , \Pi^* f)
 \\
 & =  (\Pi^* Au , \Gd (A_0^{-1}f  + h )) - (\Gd (A_0^{-1} f + h),\Pi^* Au )
 \\
 & =
  (\Pi^* Au  + N \Gd u, \Gd u) - (\Gd h,\Pi^* Au  + N \Gd u)
  \\
  & = (\Gn u, \Gd u) - (\Gd u, \Gn u),
 \end{aligned}
\]
so that Assumption~\ref{AS0} is valid as well.
\end{proof}


%%=====================================================================

As an alternative, collection~(\ref{BVPNotation}) satisfying
Assumptions~\ref{AS0} and~\ref{AS1} can be defined in terms of the
pair~$\{A,\Gd\}$.
%
Properties of~$\{A,\Gd\}$ that ensure existence of the
corresponding collection~(\ref{BVPNotation}) are easy to formulate.
%
Arguments similar to the proof of
Proposition~\ref{NullExtension} result in the following


%%==========================================================

\begin{prop}\label{PropAlternativeConstruction1}
Let $A$ be a linear operator on Hilbert space~$H$ with
domain~$\D(A)$ and $\Gd$ be an operator from $H$ to  an auxiliary
Hilbert space~$E$ with the domain~$\D(\Gd) = \D(A)$.
%
Assume that
\begin{enumerate}
  \item Operator~$A_0$ defined as a restriction of $A$ to the linear
  set~$\D(A_0) := \ker(\Gd)$  is selfadjoint and boundedly invertible.
  \item There exists a bounded mapping~$\Pi : E \to H$ such
  that~$\Pi\Gd h = h$ for any~$h \in \ker (A)$.
  \item Set~$\Gd\ker(A) $ is dense in~$E$.
\end{enumerate}
%
Then the collection~$\{A,\Gd, \Gn, H, E\}$ where the operator~$\Gn$
is defined as $\Gn := \Pi^* A + N \Gd$ with an arbitrary symmetric
operator~$N$ on the domain~$\D(N) := \Gd \ker(A)$,
satisfies both Assumptions~\ref{AS0} and \ref{AS1}.
\end{prop}




%%====================================================================
%%
%%====================================================================

\section{Minimal operator and Krein extension}

\paragraph{ 1.} Operator~$A_{00}$ defined as a restriction of~$A$ to the
set~$\D(A_{00}) := \{u \in \D(A_0) \; |\;  \Gn u = 0\}$ is called
\textbf{minimal}.
%
Obviously,
$A_{00}$ is symmetric and $A_0$  is the selfadjoint extension of~$A_{00}$.
%
Furthermore, $A_{00} : A_0^{-1} f \mapsto f$
where $ A_0^{-1} f \in \D(A_{00})$.
%
It turns out that the domain~$\D(A_{00})$ can be effectively described
in other terms, not involving directly operators~$\Gd$ and $\Gn$.
Moreover, under some quite natural assumption
$A_{00}$ is the minimal operator indeed
in the sense of extension theory of symmetric operators.



%%====================================================================

\begin{prop}\label{PropOnMinimalOperator}
The equalities hold
\begin{equation}\label{DomainOfA00}
\D(A_{00}) := \{u \in \D(A) \; |\; \Gd u = \Gn u = 0\} =
A_0^{-1}\HH^\perp.
\end{equation}
If the set~$\D(A_{00})$ is dense in $H$ then $A$ is closable and~$A_{00}^*$
coincides with its closure~$\bar A$. The domain of~$\bar A$
allows decomposition~$\D(\bar A) = \D(A_0) \dot{+} \overline\HH$
where $\overline\HH$ is the null subspace of~$\bar A$.
\end{prop}

%%====================================================================

\begin{proof}
For~$f\in H$ and $u := A_0^{-1}f \in \D(A_0)$ we
have $\Gn u = \Gn A_0^{-1} f = \Pi^* f$.
%
Therefore, the inclusion~$f\in\HH^\perp = \ker(\Pi^*)$ is equivalent
to the equality~$\Gn u =0$.


%%====================================================================

Further, in order for a densely defined operator to be closable
it is necessary and sufficient that its adjoint
be densely defined, see~\cite{BirS}, p.~70.
%
Let us consider the adjoint operator~$A^*$
and show that $A^* \supset A_{00}$. Then $A$ is closable
since $A_{00}$ is densely defined.
%
For $u := A_0^{-1} f + h \in \D(A)$
where $f\in H$, $h\in \HH$ and $ v:= A_0^{-1}f_0 \in \D(A_{00})$ where
$f_0\in \HH^\perp$ we have
\[
(Au, v) = (A (A_0^{-1} f + h), A_0^{-1}f_0)
 = (f, A_0^{-1}f_0) = (A_0^{-1} f + h, f_0) = (u, A_{00} v)
\]
so that $A^* \supset A_{00}$ as required.


%%====================================================================

Usual duality arguments yield $\bar A = A^{**} \subset A_{00}^*$,  so that
for the proof of equality~$A_{00}^* = \bar A$ we only need to show that
$\bar A   \supset A_{00}^*$.
%
We start with the observation that the
subspace~$\overline\HH$ belongs to~$\ker(\bar A)$
because the null set of a closed operator is always closed.
%
In particular, $\overline \HH\subset \D(\bar A)$.
%
Further, assuming~$x\in\D(A_{00}^*)$
by the definition of adjoint operator, $(A_{00}A_0^{-1}f_0, x ) =  (f_0, A_0^{-1}A_{00}^* x)$
for any~$f_0 \in \HH^\perp$.
%
From the other side, obviously~$(A_{00}A_0^{-1}f_0, x ) = (f_0, x)$ since
$A_{00}$ is a restriction of~$A_0$.
%
Therefore, $x - A_0^{-1}A_{00}^* x \perp \HH^\perp$, or the
vector~$\mathfrak h(x) := x - A_0^{-1}A_{00}^* x$ belongs to the
subspace~$\overline\HH$.
%
It follows that any vector~$x\in \D(A_{00}^*)$ is represented in the
form of sum~$x = A_0^{-1}A_{00}^* x + \mathfrak h(x)$
where $A_0^{-1}A_{00}^* x \in \D(A_0)$
and $\mathfrak h(x)\in\overline \HH$.
%
In other words, $\D(A_{00}^*) \subset \D(A_0) {+} \overline \HH \subset \D(\bar A)$.
%
The equality~$\D(A_{00}^*) = \D(\bar A) = \D(A_0) \dot{+} \overline \HH $ is
therefore obtained.
%
The sum here is direct because if the
intersection~$ \D(A_0) \cap \overline \HH$ was not
trivial, then it would consist of  vectors~$x \in \D(A_0)$
such that $\bar A x = A_0 x = 0$,
which contradict the assumption~$\ker(A_0) = \{0\}$.
%
Now we need to show that actions of~$A_{00}^*$ and $\bar A$ coincide
on their domain.
%
To that end note that $\overline\HH = \Ran(A_{00})^\perp = \ker(A_{00}^*)$,
so that parts of both operators~$A_{00}^*$ and~$\bar A$ in the
subspace~$\overline\HH$ is the null operator.
%
Finally, action of~$\bar A$ on the domain~$\D(A_0)$
is~$\bar A : A_0^{-1} f \mapsto f$ where $f\in H$.
%
From the other side,
for the operator~$A_{00}^*$ we obtain
\[
 (A_{00}^* A_0^{-1} f, v ) = (A_0^{-1} f , A_{00} v) = ( f , A_0^{-1}A_{00} v) = (f,v)
\]
where~$f\in H$, $v\in\D(A_{00})$.
%
Because~$\D(A_{00})$ is dense in~$H$, it follows that
$A_{00}^* : A_0^{-1} f \mapsto f$, $f\in H$.
%
Therefore actions of operators~$A_{00}^*$ and $\bar A$
coincide on their domain~$\D(A_0) \dot{+} \overline \HH$.

The proof is complete.
\end{proof}



%%=======================================================================
\paragraph{ 2.}
Operator~$A_{00}$ can have nontrivial selfadjoint parts, i.~e.
nontrivial reducing subspaces where it generates selfadjoint
operators.
%
If there are no such parts, operator~$A_{00}$ is called
\textbf{completely non-selfadjoint} or \textbf{simple}.
%
The criterion of complete non-selfadjointness of~$A_{00}$
is given in the following proposition.

%%=======================================================================

\begin{prop}
Denote $H_{nsa}$ the closed envelope of $\ker(A - zI)$,
$z\in\rho(A_0)$ and let~$H_{0} :=  H\ominus H_{nsa}$ be the
orthogonal complement. Then
\begin{equation}\label{CompleteNonSASpace}
H_{nsa} = \mathop{clos}\bigvee_{z \in \rho(A_0) }\ker(A - zI) =
\mathop{clos}\bigvee_{e \in E,\; z \in \rho(A_0) }( I -
zA_0^{-1})^{-1}\Pi e = \, \mathop{clos}\bigvee_{e \in E,\; n \geq 0}
A_0^{-n}\Pi e,
\end{equation}
and $H_{0}$ is the maximal subspace where~$A_{00}$ induces a
selfadjoint operator coinciding with~$A_{0}|_{H_{0}}$.
\end{prop}


%%=======================================================================
\begin{proof}
Equalities in~(\ref{CompleteNonSASpace}) follow from the description
of~$\ker(A -zI)$ given in Proposition~\ref{PropDirectSum},  density
of~$\Gd \HH$ in $E$, and the power expansion of resolvent~$(I -
zA_0^{-1})^{-1}$, $z\in\rho(A_0)$.


From~(\ref{CompleteNonSASpace}) we have that both $H_{nsa}$ and
$H_0$ reduce the selfadjoint operator~$A_0$ and~$H_0 \subset
\HH^\perp$.
%
Taking into consideration the definition~$\D(A_{00}) = A_0^{-1}
\HH^\perp$, we conclude that $A_0^{-1}H_0 \subset \D(A_{00}) \cap
H_0$.
%
Since~$A_{00}$ is a restriction of~$A_0$, it follows that~$A_{00}
A_0^{-1} h_0 = h_0$ for $h_0 \in H_0$.
%
Therefore $A_0|_{H_0} \subset A_{00}$, $H_0 $ reduces~$A_{00}$, and
the part of~$A_{00}$ in $H_0$ is a selfadjoint
operator~$A_0|_{H_0}$.
%
In order to show that $H_0$ is the maximal subspace where~$A_{00}$
induces a selfadjoint part, assume that~$H'$ is a reducing space
of~$A_{00}$ and $A_{00}|_{H'}$ is selfadjoint.
%
Since~$A_{00}$ is the restriction of~$A_0$, the space~$H'$ also
reduces the selfadjoint operator~$A_0$.
%
The set~$A_0^{-1}H'$ is dense in~$H'$; it is the domain of the
selfadjoint part~$A_0|_{H'} = A_{00}|_{H'}$ of~$A_{00}$. It follows
from the definition of~$\D(A_{00})$ that~$H' \subset \HH^\perp$.
Since~$H'$ is invariant for~$A_0^{-1}$, we have $0= (A_0^{-n} x ,
\Pi e )   = (x, A_0^{-n}\Pi e )$ for $x\in H'$, $e\in E$, and $n\ge
0$.
%
Therefore, $H'$ is orthogonal to the
subspace~(\ref{CompleteNonSASpace}) and~$H' \subset H_0$.
\end{proof}


%%=======================================================================
\paragraph{ 3.}
Consider the null extension~$A_K$ of operator~$A_{00}$ to the
set~$\D(A_K) := A_0^{-1}\HH^\perp \dot{+}\HH$. According to
Definition~\ref{NullExtensionsDefinition},
\[
A_K : A_0^{-1} f_0 + h \mapsto f_0, \qquad f_0 \in\HH^\perp, \quad
h\in\HH
\]
It is a simple exercise to see that~$A_K$ is symmetric, but not
necessarily densely defined. The next Theorem summarizes other
properties of~$A_K$.

%%======================================================================
\begin{thm}\label{TheoremOnKreinExtension}
\begin{enumerate}
\item
Symmetric operator~$A_K$ is a restriction of~$A$ to the domain
\[
 \D(A_0) = \{ u \in \D(A)\; | \; (\Gn - \Lambda \Gd ) u = 0\}
\]
where~$\Lambda := \Gn \Pi$ is symmetric,  densely defined and
closable on~$\D(\Gamma) := \Gd \HH$.
\item
The equation~$A_K u = f$  is solvable only when~$\Pi^* f =0$, that
is, for~$f\in \HH^\perp$. The solutions are represented in the
form~$u = A_0^{-1}f + h$, where $h\in\HH$ is arbitrary.
\item
If the domain~$\D(A_{00})$ is dense in~$H$, then~$A_K$ is
essentially selfadjoint. Its closure is the null extension of~$A_K$
to the set~$A_0^{-1}\HH^\perp \dot{+} \overline \HH$.
\end{enumerate}
\end{thm}

\begin{proof}
(1) \quad
 Let $f \in H$ and $h\in\HH$ be two arbitrary vectors
 and $u := A_0^{-1} f  + h \in \D(A) $. Then
\[
 \begin{aligned}
 ( \Gn - \Lambda \Gd ) u   & = \left(\Gn - \Gn\Pi\Gd\right)
  \left( A_0^{-1}  f   + h\right)
  \\
  & = \Gn A_0^{-1}  f + \Gn h - \Gn h = \Gn A_0^{-1} f = \Pi^* f,
 \end{aligned}
\]
where we used equalities $\Gd A_0^{-1} =0$ and $\Pi\Gd h = h$.
Since~$\ker(\Pi^*) = \Ran(\Pi)^\perp = \HH^\perp$, the inclusion~$f
\in \HH^\perp$ is equivalent to~$(\Gn -\Lambda \Gd) u =0$. If
$h\in\HH$, then
\[
 \begin{aligned}
  (\Lambda \Gd h, \Gd h ) - (\Gd h, \Lambda \Gd h)
    & = (\Gn \Pi \Gd h, \Gd h) - (\Gd h, \Gn \Pi \Gd h)
   \\
  & =
  (\Gn h, \Gd h) - (\Gd h , \Gn h ) =0
 \end{aligned}
\]
Since~$\Lambda$ is symmetric and densely defined, it is closable.


(2)\quad Obvious according to the definition~$\D(A_K)$.


(3)\quad Operator~$A_K$ is symmetric and densely defined. The
representation~$A_0^{-1}\HH^\perp \dot{+}\overline\HH$ for the
domain of its closure is obtained in the way demonstrated in the
proof of Proposition~\ref{PropOnMinimalOperator}. Let us show
that~$A_K$ is essentially selfadjoint. Since~$A_K$ is symmetric,
$\D(\overline{A_K}) \subset \D(A_K^*)$ and we need to verify
correctness of the inverse inclusion. Let~$f_0\in \HH^\perp$, $h\in
\HH$, and $x\in\D(A_K^*)$, i.~e.
\[
 (A_K(A_0^{-1}f_0 + h),x) = (A_0^{-1}f_0 + h, A_K^* x) =
(f_0,A_0^{-1} A_K^* x) + (h, A_K^* x)
\]
Since the left hand side is equal to~$(f_0,x)$, we have~$(f_0, x -
A_0^{-1}A_K^* x) = (h, A_K^* x)$.
%
For $h=0$ this equality yields the representation~$x = A_0^{-1}A_K^*
x + \mathfrak h$, where $\mathfrak h\in\overline\HH$ is some vector.
From the other side, if~$f_0 = 0$, then $(h, A_K^* x) =0$ for each
$h\in\HH$ and we see that~$A_K^* x\in \HH^\perp$.
%
Thus any vector~$x\in\D(A_K^*)$ is represented in the form~$x =
A_0^{-1}g + \mathfrak h$ with $g = A_K^* x\in\HH^\perp$ and
$\mathfrak h\in\overline\HH$. Therefore, $x\in\D(\overline{A_K})$.

The proof is complete.
\end{proof}




%%=================================================================

\begin{rem}
Operator~$A_K$ is an analogue of the Krein extension of operator~$A_{00}$,
see~\cite{AS}.
%
From the same point of view~$A_0$ is equivalent to the Friedrichs
extension of~$A_{00}$.
%
For any element~$u\in \D(A)$ we have
 $(\Gn - \Lambda\Gd) u  = ( \Gn -  \Gn \Pi \Gd ) u = \Gn (I - \Pi\Gd ) u$, so that
 the con\-di\-tion~$u\in\D(A_K)$ is equivalent to the identity~$\Gn u = \Gn \Pi \Gd u$.
%
Operator~$\Lambda$ was studied by
 M.~Vishik  in the context of elliptic
 boundary value problems  in~\cite{V}.
%
%
 Later $\Gn - \Lambda\Gd$
 was rewritten as~$\Gn (I - \Pi \Gd)$ by
 G.~Grubb~\cite{Grubb}
 and independently by A.~Alonso and B.~Simon in~\cite{AS}.
%
 See the note~\cite{Grubb2} for further references regarding
 boundary conditions for the Krein extension.
%
In our case the extension~$A_K$ is not necessarily
densely defined, nor semibounded.
%
 Extensions~$A_0$ and $A_K$ of the symmetric operator~$A_{00}$
 are transversal in the sense of~\cite{DM1},
 i.~e.~$\D(A_0)\cap \D(A_K) = \D(A_{00})$
 and~$\D(A_0)\vee \D(A_K) = \D(A)$.
%
Note at last that the operator~$A_K$ is uniquely determined
by the pair~$\{A_0, \HH\}$.
%
In particular, it does not depend on the
choice of operator~$\Gn$ in the Green formula.
\end{rem}








%%=================================================================
%%
%%=================================================================


\section{Spectral boundary value problem and Weyl function}\label{Section_BVPandWF}


Let~$\{A,\Gd,\Gn, H,E\}$ be the collection~(\ref{BVPNotation})
subject to Assumptions~\ref{AS0} and~\ref{AS1}.
%
In this section we consider the spectral boundary value problem
defined by the set~$\{A,\Gd,\Gn, H,E\}$, describe conditions of its
solvability, and introduce the notion of Weyl function.



%%==================================================================================

\begin{defn}
For given~$f \in H$ and $\varphi \in E$ we call the \textbf{spectral
boundary value problem} associated with
collection~(\ref{BVPNotation}) the problem of finding pairs~$(u,z)$
of vectors~$u\in \D(A)$ and complex numbers  $z \in \mathbb{C}$,
satisfying equations:
\begin{equation}\label{2:BVP}
\left\{
\begin{array}{l}
(A - zI) u = f \\
\Gd u = \varphi
\end{array}
\right.
\end{equation}
\end{defn}


%%==================================================================================

Let us formulate a simple result regarding solvability
of~(\ref{2:BVP}).

%%==================================================================================

\begin{thm}\label{2:BVPTheorem}
For any $z\in\rho(A_0)$, $\varphi \in \Gd\HH$, $f\in H$ the solution
 $u = u_z^{f,\varphi}$ to the problem~(\ref{2:BVP}) exists and is
unique.
%
It is represented in the form
\begin{equation}\label{2:BVPsolution}
u_z^{f,\varphi} = (A_0 - zI)^{-1}f + [ I + z (A_0 - zI)^{-1} ] \Pi
\varphi
\end{equation}
\end{thm}

%%==================================================================================

\begin{proof}
Verification  of the uniqueness of solution is more or less standard.
%
If for some $z\in\rho(A_0)$, $\varphi \in \Gd\HH$,
and $f\in H$ there exist two solutions  $u_1, u_2 \in \D(A)$
to the system~(\ref{2:BVP}),
then their difference~$u_0 := u_1 - u_2$ satisfies both equations~(\ref{2:BVP})
with $f = 0$, $\varphi = 0$.
%
Therefore the vector~$u_0$  belongs to the domain of
operator~$A_0$ because~$\Gd u_0 =0$.
%
Then it follows from the first
equation~(\ref{2:BVP}) than $(A - zI)u_0 = (A_0 - zI)u_0 = 0$ and
$u_0 = 0$ because $z \in \rho(A_0)$.


%%==================================================================================

Further, from the equality~(\ref{AA0-1}) and the inclusion $\Pi
\varphi \in \HH$ where~$\varphi \in \Gd \HH$, we obtain for
the vector~$u_z^{f,\varphi}$ defined by~(\ref{2:BVPsolution})
\[
(A - zI)u_z^{f,\varphi} = f + (A - zI)[ I + z (A_0 - zI)^{-1} ]
\Pi \varphi = f + (A - zI)\Pi \varphi + z \Pi \varphi = f
\]
Hence, the first equation in~(\ref{2:BVP}) is satisfied. Next, using
equalities $\Gd (A_0 - zI)^{-1} = 0$ and (\ref{GammaPi}) we obtain
for the vector~$u_z^{f,\varphi}$
\[
\Gd u_z^{f,\varphi} = \Gd \Pi \varphi  = \varphi.
\]
The proof is complete.
\end{proof}


%%==================================================================================

\begin{rem}\label{2:Note}
From the representation~(\ref{2:BVPsolution}) follows the norm
estimate of the so\-lu\-tion~$u_z^{f,\varphi}$:
\[
\|u_z^{f,\varphi}\| \leq C(z) (\|f\| + \|\varphi\|), \quad f\in H,
\, \varphi \in \Gd \HH, \, z \in \rho(A_0)
\]
with a constant $C(z) > 0$ depending on $z \in \rho(A_0)$.
%
Owing to the density of~$\Gd \HH $ in~$E$ (cf.~(\ref{GHDensity}))
this means that the mapping~$(f,\varphi) \mapsto u_z^{f,\varphi}$
defined on the set of pairs~$\{(f, \varphi)\; |\; f\in H, \varphi \in \Gd \HH\}$ can
be uniquely extended to a bounded operator from $H \oplus E$ to $H$.
%
Assuming $z\in\rho(A_0)$, it seems natural to interpret
vectors~$u_z^{f,\varphi}$ defined by the
formula~(\ref{2:BVPsolution}) with~$f\in H$, $\varphi\in E$ as
solutions to the problem~(\ref{2:BVP}) as well.
%
Each solution~$u_z^{f,\varphi}$ for $\varphi\in E \setminus \Gd\HH$ can
be approximated by ``regular'' solutions~$u_z^{f,\varphi_n}$, $n =
1,2,\dots$, where $\varphi_n\in\Gd\HH$.
%
A short argument reveals
that the passage from solutions~$u_z^{f,\varphi}$
where~$\varphi\in\Gd\HH$ to solutions for~$\varphi\in E$
corresponds to the null extension of operator~$A$ from the
set~$\D(A_0) \dot + \HH$ to the wider set~$\D(A_0) \dot + \Ran(\Pi)$
as described in Proposition~\ref{NullExtension}.
%
The sum is direct due to~(\ref{Ker0Strong}).
%
At last, since the vector~$u_z^{f,\varphi}$ is unambiguously defined by the right
hand side of~(\ref{2:BVPsolution}) for any~$\varphi\in E$, we can
state that for~$z\in\rho(A_0)$ the uniqueness property of
solution~(\ref{2:BVPsolution})  holds true for any~$f\in H$,
$\varphi\in E$.
\end{rem}

%%==================================================================================



\begin{rem}\label{RemWeakSolutions}
In general,
the solution~(\ref{2:BVPsolution}) does not belong to the domain~$\D(A)$
if $\varphi \in E\setminus\HH $.
%
Nevertheless, the vector~$u_z^{f,\varphi}$  defined
by~(\ref{2:BVPsolution}) is a solution to a ``weak variant'' of the
problem~(\ref{2:BVP}), thereby is a weak solution to~(\ref{2:BVP}).
%
More precisely, the vector~$u_z^{f,\varphi}$ solves the following
variational problem for unknown~$u\in H$
\begin{equation}\label{WeakEquation}
 (u, (A_{0} - \bar zI) v) = (f, v)_H + (\varphi,\Gn v)_E, \qquad
 v\in\D(A_0).
\end{equation}
%
The proof is based on direct calculations.
\[
 \begin{aligned}
 (u_z^{f,\varphi}, (A_{0} - \bar zI) v)  & = ((A_0 - zI)^{-1}f + [ I + z (A_0 - zI)^{-1} ] \Pi
\varphi, (A_{0} - \bar zI) v)
 \\
 & = (f, v )
   + ((I - zA_0^{-1})^{-1}\Pi \varphi , (A_{0} - \bar zI)v  )
  \\
 & = (f, v) + (\varphi,  \Gn A_0^{-1}(I -\bar z A_0^{-1})^{-1}(A_{0} - \bar zI)v )
  \\
  & =  (f, v) + (\varphi, \Gn v),
 \end{aligned}
\]
as required.
%
In order to show that the equation~(\ref{WeakEquation}) is the weak
version of~(\ref{2:BVP}) assume that some $w\in H$
solves~(\ref{2:BVP}). In particular, $w\in \D(A)$  and $\Gd w
=\varphi$. Then for any~$v\in\D(A_0)$ by the Green formula
\[
 \begin{aligned}
 (w, (A_0 - \bar zI) v ) & = (w,A_0 v)  - (z w, v)
 = (w, A v) + (f - Aw ,v) \\
 & = (f,v) + (w, Av ) - (A w, v)
 = (f,v) + (\Gd w, \Gn v) - (\Gn w, \Gd v)
 = (f,v) + (\varphi,\Gn
 v),
 \end{aligned}
\]
that is, the vector~$w$ solves the weak
 problem~(\ref{WeakEquation}) as well.
%
Note that if the set~$\Gn \D(A_0)$ is dense in $E$ and $\Ran(A_0 -
\bar z I)$ is dense in~$H$, then the weak solution is determined
by~(\ref{WeakEquation}) uniquely.
\end{rem}




%%===================================================================
%%
%%===================================================================




\subsection{Semi-homogeneous problems }

The classical approach of boundary value problems theory
(see~\cite{BS,Weyl} for instance)
 considers two semi-homogeneous
 problems obtained from~(\ref{2:BVP})
 by putting~$f=0$, $\varphi \neq 0$ and $f \neq 0$, $\varphi = 0$.
%
 The solution to~(\ref{2:BVP}) then is sought in the form of
 sum of solutions to these  auxiliary problems.
%
 Below we give a brief account regarding their solvability in abstract
 setting of the paper.
%


Let us start our discission with the case $f = 0$, $\varphi \in E$.
\begin{equation}\label{2:BVP0}
\left\{
\begin{array}{l}
(A - zI) u = 0 \\
\Gd u = \varphi
\end{array}
\right.
\end{equation}
According to~Theorem~\ref{2:BVPTheorem} and Remark~\ref{2:Note}, for
any $\varphi\in E$, $z \in \rho(A_0)$ there exists an unique
solution~$u_z^\varphi$ to the problem~(\ref{2:BVP0}).
%
If $\varphi \in E \setminus \Gd \HH$, then the
solution~$u_z^\varphi$ is the weak solution as explained in
Remark~{\ref{RemWeakSolutions}}.
%
The representation~(\ref{2:BVPsolution}) with~$f =0$ reduces to the
expression:
\begin{equation} \label{2:BVPsolution0}
u_z^\varphi = [I + z(A_0 - zI)^{-1}]\Pi \varphi = (I -
zA_0^{-1})^{-1} \Pi \varphi,\quad \varphi \in E
\end{equation}
It is easily seen that vectors defined in (\ref{2:BVPsolution}) and
(\ref{2:BVPsolution0}) are connected by the relation:
\begin{equation}\label{2:Odnor-neodnor}
u_z^{f,\varphi} = (A_0 - zI)^{-1} f + u_z^\varphi, \quad f\in H,
\, \varphi \in E, \, z\in\rho(A_0),
\end{equation}
%
%
Obviously, $(A_0 - zI)^{-1} f$ here is a solution to the
problem~(\ref{2:BVP}) with $f\neq 0$, $\varphi = 0$.
%
Thus, formula~(\ref{2:Odnor-neodnor}) represents the solution
to~(\ref{2:BVP}) as a direct sum of the solution to ~(\ref{2:BVP})
with $f \neq 0$, $\varphi =0$ and the solution to~(\ref{2:BVP}) with
$f = 0$, $\varphi \neq 0$.
%
It is interesting to observe that these two  problems can be
seen as conjugate to each other in the following sense.

%%========================================================================

\begin{rem} Let  $z\in\rho(A_0)$ and $v_{z}^f = (A_0 - zI)^{-1} f$
 be the solution to the
semi-homogeneous problem
\begin{equation*}
\left\{
\begin{array}{l}
(A - zI) v = f \\
\Gd v = 0
\end{array}
\right. \qquad z \in \mathbb{C}, \, f \in H.
\end{equation*}
Let~$u_z^\varphi$ be a solution to~(\ref{2:BVP0}). Then
\begin{equation*}
 (u_z^\varphi,f)_H = (\varphi, \Gn v_{\overline{z}}^f)_E,
\quad f\in H,\, \varphi \in E,\, z \in\rho(A_0).
\end{equation*}
%
%
Indeed, since
\begin{equation*}
  \left[ (I  - zA_0^{-1})^{-1}\Pi\right]^*
  = \Gn A_0^{-1} (I  - \bar{z}A_0^{-1})^{-1} = \Gn (A_0  -
  \bar{z}I)^{-1},
\end{equation*}
the assertion follows directly from the
representation~(\ref{2:BVPsolution0}).
\end{rem}

%%========================================================================

Finally, the case of~(\ref{2:BVP}) with $f = 0 $, $\varphi =0$ is in
fact the spectral problem for the eigenvalues and eigenvectors of
selfadjoint operator~$A_0$.



%%========================================================================
%%
%%========================================================================


\subsection{Weyl function}

The vector~$u_z^{\varphi}$ defined in~(\ref{2:BVPsolution0}) for
$\varphi \in \Gd \HH$
 belongs
to~$\ker(A- zI)$, consequently lies in the
domain of the operator~$\Gn$.
%
Let us calculate~$\Gn u_z^{\varphi}$
for a given pair of~$\varphi \in \Gd\HH$ and~$z\in\rho(A_0)$. We
have
\begin{align*}
\Gn u_z^{\varphi} & = \Gn [ I + z (A_0 - zI)^{-1}]\Pi \varphi   =
\Gn
\Pi \varphi + z \Gn ( A_0 - zI)^{-1} \Pi \varphi \\
&= \Gn \Pi \varphi + z \Gn A_0^{-1} ( I - z A_0^{-1})^{-1}\Pi
\varphi =
\Gn \Pi \varphi + z \Pi^* ( I - z A_0^{-1})^{-1}\Pi \varphi  \\
&= [\Gn \Pi + z \Pi^* ( I - z A_0^{-1})^{-1}\Pi]  \varphi
\end{align*}
%
Introduce an operator-function~$M(z)$, $z \in \rho(A_0)$ with
values in the set of densely defined operators on $E$ with
the domain~$\Gd\HH$:
%
\begin{equation}\label{MzDef}
M(z) \; : \;\; \varphi \longmapsto [\Gn \Pi + z \Pi^* ( I - z
A_0^{-1})^{-1}\Pi]  \varphi, \qquad \varphi \in \Gd \HH, \quad
z\in\rho(A_0)
\end{equation}
%
Since  $\varphi = \Gd u_z^\varphi$ for the solution~$u_z^\varphi$ to
the problem~(\ref{2:BVP0}), the calculations conducted above show
that
%
\begin{equation}\label{Mz}
\Gn u_z^\varphi = M(z) \Gd u_z^\varphi, \quad z \in \rho(A_0), \;
\varphi \in \Gd \HH
\end{equation}

%%========================================================================

\begin{defn}
Function $M(z)$ is called  \textbf{Weyl function} of the
problem~(\ref{2:BVP}) (or~(\ref{2:BVP0})).
\end{defn}

%%========================================================================

\begin{rem}
Note that the densely defined operator~$M(0) = \Gn \Pi$ is not
necessarily bounded, whereas the component of~$M(\cdot)$ depending
on $z$, that is, the difference~$M(z) - M(0)$, $z\in \rho(A_0)$, is an
analytic operator function with values in~$\BB(E)$.
\end{rem}
%%========================================================================

\begin{rem}
  The Weyl function~$M(z)$ can be rewritten in the form
\begin{equation}\label{MzSecondForm}
 M(z)\varphi = \Gn(I - zA_0^{-1})^{-1} \Pi \varphi,
 \quad \varphi\in \Gd \HH, z \in \rho(A_0)
\end{equation}
This representation directly follows from the definition above, since
\[
 I + z(A_0 - zI)^{-1} = A_0 (A_0 - zI)^{-1} = ( I - zA_0^{-1})^{-1}.
\]
%
Assuming $\varphi = \Gd h$ with some~$h\in\HH$ and noting
that for~$h$ and~$h_z = (I - zA_0^{-1})^{-1} h \in \ker (A- zI)$
the equality $\varphi = \Gd h = \Gd  h_z$ holds according to
Proposition~\ref{PropDirectSum}, we can
rewrite~(\ref{MzSecondForm}) in the form:
%
\begin{equation}\label{MzThirdForm}
M(z) \Gd h_z  = \Gn h_z, \quad h_z \in \ker (A - z I), \, z\in
\rho(A_0).
\end{equation}
\end{rem}


%%========================================================================


Let us formulate some properties of the constant term~$M(0) = \Gn
\Pi$ of Weyl function~$M(z)$ and its $z$-dependent part~$M(z) -
M(0)$, $z\in\rho(A_0)$.

%%========================================================================

\begin{prop}\label{PropPropertiesOfPartsOfM}
 \begin{enumerate}
\item
 Operator $M(0) = \Gn \Pi$ defined on the domain~$\Gd \HH$ is symmetric and
 closable.
\item
 The difference~$M(z) - M(0)$ is an operator $R$-function, i.~e.
 an analytic operator function taking values in~$\BB(E)$ with the positive
 imaginary part in the upper half plane.
\item
 Let~$\{A, \Gd, \Gn + N \Gd, H, E \}$
 where~$N$ is a symmetric
 operator on $E$ with domain~$\Gd \HH$
 be the collection~(\ref{BVPNotation}) obtained from~$\{A, \Gd, \Gn, H, E \}$
 according to Remark~\ref{NonUniquenessG}.
 Then the Weyl function of  boundary value problem
 associated with $\{A, \Gd, \Gn + N \Gd, H, E \}$ is the sum~$N +
 M(z)$, $z\in\rho(A_0)$.
 \end{enumerate}
\end{prop}


%%====================================================================

\begin{proof}
(1) \quad  Operator~$M(0) = \Gn \Pi$  coincides with the
operator~$\Lambda$ defined in Theorem~\ref{TheoremOnKreinExtension}.

%%====================================================================

(2) \quad  Analyticity of~$M(z) - M(0)$ for~$z\in\rho(A_0)$ is
obvious and a short calculation yields the equality
\begin{equation}\label{MzMzetaDifference}
\begin{aligned}
 (M(z) \varphi, \psi)_E - ( \varphi,M(\zeta) \psi)_E   = (z -
\overline{\zeta}) & \left((I - z A_0^{-1})^{-1}\Pi \varphi,(I -\zeta
A_0^{-1})^{-1}\Pi \psi \right)_H,
  \\
& \quad\qquad\qquad \varphi, \psi \in \Gd \HH, \quad z,\zeta \in
\rho(A_0)
\end{aligned}
\end{equation}
where the right hand side is defined correctly for all~$\varphi, \psi \in
E$.  In the special case of~$\zeta =  z$, $z\notin \Real$ and
$\varphi = \psi$ we obtain
\[
 \IM (M(z)\varphi, \varphi) =
 (\IM z)\cdot \|(I -zA_0^{-1})\Pi\varphi\|^2, \qquad \varphi\in E,
 \quad z\notin\Real
\]
showing that the function~$M(z) - M(0)$ is an
operator~$R$-function.

%%====================================================================

(3) \quad Informally, the assertion is an consequence of the Weyl
function definition~(\ref{MzThirdForm}).
%
The formal proof reads as follows.
%
Firstly, the operator~$\Pi$ does not depend on the
particular choice of operator~$\Gn$ in the Green
formula~(\ref{GreenFormula}).
%
Indeed,  for~$\widetilde \Gamma_{\!1}:= \Gn + N \Gd$
we have $\Pi^*  = \Gn A_0^{-1}  =  ( \Gn + N \Gd)
A_0^{-1} = \widetilde \Gamma_{\!1} A_0^{-1}$.
%
Therefore, the $z$-dependent part of the Weyl function~$M(z)- M(0) =
z \Pi^* (I -zA_0^{-1})^{-1}\Pi$ is not affected by the choice
of~$\Gn$.
%
Moreover, for $x \in \Gd \HH$ we have $\widetilde \Gamma_{\!1} \Pi x = ( \Gn
+ N\Gd) \Pi x=  \Gn \Pi x + N\Gd \Pi  x =  ( M(0) + N ) x$, as required.


\smallskip


The proof is complete.
\end{proof}


%%====================================================================


\begin{rem}\label{BoundedWeylFunction}
Assuming~$N =  - M(0)$ in the statement~(3) of the Proposition,
we conclude that the Weyl function of the boundary
 value problem corresponding
 to collection~$\{A, \Gd, \Gn  - M(0)  \Gd, H, E \}$ is a bounded analytic
 operator function~$M(z) - M(0)$ equal to the null operator at the origin.
%
Ope\-ra\-tor~$\widetilde{\Gamma}_{\! 1} = \Gn - M(0) \Gd$ considered as
an alternative to~$\Gn$ was introduced in the context of boundary value
problems for elliptic partial differential equations by Vishik in~\cite{V}.
\end{rem}




\begin{rem}\label{RemarkShiftedWeylFunction}
Let $A_t := A + t I$ where $t \in \Real$. Then Assumption~\ref{AS0}
is obviously satisfied for the operator~$A_t$.
Definition~(\ref{Mz}) makes it possible
to introduce the concepts of Weyl function for the operator~$A_t$.
Since $\ker(A - zI) = \ker(A_t - \zeta
I)$, where $\zeta = t + z$ and operators $\Gd$, $\Gn$ do not depend
on~$t\in \Real$, the Weyl function~$M_t(\cdot)$ for operator~$A_t$
can be defined in a quite natural way as $M_t(\zeta) := M(\zeta - t)$,
$\zeta \in \rho(A_0+ tI)$.

\end{rem}


%%====================================================================
%%
%%====================================================================


\subsection{Uniqueness}

This subsection is dedicated to the question of uniqueness as to
what extent a boundary value problem can be recovered from its Weyl
function.
%
It turns out that the answer depends on the assumption of complete
nonselfadjointness of the minimal operator~$A_{00}$.
%
If~$A_{00}$ is completely nonselfadjoint, then all components of the
collection~$\{A, \Gd, \Gn, H, E\}$ are determined by the Weyl
function uniquely up to an unitary equivalence.

%%====================================================================


\begin{thm}\label{TheoremUniqueness}
 Let $E$, $H$ be two Hilbert spaces and $M_1(\cdot)$, $M_2(\cdot)$
are the Weyl functions of two boundary value problems~$\{A^j, \Gd^j,
\Gn^j, H, E\}$, $j = 1,2$ defined in $H$ with the boundary space~$E$
whose minimal operators are completely nonselfadjoint.
%
Other objects that corresponds to these problems, such as
operators~$\Pi$ and $A_0$, will be distinguished similarly by
employing the superscript based notation.
%
Assume that symmetric operators~$\Lambda^j = M_j(0)$, $j=1,2$  are
defined on the common domain~$\mathscr D := \D(M_1(\cdot)) =
\D(M_2(\cdot))$ dense in~$E$ and the differences~$m_j(z) := M_j(z) -
M_j(0)$, $j = 1, 2$ are bounded operator functions analytic in some
neighborhood of the origin~$z=0$ where~$m_1(z) = m_2(z)$.
%
Then there exists an unitary~$U : H\to H$ such that $ U A_0^1  =
A_0^2 U $,  $ U \Pi^1 = \Pi^2$, $UA^1 = A^2 U$, $\Gd^1   = \Gd^2 U
$, and
 $\Gn^1 = (\Gn^2
+ N \Gd^2 ) U$ where~$N = \Lambda^1 - \Lambda^2 $ is a symmetric
operator on~$\mathscr D$.
%
\end{thm}

%%====================================================================


\begin{proof}
From the condition~$m_1(z) = m_2(z)$ by using expansion~$(I -
zT)^{-1} = \sum_{n=0}^\infty (zT)^n$, $T\in \BB(H)$ with $T =
A_0^j$,  we obtain
\[
 ((A_0^1)^{-n}\Pi^1 e, (A_0^1)^{-m}\Pi^1 e) = ((A_0^2)^{-n}\Pi^2 e, (A_0^2)^{-m}\Pi^2
 e),
\qquad n,m = 0,1,2,\dots, \quad e\in E
\]
%
Define the mapping~$\widetilde U$ on the set~$\{ (A_0^1)^{-n}\Pi^1 e
\; |\; n \geq 0, \, e\in E\}$ dense in~$H$ due to complete
nonselfadjointness of~$A_{00}^1$ as $\widetilde U :
(A_0^1)^{-n}\Pi^1 e \mapsto (A_0^2)^{-n}\Pi^2 e$.
%
The operator~$\widetilde U$ is isometric and its range is dense
in~$H$.
%
We extend~$\widetilde U$ to the unitary~$U$ defined everywhere
on~$H$.
%
Then~$U (A_0^1)^{-n}\Pi^1 = (A_0^2)^{-n}\Pi^2$.
%
For $n =0$ we obtain~$U \Pi^1 = \Pi^2$.
%
Moreover, $U A_0^1 (A_0^1)^{-n}\Pi^1 =A_0^2 (A_0^2)^{-n}\Pi^2 =
A_0^2 U (A_0^1)^{-n}\Pi^1$ for any~$n\ge 0$; therefore $U A_0^1 =
A_0^2 U $.
%
A similar computation can be found in~\cite{Bro}.


%%====================================================================
Now we can define the operator~$A^j$ on its domain~$\D(A^j) :=
\D(A_0^j) \dot{+} \Pi^j \mathscr D$ as follows.
%
Let the action of $A^j$ on the set~$\D(A_0^j)$ coincide with that
of~$A_0^j$ and $A^j h = 0$ for $h\in \Pi^j\mathscr D$.
%
Consequently, $U A^1 = A^2 U$.
%
Operator~$\Gd^j$ is defined as null on the set~$\D(A_0^j)$ and as
 the left inverse to~$\Pi^j$ on~$\Pi^j\mathscr D$.
%
 Therefore,
 $\D(\Gd^j) = \D(A^j)$. For $h := \Pi^1 \mathscr D$ the
 relation~$U \Pi^1 = \Pi^2$
 yields~$ \Gd^1 h = \Gd^2\Pi^2\Gd^1 h= \Gd^2U \Pi^1\Gd^1 h  = \Gd^2 U h$.
%
For~$f \in \D(A_0^1)$ obviously $0 = \Gd^1 f = \Gd^2 U f$ since $Uf
\in \D(A_0^2)$. Therefore, $\Gd^1 u = \Gd^2 U u$ for $u \in
\D(A^1)$.


%%====================================================================

Having obtained pairs~$\{A^j, \Gd^j\}$ we can employ
Proposition~\ref{PropAlternativeConstruction1} and conclude that
boundary value problems~$\{A^j, \Gd^j, \Gn^j, H , E\}$ corresponding
to the Weyl functions~$M^j(\cdot)$ are uniquely determined with the
exception of operators~$\Gn^j$. From the representation~$\Gn^j =
(\Pi^j)^* A^j + \Lambda^j \Gd^j$ we have~$\Gn^1 = ( (\Pi^2)^* A^2 +
\Lambda^1 \Gd^2 )U  = \Gn^2 U + (\Lambda^1 - \Lambda^2)\Gd^2 U =
(\Gn^2 + N \Gd^2)U$.

The proof is complete.
\end{proof}


%%====================================================================

Note that the statements of Theorem regarding operators~$\Gd$, $\Gn$
are only valid if~$\D(\Lambda^1) \cap \D(\Lambda^2)$ is dense
in~$E$. With this assumption omitted, details of the proof allows us
to formulate the following Corollary.

%%====================================================================

\begin{cor}\label{CorUniqueness}
In the notation of Theorem~\ref{TheoremUniqueness} if $m_1(z) =
m_2(z)$ for~$z\in\Complex$ in some neighborhood of the origin, then
there exists an unitary~$U : H\to H$ such that $ U A_0^1  = A_0^2 U
$ and $ U \Pi^1 = \Pi^2$.
\end{cor}


%%====================================================================


Theorem~\ref{TheoremUniqueness} sets forth  questions of finding
objects that would be sufficient to serve as the ``inverse data'' in
the process of reconstructing the boundary value problem, as well as
finding an explicit recipes for such reconstructions.
%
One relevant result regarding the former is given below for the case
when operator~$A_0$ has no continuous spectrum.
%
In a sense, it is a version of the Borg-Levinson theorem for the
Schr\"odinger operator (see~\cite{NSU}) generalized to the case of
boundary value problems under consideration.

%%====================================================================


\begin{thm}
Let the spectrum of operator~$A_0$ be purely discrete.
Denote~$\{z_j\}_1^\infty$, $\{f_j\}_1^\infty$ the eigenvalues and
the orthonormal basis of corresponding eigenvectors of~$A_0$.
%
If the symmetric operator~$A_{00}$ is completely nonselfadjoint,
then operators~$A_0$ and $\Pi$ are determined uniquely up to an
unitary equivalence by the collections~$\{z_j\}_1^\infty$ and~$\{\Gn
f_j\}_1^\infty$.
\end{thm}

%%====================================================================

\begin{proof}
We only need to show that the function~$m(z) = M(z) - M(0)$, $z\in
\rho(A_0)$ can be recovered from the given collections~$\{z_j\}$,
$\{\Gn f_j\}$. The reconstruction of $A_0$ and $\Pi$ knowing the
function~$m(z)$ is ensured by Corollary~\ref{CorUniqueness}.
%
Since $m(\cdot)$ is bounded, it is sufficient to consider the
quadratic form~$(m(z)e, e)$, $e\in E$ only. We have
\[
 \begin{aligned}
   (m(z)e, e) &  = (z \Pi^* (I -zA_0^{-1})^{-1}\Pi e, e ) = (z (I -zA_0^{-1})^{-1}\Pi e, \Pi e )
   \\
   & = z\sum_j \langle(I -zA_0^{-1})^{-1}\Pi e, f_j\rangle\langle f_j, \Pi e\rangle
     = z\sum_j \langle e, \Gn A_0^{-1}( I - \bar z A_0^{-1})^{-1} f_j\rangle
              \langle \Gn A_0^{-1}f_j,  e\rangle
   \\
   & = z\sum_j \frac{1}{z_j^{2}}\cdot \frac{\langle e, \Gn f_j \rangle
         \langle \Gn f_j, e \rangle}{1 - z z_j^{-1}}
     = \sum_j     \frac{z}{z_j}\cdot \frac{|(e, \Gn f_j)|^2}{z_j - z}
 \end{aligned}
\]
Therefore, the function~$m(z)$ is uniquely determined by the
sets~$\{z_j\}$ and $\{\Gn f_j\}$.
\end{proof}





%%====================================================================
%%
%%====================================================================




\subsection{General boundary conditions}

Properties of the Weyl function play an
important role in the study of
spectral problems for restrictions of operator~$A$
to the domains described by
various boundary conditions.
%
 Let $B_0, B_1$ be two
linear operators  on the space~$E$
such that $\D(B_0) \supset \Gd \HH$ and $B_1 \in \BB(E)$.
%
The problem of our interest is to find the vector~$w \in \D(A)$
satisfying following equations:
\begin{equation}\label{GBVPSection3}
\left\{
\begin{array}{l}
(A - zI) w = f \\
( B_0 \Gd + B_1 \Gn) w = \varphi
\end{array}
\right.
\end{equation}
for given $f \in H $, $\varphi \in E$, and  $z \in \Complex$.
%
Note that we can easily assume that the operators~$B_0$ and $B_1$
in~(\ref{GBVPSection3}) depend on the complex number~$z\in\Complex$.
%
In this case the problem~(\ref{GBVPSection3}) is an example of the
boundary value problem with the spectral parameter in the boundary
condition.

%%===================================================================

The following theorem establishes a
one-to-one correspondence between solutions to the
system~(\ref{GBVPSection3}) and solutions to a certain ``boundary
equation'' in the space~$E$ that involves the Weyl function.
%
Such a reduction of the boundary value problem to the ``equation on
the boundary'' is well known in the literature, see~\cite{Ag,KK,ML}
for examples.
%
The theorem can be seen as an abstract variant of this procedure.
%
Note that its value is fully realized only if~$B_1$ is not equal to
the null operator.


%%====================================================================

\begin{thm}\label{MainTheoremSection3}
Let  $w_z^{f,\varphi} \in \D(A)$ be a solution to the
problem~(\ref{GBVPSection3}) for given $z \in \rho(A_0)$, $f \in H$
and $\varphi \in E$.
%
Then the vector
\begin{equation}\label{PsiFromWSection3}
\Psi_z^{f,\varphi} := \Gd w_z^{f,\varphi}
\end{equation}
is a solution to the equation
\begin{equation}\label{GBVPonBoundarySection3}
 (B_0 + B_1 M(z))\Psi = \varphi - B_1 \Pi^* (I - zA_0^{-1})^{-1} f,
 \quad \Psi \in \Gd \HH
\end{equation}
%
Conversely, if  $\Psi_z^{f,\varphi}\in \Gd \HH$ is a solution
to~(\ref{GBVPonBoundarySection3}) for  given $z \in \rho(A_0)$, $f
\in H$ and $\varphi \in E$, then the vector
\begin{equation}\label{WFromPsiSection3}
w_z^{f,\varphi} := (A_0 - z I)^{-1} f + \big[ I + z (A_0 -
zI)^{-1}\big] \Pi \Psi_z^{f,\varphi}
\end{equation}
is a solution to the problem~(\ref{GBVPSection3}).
\end{thm}

%%====================================================================

\begin{proof}
Let $ w_z^{f,\varphi}\in \D(A)$ solves the
problem~(\ref{GBVPSection3}).
%
Then
\begin{equation*}
 \begin{aligned}
  \lefteqn{
   \varphi - B_1 \Pi^* (I - zA_0^{-1})^{-1} f
  }\\
  & = (B_0\Gd + B_1 \Gn) w_z^{f,\varphi} - B_1 \Gn A_0^{-1} (I - zA_0^{-1})^{-1} f
  \\
  & = B_0\Gd w_z^{f,\varphi} +  B_1 \Gn \left[ w_z^{f,\varphi} - (A_0 -zI)^{-1}f\right]
  \\
  & = B_0\Gd w_z^{f,\varphi} +
    B_1 \Gn \left [ I - (A_0 - zI)^{-1} (A - zI) \right ]  w_z^{f,\varphi}
  \\
  & = B_0\Gd w_z^{f,\varphi} +
    B_1 M(z) \Gd \left [ I - (A_0 - zI)^{-1} (A - zI) \right ]  w_z^{f,\varphi}
  \\
  & = \left [ B_0 + B_1 M(z) \right ] \Gd w_z^{f,\varphi}
 \end{aligned}
\end{equation*}
where we made use of the inclusion~$[I - (A_0 - zI)^{-1} (A -
zI)]w_z^{f,\varphi} \in \ker(A - zI)$ and
property~(\ref{MzThirdForm}) of Weyl function.
%
We see that~(\ref{PsiFromWSection3}) is a solution to the
problem~(\ref{GBVPonBoundarySection3}).
%
The required inclusion~$\Gd w_z^{f,\varphi}\in \Gd \HH$
follows from the decomposition
\begin{equation*}
w_z^{f,\varphi} = (A_0 - zI)^{-1}f + h_z = (A_0 - zI)^{-1}f + (I -
zA_0^{-1})^{-1}h,
\end{equation*}
where $h_z \in \ker(A - zI)$ and $h := (I -zA_0^{-1})h_z\in \HH$,
so that $\Gd w_z^{f,\varphi} = \Gd h_z = \Gd h\in \Gd \HH$
according to Proposition~\ref{PropDirectSum}.

%%====================================================================

To prove the inverse, let $\Psi_z^{f,\varphi}\in \Gd \HH$ be a
solution to the equation~(\ref{GBVPonBoundarySection3}).
%
Then since
the second summand in~(\ref{WFromPsiSection3}) belongs to $\ker(A -
zI)$, the vector~$w_z^{f,\varphi}$ defined
by~(\ref{WFromPsiSection3}) lies in the domain of the operator~$A$
and obviously, $(A - zI)w_z^{f,\varphi} = f$.
%
Let us verify that~$w_z^{f,\varphi}$ is also a solution
to the second equation~(\ref{GBVPSection3}).
%
We have
\begin{equation*}
 \begin{aligned}
\lefteqn{
 (B_0\Gd + B_1 \Gn) w_z^{f,\varphi}
 }
 \\
 & = (B_0\Gd + B_1 \Gn)\left[
 (A_0 - z I)^{-1} f +
 \big( I + z (A_0 - zI)^{-1}\big) \Pi \Psi_z^{f,\varphi}
 \right]
 \\
 & = B_0 \Gd \Pi \Psi_z^{f,\varphi}
  +  B_1 \Gn (A_0 - zI)^{-1} f
   + B_1 \Gn (I - zA_0^{-1})^{-1}\Pi \Psi_z^{f,\varphi}
 \\
 & = \left[ B_0 + B_1 M(z)\right] \Psi_z^{f,\varphi}
 + B_1 \Gn (A_0 - zI)^{-1} f
 \\
 &
  = \varphi - B_1 \Pi^* (I - zA_0^{-1})^{-1} f
   + B_1 \Gn (A_0 - zI)^{-1} f
 \\
 & = \varphi - B_1 \Gn
 \left[  A_0^{-1} (I - zA_0^{-1})^{-1}  -  (A_0 - zI)^{-1}  \right]f
 \end{aligned}
\end{equation*}
where we used  representation~(\ref{MzSecondForm}) for the Weyl
function, equality~$\Gd \Pi\Psi_z^{f\varphi} = \Psi_z^{f\varphi}$
and assump\-ti\-on~(\ref{GBVPonBoundarySection3}).
%
Now the required result~$(B_0\Gd + B_1 \Gn) w_z^{f,\varphi} =
\varphi$ follows from the identity~$A_0^{-1}(I -zA_0^{-1})^{-1} =
(A_0 - zI)^{-1}$.

The proof is complete.
\end{proof}



%%====================================================================



\begin{rem}\label{ConditionOfSovability}
 A simple condition that
guarantees solvability of~(\ref{GBVPonBoundarySection3}) for any
$f\in H$, $\varphi\in E$ is the closedness of the operator~$(B_0 +
B_1 M(z))$ on its domain~$\Gd \HH$ and its invertibility, so that the
range of its inverse~$[ B_0 + B_1 M(z)]^{-1}$ coincides with~$\Gd
\HH$. As easily seen from~(\ref{WFromPsiSection3}), in this case the
norm estimate~$\|w_z^{f,\varphi}\| \leq C(z) (\|f\| + \|\varphi\|)$ holds
with some positive constant~$C(z) < \infty$ depending on
$z\in\rho(A_0)$.
\end{rem}

%%====================================================================

\begin{rem}
 Comparison of (\ref{WFromPsiSection3}) and (\ref{2:BVPsolution})
 shows that the solution~$w_z^{f,\varphi}$
 can be obtained by solving the problem~(\ref{2:BVP})
 with $\varphi\in E$  replaced with the solution~$\Psi_z^{f,\varphi}$
 to~(\ref{GBVPonBoundarySection3}) provided that the latter exists.
 In case if~(\ref{GBVPonBoundarySection3}) is
 solvable and~$\Psi_z^{f,\varphi} \in E$, but not
 necessarily~$\Psi_z^{f,\varphi} \in \Gd \HH$
 as required in conditions of Theorem~\ref{MainTheoremSection3},
 solution~$w_z^{f,\varphi}$ still can be defined as a weak
 solution~$u_z^{f,\Psi}$ where $\Psi := \Psi_z^{f,\varphi}$
 to the problem~(\ref{2:BVP}) described in Remark~\ref{RemWeakSolutions}.
\end{rem}


%%==================================================================
%%
%%==================================================================


\section{An example}

This section illustrates the formalism developed above by means of
an example.
%
The setting can be extended to more general cases of operators and
domains, most notably to the general strongly elliptic partial
differential operators and domains of lesser smoothness.
%
Such generalizations would be based upon the potential theory
developed to the degree that embraces more generic domains and
operators.
%
Relevant results are partially available in the literature,
see~\cite{DU, ML}.
%
However, we keep the exposition simple in hope to dedicate a
separate publication to more advanced situations.


%%=======================================================================

 Let ~$\Omega\subset\Real^3$ be a bounded simply connected domain
 with the $C^\infty$-boundary~$\Gamma$, and  $H := L_2(\Omega)$,
 $E:= L_2(\Gamma)$.
%
 Operator~$A$ is the
 Laplacian~$\Delta =\frac{\partial^2}{\partial x_1^2} +
 \frac{\partial^2}{\partial x_2^2} +
 \frac{\partial^2}{\partial x_3^2}$
 on domain~$\D(A) = H^2(\Omega)$
 where $H^2(\Omega)$ is the Sobolev space of
 functions~$u \in L_2(\Omega)$
 such that $\partial^2 u/\partial x_i^2 \in L_2(\Omega)$, $i = 1,2,3$.
%
 Denote~$\partial_\nu := \frac{\partial}{\partial \nu}$ the
 derivative in the direction of the outer normal~$\nu$ to the
 boundary~$\Gamma$. For any function~$u\in \D(A)$ there exist its
 trace~$\left. u\right|_{\Gamma} $ and the trace of its
 derivative~$\left. (\partial_\nu u)\right|_{\Gamma}$ on~$\Gamma$.
%
 We define the mappings~$\Gd$ and $\Gn$
 on the domain~$\D(A) = H^2(\Omega)$ as follows
\[
 \Gd \, : \; u \mapsto  \left. u\right|_{\Gamma}, \qquad
  \Gn  \, : \;  u \mapsto  \left.
    ({\partial_\nu u})\right|_{\Gamma}, \qquad u \in
  \D(A)
\]
It is known that $\Gd$ and $\Gn$ are bounded operators acting
from~$H^2(\Omega)$ into~$L_2(\Gamma)$.
%
More precisely, the following equalities for their ranges hold:
$\Ran (\Gd ) = H^{3/2}(\Gamma)$ and $\Ran (\Gn) = H^{1/2}(\Gamma)$,
where~$H^{3/2}(\Gamma)$ and~$H^{1/2}(\Gamma)$ are fractional Sobolev
spaces on the surface~$\Gamma$, see \cite{ML, LM}.
%
For any vectors~$u, v \in \D(A)$ the Green formula holds:
\[
 (Au , v)_H - (u, Av)_H = (\Gn u, \Gd v )_E - (\Gd u, \Gn v )_E,
 \qquad u, v \in \D(A)
\]

%%====================================================================

The operator~$A_0$ defined as a restriction of~$A = \Delta$ to the
domain~$\D(A_0) := \{u\in H^2(\Omega)\;|\;\Gd u = 0 \}$ is the
operator of Dirichlet boundary value problem in
the space~$L_2(\Omega)$.
%
It is a closed boundedly invertible selfadjoint operator with
discrete spectrum. (See~\cite{Br} for the proof of its closedness.)


%%====================================================================

The null set~$\HH$ of~$A = \Delta$ consists of all harmonic
functions~$h$ in $\Omega$, $\Delta h = 0$ that belong to the
space~$H^2(\Omega)$.
%
Note that~$\Gd\HH\subset H^{3/2}(\Gamma)$ and~$\Gn\HH\subset
H^{1/2}(\Gamma)$, since $\HH\subset H^2(\Omega)$. The closure
of~$\HH$ in~$L_2(\Omega)$ is wider that the Sobolev
class~$H^2(\Omega)$ and includes square integrable functions
on~$\Omega$ that don't possess boundary values on~$\Gamma$ and don't
belong to the domain~$\D(A)$.



%%====================================================================



The intersection~$\D(A_0)\cap\HH $ is trivial because any harmonic
function from~$\HH$ with the zero boundary values on~$\Gamma$
is equal to zero almost everywhere in~$\Omega$. The
operator~$\Pi$ is the left inverse to the mapping $h\mapsto
h|_{\Gamma}$, $h\in\HH$.  It is the operator of harmonic
continuation from the boundary~$\Gamma$ that maps a
function~$\varphi\in H^{3/2}(\Gamma)$ to the harmonic function~$h$ from~$\HH$
such that~$h|_\Gamma = \varphi$.
 Noticing that, the decomposition~$\D(A) =\D(A_0) \dot {+}
\HH$ becomes almost obvious. Indeed, for any~$u\in
H^2(\Omega)$ there exists a unique harmonic function~$h = \Pi
(\left.u\right|_{\Gamma})$
 with the same boundary values~$ h|_\Gamma =  u|_\Gamma$.
%
Therefore, the difference~$u -h$ belongs to~$H^2(\Omega)$ and
vanishes at the boundary: $\left.(u-h)\right|_{\Gamma} =0$. Hence,
$u-h$ lies in the domain~$\D(A_0)$. This decomposition of the
space~$H^2(\Omega)$ and the equality~$\Gd H^2(\Omega) =
H^{3/2}(\Gamma)$ given above show that $\Gd \HH = H^{3/2}(\Gamma)$.

%%====================================================================


Consequently, Assumptions~\ref{AS0} and~\ref{AS1} hold true for the
collection~$\{\Delta, u\mapsto u|_\Gamma, u\mapsto \partial_\nu u
|_\Gamma, L_2(\Omega) , L_2(\Gamma)\}$ and we can apply the theory
developed in preceding sections to the semi-homogeneous spectral
boundary value problem
\begin{equation}\label{BPPotential}
\left \{ \;
 \begin{aligned}
 & (\Delta  - z)u  = 0 \quad \text { in } \Omega, \\
 &  \left. u\right|_{\Gamma}   = \varphi
 \end{aligned}\quad
 \right .
\end{equation}
where~$\varphi \in H^{3/2}(\Gamma)$.




%%====================================================================



We start with the following
representation of any vector $\mathfrak u_z\in H^2(\Omega)$ satisfying the
equation~$ (\Delta - z) u = 0$, $ z \in \Complex$ in~$\Omega$
\[
 \mathfrak u_z(x) = \int_{\Gamma}
 \left[
 G_z(x,y)\cdot \frac{\partial \mathfrak u_z }{\partial \nu}
   - \frac{\partial G_z(x,y)}{\partial \nu_y} \cdot \mathfrak u_z(y)
 \right] d S_y, \qquad x \in \Omega
\]
where
\[
 G_z(x,y) := \frac{1}{4\pi}\frac{\exp{(ik|x-y|)}}{|x-y|}, \qquad x, y
 \in \Real^3,  \quad k := \sqrt { -z} ,\quad \IM k \ge 0
\]
is the fundamental solution to the Helmholz equation~$(\Delta - z) u
=0$ in $\Real^3$ and~$d S_y$ is the euclidian surface measure on~$\Gamma$.
%
This formula is well known for functions~$\mathfrak u_z$
continuous in the closure of the domain~$\Omega$ (see \cite{CK}). For
the general case of $\mathfrak u_z\in H^1(\Omega)$ the relevant result can be
found in~\cite{ML}. We rewrite this representation in the form
\begin{equation}\label{GeneralSolution}
 \mathfrak u_z =  \mathscr S_z (\left.\partial_\nu  \mathfrak u_z\right|_{\Gamma}) -
 \mathscr D_z (\left. \mathfrak u_z\right|_{\Gamma}) =
  \mathscr S_z \Gn \mathfrak u_z -  \mathscr D_z \Gd \mathfrak u_z
\end{equation}
where operators~$\mathscr S_z$ and~$\mathscr D_z$ are the usual single and
double layer potentials. For
functions~$w \in L_2(\Gamma) $ they are defined as mappings
\begin{equation*}
 (\mathscr S_z  w )(x):= \int_{\Gamma}
 G_z(x,y) w(y) dS_y, \qquad
 (\mathscr D_z  w)(x) := \int_{\Gamma}
 \frac{\partial G_z(x,y)}{\partial \nu_y} w(y) dS_y, \qquad x\in\Omega
\end{equation*}
%
Operators~$\mathscr S_z$ and $\mathscr D_z$ acting from the
space~$L_2(\Gamma)$ into $L_2(\Omega)$ are compact for~$\IM
z\neq 0$, see \cite{ML}. Boundary values of~(\ref{GeneralSolution})
on the surface $\Gamma$ can be
calculated using well known formulae of potential theory.
%
Namely, for any $w\in C^2(\overline{\Omega})$ the following
equalities hold \cite{CK, MA}:
\begin{equation}\label{BoundayValuesOfPotentials}
 \begin{aligned}
 \Gd \mathscr S_z w & = \mathrm   S_z w, &
 \Gd \mathscr D_z w & =  - \frac{1}{2}(  w -\mathrm  K_z w),
 \\
 \Gn \mathscr S_z w & =   \frac{1}{2} (  w +\mathrm K'_z w ), &
 \Gn \mathscr D_z w & =\mathrm  R_z w
 \end{aligned}
\end{equation}
 where for $x\in\partial\Omega$
\[
 (\mathrm S_z w)(x) := \int_{\Gamma}
 G_z(x,y) w(y) dS_y, \qquad
(\mathrm K_z  w)(x) :=  2 \int_{\Gamma}
 \frac{\partial G_z(x,y)}{\partial \nu_y} w(y) dS_y
\]
are direct values of the single and double layer potentials on the
surface~$\Gamma$, and
\[
 (\mathrm K'_z w)(x) := 2 \int_{\Gamma}
 \frac{\partial G_z(x,y)}{\partial \nu_x} w(y) dS_y,  \qquad
 (\mathrm R_z w)(x) := \frac{\partial}{\partial \nu_x}
 \int_{\Gamma}
 \frac{\partial G_z(x,y)}{\partial \nu_y} w(y) dS_y
\]
%
Operators $\mathrm K'_z$ and $\mathrm R_z$
are sometimes called the adjoint double layer potential
and the hypersingular operator, respectively.
%
It turns out that under our assumptions about the boundary~$\Gamma$
the mappings $\mathrm S_z$, $\mathrm K_z$, $\mathrm K'_z$ are
compact operators on the space~$L_2(\Gamma)$~(\cite{FJR, MA}).
%
Operator~$\mathrm R_z$ is unbounded on $L_2(\Gamma)$.
%
It can be shown, however, that $\mathrm R_z  : H^1(\Gamma) \rightarrow
L_2(\Gamma)$, see \cite{ML}.
%
Consequently, values~$\Gd \mathfrak u_z$
and $\Gn \mathfrak u_z$ for the vector~$\mathfrak u_z$ from~(\ref{GeneralSolution})
belong to the domain of operators~$\mathrm S_z$, $\mathrm K_z$,
$\mathrm K'_z$, and in addition, $\Gd \mathfrak u_z $ as an
element of~$H^{3/2}(\Gamma)\subset
H^{1}(\Gamma)$, lies in the domain of~$\mathrm R_z $.
%
Thus, we have
\[
 \begin{aligned}
& \Gd \mathfrak u_z & = \Gd (  \mathscr S_z \Gn  - \mathscr D_z \Gd) \mathfrak u_z & =
\mathrm  S_z \Gn \mathfrak u_z + \frac{1}{2}
(  I - \mathrm K_z) \Gd  \mathfrak u_z \\
& \Gn \mathfrak u_z & = \Gn ( \mathscr S_z \Gn  - \mathscr D_z \Gd) \mathfrak u_z & =
   \frac{1}{2}(I +\mathrm  K'_z)\Gn \mathfrak u_z
- \mathrm  R_z\Gd  \mathfrak u_z
 \end{aligned}
\]
%
An elementary rearranging of summands yields:
\begin{equation*}
( I +  \mathrm K_z )\Gd  \mathfrak u_z =  2 \mathrm S_z \Gn \mathfrak u_z, \qquad
 ( I -  \mathrm K'_z  ) \Gn \mathfrak u_z  = - 2 \mathrm R_z\Gd \mathfrak u_z
\end{equation*}
%
Recall now that operators~$\mathrm K_z$ and $\mathrm K'_z$ are in
fact analytic operator-functions of the complex parameter~$z$.
%
Since $\mathrm K_z$ and $\mathrm K'_z$ are compact, the operators $I
+ \mathrm K_z $ and $I - \mathrm K'_z$ are boundedly invertible for
all $z\in\Complex$ except for a countable set of points accumulating
to the infinity, provided they are invertible at least for one value
of~$z\in\Complex$.
%
(See~\cite{Bro} for example.).
%
More accurate result from~\cite{CK} says that poles of the
operator-function~$(I + \mathrm K_z)^{-1}$ coincide with the
eigenvalues of the Neumann boundary problem in $L_2(\Omega)$. This
problem is defined as~$(\Delta - z) u = 0$, $\left. \partial_\nu u
\right|_{\Gamma} =0$. Analogously, poles of~$(I - \mathrm
K'_z)^{-1}$ are the eigenvalues of the Dirichlet boundary
problem~$(\Delta  - z) u = 0$, $\left. u \right|_{\Gamma}=0$ in
$L_2(\Omega)$, that is, the spectrum of operator~$A_0$.
%
 Since all these
eigenvalues are real, it follows that the Weyl function~$M(\cdot)$
of the boundary value problem~(\ref{BPPotential}) and its
inverse~$M^{-1}(\cdot)$ are
\begin{equation}\label{WeylFinctionAndItsInverse}
 M(z) = -2 (I -\mathrm K'_z)^{-1}\mathrm R_z,
 \qquad
 M^{-1}(z) = 2  (I + \mathrm K_z)^{-1} \mathrm S_z, \qquad z\in\Complex_\pm
\end{equation}
Function~$M(\cdot)$ is defined correctly on the
domain~$H^{1}(\Gamma)$, whereas values of its
inverse~$M^{-1}(\cdot)$ are compact operators on~$L_2(\Gamma)$.




%%=============================================================================




According to the theory above, the solution to
problem~(\ref{BPPotential}) is the vector~$(I -z
A_0^{-1})^{-1}\Pi\varphi$, see formula~(\ref{2:BVPsolution0}).
%
Now the representation~(\ref{BoundayValuesOfPotentials})
for~$\Gd \mathscr D_z$
gives rise to the following Proposition.


%%=============================================================================


\begin{prop}\label{InternalStateProp}
Solutions~$u_z^\varphi = (I -z A_0^{-1})^{-1}\Pi \varphi$
 of the problem~(\ref{BPPotential}) are given by the formula
\begin{equation}\label{InternalState}
 u_z^\varphi = -2 \mathscr D_z (I - \mathrm K_z)^{-1} \varphi, \qquad z\in\rho(A_0)
\end{equation}
where $\varphi \in H^{3/2}(\Gamma)$.
\end{prop}


%%=============================================================================



\begin{proof}
The fact that~(\ref{InternalState})
is a solution to the problem~(\ref{BPPotential}) is the well known result
of the potential theory.
%
Taking into consideration~(\ref{BoundayValuesOfPotentials}) we need
to show that~$(I - \mathrm K_z)^{-1} \varphi$, $z\in\rho(A_0)$ is a
vector from~$H^{3/2}(\Gamma)$ for~$\varphi \in H^{3/2}(\Gamma)$.
%
This follows directly from the fact that~$\mathrm K_z$ is a pseudodifferential
operator of order not greater than~$-1$ (see~\cite{Ag}).
%
Hence, $(I - \mathrm K_z)^{-1}$ is an
invertible pseudodifferential operator or order zero, so that
 $(I - \mathrm K_z)^{-1} H^{3/2}(\Gamma) =H^{3/2}(\Gamma)$.
%
Finally, uniqueness of  representation~(\ref{InternalState}) for
vectors~$u_z^\varphi = (I -zA_0^{-1})^{-1}\Pi \varphi$ is a
consequence of Proposition~\ref{PropDirectSum} which guarantees that
if two vectors~$u_1$, $u_2$ from~$\ker(A -zI)$ have the same
boundary values~$\Gd u_1 = \Gd u_2$ then necessarily~$u_1 = u_2$.

%
 The proof is complete.
\end{proof}


%%==========================================================


\begin{rem}\label{InvertibilityOfDoubleLayerPotential}
According to results above, the set of all
solutions~(\ref{InternalState}) coincides with~$\ker(A -zI)$ and the
mapping~$\varphi \mapsto u_z^{\varphi} = (I -zA_0^{-1})^{-1}\Pi
\varphi $ establishes a one-to one correspondence between
$\varphi\in H^{3/2}(\Gamma)$ and the respective
solution~$u_z^\varphi$.
%
This result combined with formula~(\ref{InternalState})
and properties of~$(I
-\mathrm K_z)^{-1}$ cited in the proof of Proposition~\ref{InternalStateProp}
implies that the double layer potential~$\mathscr D_z$
restricted to the set $H^{3/2}(\Gamma)$ is an invertible operator.
%
More precisely, its range coincides with $\ker(A-zI)$ and the
inverse mapping is given by the formula
\begin{equation*}
(\mathscr D_z )^{-1} \mathfrak u_z =  -2 (I - \mathrm K_z)^{-1}\Gd\mathfrak u_z,
\qquad\mathfrak u_z \in\ker(A-zI).
\end{equation*}
\end{rem}


%%===============================================================



\begin{rem}
From the representation~(\ref{InternalState}) we obtain for $\varphi = \Gd u_z^\varphi$
\[
 \Gn u_z^\varphi  = - 2 \Gn \mathscr D_z (I - \mathrm
K_z)^{-1} \varphi = - 2 \mathrm R_z (I -\mathrm K_z)^{-1} \varphi
\]
since~$(I -\mathrm K_z)^{-1} \varphi \in H^{3/2}(\Gamma)$
for~$\varphi \in H^{3/2}(\Gamma)$.
%
It means that the Weyl function~$M(\cdot)$ can be rewritten in the
form
\[
 M(z) = - 2 \mathrm R_z (I -\mathrm K_z)^{-1}, \qquad
 z\in\rho(A_0)
\]
\end{rem}


%%===========================================================================

\begin{rem}
 Operator~$M(0) = -2 \mathrm R_0 (I -\mathrm K_0)^{-1} = - 2 (I -\mathrm K'_0)^{-1} \mathrm R_0 $
 is the so-called Dirichlet-to-Neumann map for the Laplace
 equation~$\Delta u = 0$.
%
 It maps a smooth function~$\varphi$ on the boundary~$\Gamma$ to the
 trace of the normal derivative on~$\Gamma$ of the solution~$u^\varphi_0$ to
 the Laplace equation~$\Delta u =0$
 with the boundary condition~$u|_\Gamma = \varphi$.
%
 In other words, $M(0)$ maps the
 boundary values of the Dirichlet  boundary problem for the Laplacian
 into those of the Neumann problem,
 $M(0) : \varphi \mapsto \left. \partial_\nu u^\varphi_0 \right|_\Gamma$.
%
 Representation~$M(0) = \Gn \Pi$ established above for the general
 case now becomes self-evident.
%
 Properties of the Dirichlet-to-Neumann map
 of various partial differential operators were investigated in
 a number of papers, starting from~\cite{V}.
 (See as well the survey~\cite{SU}.)
%
 In the case under consideration $M(0)$ is a positive selfadjoint
 pseudodifferential operator
 of order~$1$ (see \cite{Ag, V}).
%
 Further, since the Weyl
 function~$M(z)$, $z\in\rho(A_0)$ maps $\varphi$ to
 ${\left. u_z^\varphi\right|}_\Gamma$ where $u_z^\varphi$ is the solution to the
 equation~$(\Delta - z) u = 0$, it can be said that the
 operator~$M(z)$ is the Dirichlet-to-Neumann map for the
 equation~$(\Delta - z) u = 0$, cf.~\cite{SU}.
\end{rem}

%%===========================================================================

\begin{rem}
 Invertibility of the double layer potential~$\mathscr D_z$,
 $z\in\rho(A_0)$ stated in Remark~\ref{InvertibilityOfDoubleLayerPotential}
 shows that~$\mathscr D_z$ maps~$H^{3/2}(\Gamma)$ into a subset of~$H^2(\Omega)$.
%
 Applying interpolation arguments~\cite{LM} we conclude
 that~$\mathscr D_z$ is the operator from~$L^2(\Gamma)$
 to the space~$H^{1/2}(\Omega)$, therefore
 $\mathscr D_z$ is compact as a mapping
 from~$E = L_2(\Gamma)$ to $H = L_2(\Omega)$.
%
 Thus, according to Proposition~\ref{InternalStateProp}, the
 transformation~$\varphi \mapsto (I - zA_0^{-1})^{-1}\Pi\varphi$, $\varphi \in E$
 is compact.
%
 Furthermore, since $M(z) - M(\zeta) =(z - \zeta)
 \Pi^*(I -\zeta A_0^{-1})^{-1}(I - z A_0^{-1})^{-1}\Pi $ for~$z,\zeta \in \rho(A_0)$,
 see~(\ref{MzMzetaDifference}), the difference~$M(z) - M(\zeta)$ is
 a  compact operator on~$E$.
%
 In particular, it holds for~$M(z) - M(0)$, thereby $M(z)$, $z\in \rho(A_0)$
 is a compact perturbation of the
 Dirichlet-to-Neumann map~$\Lambda = M(0)$ of the Laplacian.
\end{rem}


\begin{rem}
 Theorem~\ref{TheoremUniqueness}
 shows that if one knows the  function~$M(z)$,
  then one can determine the operator $A$ and boundary maps~$\Gd$,
  $\Gn$ up to isomorphism.
%
   We know from the case of
   the Sturm-Liouville operator in one dimension
   that this is the best
   possible result in the abstract setting.
%
   However for the multidimensional case of differential
   operators one can recover its coefficients
   knowing just $M(z_0)$ for a single point $z_0\in \rho(A_0)$, see
   e.~g.~\cite{NSU, SU}.
%
 In fact even weaker
   results are now known such as recovery of the coefficients
   knowing the function~$M(z)$ on just a
   part of a smooth boundary, see~\cite{Uh}
   for the survey.
%
%
\end{rem}

%%=======================================================================
%%
%%=======================================================================




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\vspace{\stretch{1}} Vladimir Ryzhov \\ %% [-10pt]
67-6400 Spencer Road \\ %% [-10pt]
Kelowna, BC \\ %% [-10pt]
V1X 7T6 \\ %% [-10pt]
Canada \\ %% [-10pt]
vryzhov@bridges.com \\ %% [-10pt]
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