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invariant subspaces, operator Riccati equation, 
operator matrix, operator angle, factorization theorem
---------------0210032148468
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\begin{document}

\begin{frontmatter}

% Title, authors and addresses

% use the thanksref command within \title, \author or \address for footnotes;
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\title{On the existence of solutions to the operator Riccati equation and the tan $\Theta$
theorem}

% use optional labels to link authors explicitly to addresses:
\author{Vadim Kostrykin}
\address{Fraunhofer-Institut f\"{u}r Lasertechnik, Steinbachstra{\ss}e 15, D-52074, Aachen,
Germany\\ E-mail: kostrykin@t-online.de, kostrykin@ilt.fraunhofer.de}

\author{Konstantin A. Makarov}
\address{Department of Mathematics, University of
Missouri, Columbia, MO 65211, USA\\ E-mail: makarov@math.missouri.edu}

\author{Alexander K. Motovilov\thanksref{label4}}
\address{Department of Mathematics, University of
Missouri, Columbia, MO 65211, USA\\ E-mail: motovilv@thsun1.jinr.ru}
\thanks[label4]{On leave of
absence from the Joint Institute for Nuclear Research, 141980
Dubna, Moscow Region, Russia}

\runauthor{V. Kostrykin, K. A. Makarov, A. K. Motovilov}
\runtitle{Existence of Solutions to the Operator Riccati Equation}

\date{October 2, 2002}


\begin{abstract}
Let $A$ and $C$ be self-adjoint operators such that the
spectrum of $A$ lies in a gap of the spectrum of $C$ and let
$d>0$ be the distance between the spectra of $A$ and $C$. We
prove that under these assumptions the sharp value of the
constant $c$ in the condition $\|B\|<c d$ implying the
solvability of the operator Riccati equation
$XA-CX+XBX=B^\ast$ is equal to $\sqrt{2}$. We also prove an
extension of the Davis-Kahan $\tan\Theta$ theorem.
\end{abstract}

\begin{keyword}
invariant subspaces \sep operator Riccati equation \sep
operator matrix \sep operator angle \sep factorization theorem

\MSCC 47A15 \sep 39B42 \sep 47A56
\end{keyword}
\end{frontmatter}

\pagestyle{headings}

% main text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{Intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this paper we consider the operator Riccati equation
\begin{equation}
\label{Ric0} XA-CX+XBX=B^*
\end{equation}
associated with a $2\times 2$ block operator matrix
\begin{equation}
\label{cHintro} {\bH}=\begin{pmatrix}
  A     &\,&   B  \\
  B^*     &&   C
\end{pmatrix}
\end{equation}
on the orthogonal sum $\cH=\cH_A \oplus \cH_C$ of separable
Hilbert spaces. Here $A$ is a bounded self-adjoint operator on
the Hilbert space $\cH_A$, $C$ possibly unbounded
self-adjoint operator on the Hilbert space $\cH_C$, and $B$ is
a bounded operator from $\cH_C$ to $\cH_A$.

Solving the Riccati equation appears to be an adequate tool in
the study of the invariant subspaces of the operator $\bH$
which are the graphs of bounded operators from $\cH_A$ to
$\cH_C$. More precisely, given a bounded solution $X$ to the
Riccati equation \eqref{Ric0} such that $\Ran
X\subset \Dom(C)$, the graph
\begin{equation*}
\cG(X)=\left\{x\oplus X x|\ x\in\cH_A \right\}
\end{equation*}
of the operator $X$ reduces the operator $\bH$. In the framework of this approach the
following two problems naturally arise. The first problem is to study the spectrum of
the part of $\bH$ associated with the reducing subspace $\cG(X)$ (respectively its
orthogonal complement $\cG(X)^\perp$), and the second one is to estimate the operator
angle $\Theta$ (see, e.g., \cite{KMMa} for discussion of this notion)
 between the subspaces $\cG(X)$ and $\cH_A$ (respectively $\cG(X)^\perp$ and
$\cH_C$). Both of these problems can efficiently be solved if a bounded solution $X$
to
\eqref{Ric0} is known: the operator $\bH$ appears to be similar to the diagonal block
operator matrix
\begin{equation*}
\begin{pmatrix} A + B X &\,& 0 \\ 0 && C- B^\ast X^\ast \end{pmatrix}
\end{equation*}
associated with the decomposition $\cH=\cH_A\oplus \cH_C$, and
the operator angle $\Theta$ between the subspaces $\cG(X)$ and
$\cH_A$ has the representation
\begin{equation*}
\Theta = \arctan \sqrt{X^\ast X}.
\end{equation*}

If the spectra of $A$ and $C$ overlap, then the Riccati
equation \eqref{Ric0} may  have no solution at all (cf., e.g.,
\cite[Example 3.2]{AMM}). At the same time the spectra
separation condition alone does not guarantee the existence of
solutions either. Under the spectra separation hypothesis
\begin{equation*}
\dist(\sigma(A), \sigma(C)) > 0,
\end{equation*}
a natural sufficient condition for the existence of solutions
to the Riccati equation \eqref{Ric0} requires a smallness
assumption on the operator $B$ of the form
\begin{equation}\label{cdest}
\|B\|<c\,\dist\bigl(\spec(A),\spec(C)\bigr)
\end{equation}
with a  constant $c>0$ independent of the distance between the
spectra $\sigma(A)$ and $\sigma(C)$ of the operators $A$ and
$C$, respectively. The best possible constant
$c_{\mathrm{best}}$ in \eqref{cdest} is still unknown.
However, $c_{\mathrm{best}}$ is known to be within the
interval $\left[{\pi}^{-1},\sqrt{2}\right]$ (see \cite{AMM})
and for bounded $A$ and $C$ to satisfy
$c_{\mathrm{best}}\in\left[\frac{3\pi-\sqrt{\pi^2+32}}{\pi^2-4},\sqrt{2}\right]$
(see \cite{KMMgamma}). In \cite{KMMgamma} the best possible
constant $c_{\mathrm{best}}$ has been conjectured to be
$\sqrt{3}/2$. Some earlier results in this direction can be
found in \cite{AdLT}, \cite{MeMo99}, \cite{Motovilov:SPb:91},
\cite{Motovilov:95}.

The best possible constant in inequality \eqref{cdest} may be different under additional
assumptions upon mutual disposition of the spectra of $A$ and $C$. For instance, if the spectra of
$A$ and $C$ are subordinated, e.g.,
\begin{equation}
\label{subord} \sup\spec(A)<\inf\spec(C),
\end{equation}
the Riccati equation \eqref{Ric0} is known to have a strictly
contractive solution for any bounded $B$ (see, e.g.,
\cite{AL95}). To some extent abusing the terminology one may
say that in this case the best possible constant in inequality
\eqref{cdest} is infinite: No smallness assumptions on $B$ are
needed.

In the limiting case of \eqref{subord},
\begin{equation*}
\sup\spec(A)=\inf\spec(C),
\end{equation*}
the existence of contractive solutions has been established in
\cite{AdLT} under some additional assumptions which has been
dropped in \cite{alpha}. See also \cite{MenShk} where the
spectra separation condition has been somewhat relaxed and the
existence of a bounded but not necessarily contractive
solution has been established.

Our \emph{first principal result} concerns  the case where the
operator $C$ has a finite spectral gap  containing the
spectrum of $A$. Recall that  by a finite spectral gap of a
self-adjoint operator $T$ one  understands an \emph{open} finite
interval on the real axis lying in the resolvent set of $T$
such that both of its end points belong to the spectrum of
$T$.

\begin{introtheorem}\label{thm:1}
Assume that the self-adjoint operator $C$ has a finite
spectral gap $\Delta$ containing the spectrum of bounded
self-adjoint operator $A$. Assume in addition that
\begin{equation}
\label{korint} \|B\|<\sqrt{d|\Delta|} \quad \text{where}\quad d=\dist(\spec(A),\spec(C)),
\end{equation}
with $|\cdot|$ denoting Lebesgue measure on $\bbR$. Then the spectrum of the operator $\bH$ in the
gap $\Delta$ is a proper closed subset of $\Delta$. The spectral subspace of the operator $\bH$
associated with the interval $\Delta$ is the graph of a bounded solution $X:\
\cH_A\rightarrow\cH_C$ to the Riccati equation \eqref{Ric0}. The operator $X$ is the unique
solution to the Riccati equation in the class of bounded operators possessing the properties
\begin{equation}
\label{Uniq}
\begin{array}{l}
\spec(A+B X)\subset\Delta,\\
\spec(C-B^* X^*)\subset\R\setminus\Delta, \quad \dom(C-B^*
X^*)=\dom(C).
\end{array}
\end{equation}
\end{introtheorem}

Moreover,  under the assumption of the theorem that the operator $C$ has
a finite spectral gap $\Delta$ containing the spectrum of $A$, we prove that
$c=\sqrt{2}$ is best possible in
\eqref{cdest} ensuring the existence of a bounded solution to
the Riccati equation \eqref{Ric0} (see Remark \ref{coptim}).

Our \emph{second principal result} holds with no assumptions
upon the mutual disposition of the spectra of $A$ and $C$ (in
particular the spectra of $A$ and $C$ may overlap).

\begin{introtheorem}\label{thm:2}
Assume that the self-adjoint operator $C$ has a spectral gap $\Delta$ (finite or
infinite) and the self-adjoint operator $A$ is bounded. Assume that the Riccati
equation \eqref{Ric0} has a bounded solution $X$ and hence the graph subspace
$\cG(X)$ reduces the block operator matrix $\bH$. Suppose that the spectrum of the
part $\bH|_{\cG(X)}$ of the operator $\bH$ associated with the reducing subspace
$\cG(X)$ is a closed subset of $\Delta$. Then the operator $X$ satisfies the norm
estimate
\begin{equation}
\label{NXest} \|X\|\leq \frac{\|B\|}{\delta} \quad \text{with} \,\,\,
\delta=\dist\bigl(\spec(\bH|_{\cG(X)}),\spec(C)\bigr).
\end{equation}
Equivalently,
\begin{equation}\label{TanTheta}
\|\tan\Theta\|\leq\frac{\|B\|}{\delta},
\end{equation}
where $\Theta$ is the operator angle between the subspaces $\cH_A$ and $\cG(X)$.
\end{introtheorem}

Estimate \eqref{TanTheta} generalizes the Davis-Kahan
$\tan\Theta$ theorem \cite{Davis:Kahan} previously known only
in the case where the operator $C$ is semibounded and the
spectrum of the part $\bH|_{\cG(X)}$ lies in the
\emph{infinite} spectral gap of $C$, i.e.,  the spectra of $C$
and $\bH|_{\cG(X)}$ are subordinated. This generalization extends the list of the
celebrated $\sin\Theta$ and $\sin 2\Theta$ theorems, proven in the case where the
operator $C$ has a gap of finite length \cite{Davis:Kahan}.

Our main techniques are based on applications of the Virozub-Macaev
factorization theorems for analytic operator-valued functions \cite{ViMt} in the
spirit of the work \cite{MenShk} and the Daletsky-Krein factorization formula
\cite{DK}. Under the hypothesis of Theorem \ref{thm:1} we prove that
\begin{itemize}

\item for $\lambda\notin\spec(C)$  the operator-valued Herglotz function
$M(\lambda)=\lambda I-A+B(C-\lambda I)^{-1}B^*$ admits a factorization
\begin{equation}
\label{ZAint} M(\lambda)=W(\lambda)(Z-\lambda I)
\end{equation}
with $W$ being an operator-valued function holomorphic on the resolvent set of the
operator $C$ and $Z$ a bounded operator with the spectrum in the spectral gap
$\Delta$ of the operator $C$,

\item the Riccati equation \eqref{Ric0} has
a bounded solution given by
\begin{equation*}
X=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\zeta(C-\zeta I)^{-1}B^* (Z-\zeta I)^{-1},
\end{equation*}
where $\Gamma$ is an appropriate Jordan contour encircling the spectrum of the
operator~$Z$,

\item the spectral subspace of the $2\times 2$ operator
matrix $\bH$ \eqref{cHintro} associated with the interval $\Delta$ is a graph of the
operator $X$,

\item the spectrum of the operator $\bH$ in the interval
$\Delta$ coincides with that of the operator $Z$, i.e.,
$\spec(H)\cap\Delta=\spec(Z)$.
\end{itemize}

In Section~\ref{sec:fac} we prove factorization formula
\eqref{ZAint} and give bounds on the location of the spectrum
of the operator $Z$ (see Theorem \ref{Zexist}). In Section
\ref{sectan} we prove Theorem \ref{thm:1}. Finally, in Section
\ref{SecEst} we prove Theorem \ref{thm:2} and, combining the
results of Theorems \ref{thm:1} and \ref{thm:2}, under
condition \eqref{korint}  we estimate the operator angle
between the spectral subspaces of the operators
$\bH_0=\begin{pmatrix} A &\,\,& 0 \\ 0 & & C
\end{pmatrix}$ and $\bH=\begin{pmatrix} A &\,\,& B \\ B^*& & C
\end{pmatrix}$ corresponding to the interval $\Delta$ (see Corollary
\ref{TanFin}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Factorization theorems for operator-valued Herglotz functions}
\label{sec:fac}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}

The principal purpose of this section is to obtain
factorization results for a class of operator-valued Herglotz
functions associated with the Riccati equation \eqref{Ric0}.

Given a Hilbert space $\cK$ by $I_{\cK}$ we denote the identity operator on $\cK$. If
it does not lead to any confusion we will simply write $I$ instead of
 more pedantic notation $I_\cK$. The set of all linear bounded
operators on $\cK$ will be denoted by $\cB(\cK)$. We also adopt the following
notation. Let $K$ and $L$ be self-adjoint operators on a Hilbert space $\cK$. We say
$K< L$ (or, equivalently, $L> K$) if there is a number $\gamma>0$ such that
$L-K>\gamma I$. If $H$ is a closed operator on a Hilbert space $\cK$, by $\rho(H)$ we
denote the resolvent set of $H$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For notational setup we assume the following
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{hypo}
\label{ABCdiag}
Assume that $A$ is a bounded self-adjoint operator on a
Hilbert space $\cH_A$ and $C$ possibly unbounded self-adjoint
operator with domain $\Dom(C)$ on a Hilbert space $\cH_C$.
Suppose that $C$ has a spectral gap $\Delta =(\alpha,\beta)$,
$\alpha<\beta$ and the spectrum of $A$ lies in $\Delta$, i.e.,
$\spec(A)\subset \Delta$, and let
\begin{equation*}
d=\dist\bigl(\spec(A),\spec(C)\bigr)>0.
\end{equation*}
Assume in addition that $B$ is a bounded operator from $\cH_C$ and $\cH_A$.
\end{hypo}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Under Hypothesis \ref{ABCdiag} introduce the operator-valued Herglotz function
\begin{equation}
\label{M1def} M(\lambda)=\lambda I -A+B(C-\lambda I)^{-1}B^*, \quad \lambda\in\rho(C).
\end{equation}

By definition the spectrum $\spec(M)$ of the function $M$
coincides with the set of all $\lambda\in\bbC$ such that either
the operator $M(\lambda)$ is not invertible or the inverse
$[M(\lambda)]^{-1}$ is an unbounded operator.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}\label{3islroot}
Under Hypothesis \ref{ABCdiag}  the function $M(\lambda)$ given by
\eqref{M1def} is holomorphic in $\rho(C)$. In particular, the holomorphy domain of
$M$ contains the interval $\Delta$ and
\begin{equation}
\label{Der} \displaystyle\frac{d}{d\lambda}M(\lambda) >
0\quad\text{for}\quad \lambda\in\Delta.
\end{equation}
Assume in addition that
\begin{equation}
\label{Bq2d}
%
\|B\|<\sqrt{d|\Delta|}.
%
\end{equation}
Then
\begin{equation}
\label{Ier}
M(\lambda)< 0 \quad \mbox{for}\quad
\lambda\in\bigl(\alpha ,\inf\spec(A)-\delta^-_B\bigr)
\end{equation}
and
\begin{equation}
\label{Iel} M(\lambda)> 0 \quad \mbox{for}\quad
\lambda\in\bigl(\sup\spec(A)+\delta^+_B,\beta\bigr),
\end{equation}
where
\begin{align}
\label{dBm}
\delta^-_B&=\|B\|\tan\left(\frac{1}{2}\arctan\frac{2\|B\|}
{\beta-\inf\spec(A)}\right)<\inf\spec(A)-\alpha ,\\
\label{dBp}
\delta^+_B&=\|B\|\tan\left(\frac{1}{2}\arctan\frac{2\|B\|}
{\sup\spec(A)-\alpha }\right)<\beta-\sup\spec(A).
\end{align}
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
By the spectral theorem
\begin{equation*}
B(C-\lambda I)^{-1}B^*=\int\limits_{\R\setminus \Delta}
%
B\EE_C(d\mu)B^*\frac{1}{(\mu-\lambda)^2},\quad\lambda\in\rho(C),
\end{equation*}
where $\EE_C(\mu)$ stands for the spectral family of the
self-adjoint operator $C$. Hence
\begin{equation}
\label{DerE} \displaystyle\frac{d}{d\lambda}M(\lambda)=I +
\int\limits_{\R\setminus \Delta}
%
B\EE_C(d\mu)B^*\frac{1}{(\mu-\lambda)^2}, \qquad\lambda\in\rho(C).
\end{equation}
For $\lambda\in\Delta$ the integral in \eqref{DerE} is a non-negative operator.
Therefore,
\begin{equation*}
 \displaystyle\frac{d}{d\lambda}M(\lambda)\geq I,
 \quad \lambda\in \Delta,
\end{equation*}
proving \eqref{Der}.

Next we estimate the quadratic form of $M(\lambda)$. Let
$f\in\cH_A$,  $\|f\|=1$. Then
\begin{align}\label{Fflam}
\lal M(\lambda)f,f\ral&=
   \lambda-\lal A f,f\ral+\lal(C-\lambda)^{-1}B^*f,B^*f\ral
\nonumber\\
&=\lambda-\lal A f,f\ral
 +\int\limits_{-\infty}^{\alpha }
\frac{1}{\mu-\lambda}\lal \EE_C(d\mu)B^*f,B^*f\ral \\
&\quad+\int\limits_{\beta}^{+\infty} \frac{1}{\mu-\lambda}\lal
\EE_C(d\mu)B^*f,B^*f\ral, \quad\lambda\in\rho(C).
\nonumber
\end{align}
Since for $\lambda\in \Delta$ the integral in the second line of \eqref{Fflam} is
non-positive and that in the third line  is non-negative, one obtains
the two-sided estimate
\begin{equation*}
\bigg ( \lambda-\sup
\spec(A)-\frac{\|B\|^2}{\lambda-\alpha }\bigg )I \le
 M(\lambda)
\le \bigg ( \lambda-\inf \spec(A)-\frac{\|B\|^2}{\lambda-\beta} \bigg )I
\end{equation*}
for any $\lambda\in \Delta$. Now, a simple calculation  shows that
\eqref{Ier} and \eqref{Iel} hold, completing the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For convenience of the reader we reproduce here the Virozub-Matsaev factorization
theorem  \cite{ViMt} (also see \cite{MrMt}). We present the statement
following Propositions 1.1 and 1.2 of \cite{MenShk}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
\label{Matsa}
Let $\cK$ be a Hilbert space and  $F(\lambda)$ a
holomorphic $\cB(\cK)$-valued function on a simply connected
domain $\Omega\subset\bbC$. Assume that $\Omega$ includes an
interval $[a,b]\subset\bbR$ such that
\begin{equation*}
F(a)< 0, \quad F(b)> 0,\quad\text{and}\quad\frac{d}{d\lambda}F(\lambda)> 0\quad
\text{for all}\quad\lambda\in[a,b].
\end{equation*}
Then there are a domain $\widetilde{\Omega}\subset\Omega$
containing $[a,b]$ and a unique bounded operator $Z$ on $\cK$
with $\spec(Z)\subset(a,b)$ such that $F(\lambda)$ admits the
factorization
\begin{equation*}
F(\lambda)=G(\lambda)(Z-\lambda I), \quad \lambda\in\widetilde{\Omega},
\end{equation*}
where $G(\lambda)$ is a holomorphic operator-valued function
on $\widetilde{\Omega}$ whose values are bounded and boundedly
invertible operators in $\cK$, that is,
\begin{equation*}
G(\lambda)\in\cB(\cK)\quad\text{and}\quad [G(\lambda)]^{-1}\in\cB(\cK)\quad\text{for}
\quad\lambda\in\widetilde{\Omega}.
\end{equation*}
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As a direct corollary of Theorem \ref{Matsa}, under Hypothesis
\ref{ABCdiag} we get the following factorization result for
Herglotz functions of the form \eqref{M1def}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
\label{Zexist} Assume hypothesis of Lemma \ref{3islroot} and
let $\delta^\pm_B$ be given by \eqref{dBm} and \eqref{dBp}.
Then there is a unique operator $Z\in\cB(\cH_A)$ with
\begin{equation}\label{inclusion}
\spec(Z)\subset[\inf\spec(A)-\delta^-_B,\sup\spec(A)+\delta^+_B]
\end{equation}
 such that the function \eqref{M1def} admits the factorization
\begin{equation}
\label{M1fact}
M(\lambda)=W(\lambda)(Z-\lambda I),\quad\lambda\in\rho(C).
\end{equation}
Here $W$ is a holomorphic $\cB(\cH_A)$-valued function on
$\rho(C)$ such that for any
\begin{equation*}
\lambda\in\bigl(\bbC\setminus \spec(M)\bigr)\cup\Delta
\end{equation*}
the operator $W(\lambda)$ has a bounded inverse.
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
By Lemma \ref{3islroot}, the function $F(\lambda)=M(\lambda)$
satisfies assumptions of Theorem \ref{Matsa} for any
$a\in(\alpha ,\inf\spec(A)-\delta^-_B)$ and any
$b\in(\sup\spec(A)+\delta^+_B,\beta)$, proving the existence
of the unique bounded operator $Z\in\cB(\cH_A)$ satisfying
\eqref{M1fact} and \eqref{inclusion}. Theorem \ref{Matsa} also
states that the factor $W(\lambda)$ in \eqref{M1fact} has a
bounded inverse in a complex neighborhood $U\subset\bbC$ of
the interval
$[\inf\spec(A)-\delta^-_B,\sup\spec(A)+\delta^+_B]$. The last
assertion of the theorem then follows from the representation
\begin{equation*}
[W(\lambda)]^{-1}=(Z-\lambda I)[M(\lambda)]^{-1}, \quad
\lambda\in U\setminus\spec(M)
\end{equation*}
by analytic continuation to the domain $ \bigl(\bbC\setminus
\spec(M)\bigr)\cup\Delta$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will call the operator $Z$ referred to in Theorem
\ref{Zexist} the operator root of the Herglotz function $M$
(cf.~\cite[Remark 4]{ViMt}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existence and uniqueness results. Proof of Theorem 1}
\label{sectan}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}

We start this section by an existence result for the operator
Riccati equation \eqref{Ric0}.

Under Hypothesis \ref{ABCdiag}, a bounded operator $X$ from
$\cH_A$ to $\cH_C$ is said to be an operator solution to
\eqref{Ric0} if $\Ran(X)\subset\Dom(C)$ and \eqref{Ric0} holds
as an operator equality.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
\label{RicSol} Assume Hypothesis \ref{ABCdiag} and suppose that
\begin{equation*}
\|B\|<\sqrt{d|\Delta|}.
\end{equation*}
Then the operator Riccati equation \eqref{Ric0} has a
bounded operator solution.
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $Z$ be the operator root of the operator-valued function
$M(\lambda)$ \eqref{M1def} referred to in Theorem \ref{Zexist}
and $\Gamma$ an arbitrary bounded Jordan contour encircling
the spectrum of $Z$ in the clockwise direction with winding
number $0$ with respect to the spectrum of the operator $C$.
Put
\begin{equation}
\label{Xdef} X=-\frac{1}{2\pi{\mathrm i}} \int\limits_\Gamma d\zeta(C-\zeta I)^{-1}B^* (Z-\zeta
I)^{-1},
\end{equation}
Clearly, for $\zeta\in\Gamma$ we have $\Ran(C-\zeta
I)^{-1}\subset\dom(C)$ and hence
\begin{equation*}
\Ran(X)\subset\Dom(C),
\end{equation*}
which immediately follows from  \eqref{Xdef}. Multiplying both sides of
\eqref{Xdef} by $B$ from the left yields
\begin{equation} \label{BX1}
BX=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\zeta
%
B(C-\zeta I)^{-1}B^*(Z-\zeta I)^{-1}.
\end{equation}
Meanwhile,
\begin{equation*}
B(C-\zeta I)^{-1}B^*=A-\zeta I+M(\zeta)= A-\zeta I +W(\zeta)(Z-\zeta I), \quad
\zeta\in\Gamma,
\end{equation*}
and, hence, using \eqref{BX1}
\begin{align}
\label{BXint} BX&=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\zeta
%
\left[W(\zeta)+A(Z-\zeta I)^{-1}-\zeta(Z-\zeta I)^{-1}\right].
\end{align}
The function $W(\zeta)$ is holomorphic in the domain bounded
by the contour $\Gamma$ and, thus, the first term in the
integrand on the r.h.s.\ of \eqref{BXint} gives no
contribution. Since $\Gamma$ encircles the
spectrum of $Z$, the integration of the remaining two terms in
\eqref{BXint} can be performed explicitly using the operator
version of the residue theorem, which yields $BX=-A+Z$ and
hence
\begin{equation}
\label{Xgr} Z=A+BX.
\end{equation}

Since the spectra of the operators $C$ and $Z$ are disjoint
and $Z$ is a bounded operator, it is straightforward to show
(see, e.g., \cite{D53} or \cite{R56})  that the
operator $X$ given by \eqref{Xdef} is the unique operator
solution to the Sylvester equation
\begin{equation*}
XZ-CX=B^*,
\end{equation*}
which by \eqref{Xgr} proves that $X$ solves the Riccati  equation
\eqref{Ric0}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The existence of solutions to the Riccati equation
\eqref{Ric0} is directly related to the possibility of block
diagonalization of the $2\times2$ self-adjoint  block operator
matrix
\begin{equation}
\label{twochannel} {\bH}=\begin{pmatrix}
  A    &\,\,&   B \\
  B^*    & &   C
\end{pmatrix},\quad\Dom(\bH)=\cH_A\oplus\Dom(C),
\end{equation}
in the Hilbert space
\begin{equation}
\label{decom} \cH=\cH_A\oplus\cH_C.
\end{equation}
The precise statement describing this  connection is as follows (see
Lemma 5.3 and Theorem 5.5 in \cite{AMM}; cf.~\cite{AdLT}, \cite{Daughtry},
 \cite{KMMa}, \cite{Motovilov:95}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
\label{thHi2}
Assume that $A$ is a bounded self-adjoint
operator on a Hilbert space $\cH_A$, $C$ possibly unbounded
self-adjoint operator with domain $\Dom(C)$ on a Hilbert space
$\cH_C$, and $B$ a bounded operator from $\cH_C$ to $\cH_A$.
Then a bounded operator $X$ from $\cH_A$ to $\cH_C$ is a
solution to the Riccati equation \eqref{Ric0} iff the graph
$\cG(X)$ of $X$ is a reducing subspace for the $2\times2$
block operator matrix \eqref{twochannel}. Moreover, if
$X\in\cB(\cH_A,\cH_C)$ is a solution to \eqref{Ric0} then:

\medskip

\noindent{\rm(i)} The operator $\bV^{-1}\bH\bV$  with
\begin{equation*}
\bV=\begin{pmatrix}
I &\,& -X^* \\
 X  & & I
\end{pmatrix}.
\end{equation*}
is block diagonal with respect to decomposition \eqref{decom}.
Furthermore,
\begin{equation*}
\bV^{-1}\bH\bV=\begin{pmatrix}
%
Z     &\,\,&   0      \\
0     & &  \widehat{Z}
%
\end{pmatrix},
\end{equation*}
where $Z=A+BX$ and $ \widehat{Z}=C-B^*X^*$ with $\Dom(\widehat{Z})=\dom(C)$.
\medskip

\noindent{\rm(ii)} The operators
\begin{equation}\label{HA}
%
\Lambda =(I+X^*X)^{1/2}Z (I+X^*X)^{-1/2}
\end{equation}
and
\begin{equation*}
\widehat{\Lambda} = (I+XX^*)^{1/2}\widehat{Z}
(I+XX^*)^{-1/2}
\end{equation*}
with $\dom(\widehat \Lambda)=(I+XX^*)^{1/2}(\dom(C))$ are self-adjoint operators in
$\cH_A$ and $\cH_C$, respectively.
\end{thm}

Theorem \ref{thHi2} yields the following uniqueness result as
a corollary.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor}\label{Xuniq}
Assume Hypothesis of Theorem \ref{thHi2}. Suppose that
$\Sigma$ and $\widehat{\Sigma}$ are disjoint Borel subsets of
$\bbR$ such that $\dist(\Sigma,\widehat{\Sigma})>0$. Let
$\cX=\cX(A,B,\Sigma,\widehat{\Sigma})$ be the set of all
bounded operators $X$ from $\cH_A$ to $\cH_C$ with the
properties
\begin{align}
\label{sZA}
&\spec(A+BX)\subset\Sigma ,\\
\label{sZC}
&\spec(C-B^*X^*)\subset\widehat{\Sigma}, \quad
\dom(C-B^*X^*)=\dom(C).
\end{align}
Then if $X,Y\in\cX$ satisfy the Riccati equation \eqref{Ric0}, then $X=Y$.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Suppose $X$ and $Y$ are two bounded solutions to \eqref{Ric0} both satisfying
\eqref{sZA} and \eqref{sZC}. Then by Theorem \ref{thHi2} the graphs of $X$ and $Y$
both coincide with the spectral subspace of the $2\times2$ operator matrix
\eqref{twochannel} associated with the set $\Sigma$, and
hence, $X=Y$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we are ready to prove the key result of the section.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
\label{strudom}
Assume Hypothesis \ref{ABCdiag} and suppose  that
\begin{equation*}
\|B\|<\sqrt{d|\Delta|}.
\end{equation*}
Then  the spectral subspace of the operator \eqref{twochannel}
associated with the interval $\Delta$ is the graph of
a bounded operator $X$ from $\cH_A$ to $\cH_C$.
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
It is well known (see, e.g., \cite{MeMo99}) that the resolvent
of the operator $\bH$ can be represented as the following
$2\times2$ operator matrix
\begin{align}
\nonumber
(\bH-\lambda I)^{-1}=&\begin{pmatrix} 0 & &  0 \\ 0 &\,\,& (C-\lambda I)^{-1}
\end{pmatrix}\\
\label{ResH}
&-\left(\begin{array}{c} I \\
-(C-\lambda I)^{-1}B^*
\end{array}\right)
M^{-1}(\lambda)
\begin{pmatrix}
I &\,\quad  -B(C-\lambda I)^{-1}
\end{pmatrix}, \\
\nonumber
&\quad \lambda\in\rho(\bH),
\end{align}
where $M$ is the Herglotz function given by \eqref{M1def}.  By Theorem
\ref{Zexist} the Herglotz function $M(\lambda)$ admits the factorization
\begin{equation}
\label{facWW}
M(\zeta)=W(\zeta)(Z-\zeta I),\quad
\zeta\in\rho(C),
\end{equation}
where $Z$ is a bounded operator with
\begin{equation}
\label{zvezd}
\spec(Z)\subset \Delta.
\end{equation}
Representation \eqref{ResH} shows that for $\lambda\in\Delta$ the operator
$\bH-\lambda I$ has a bounded inverse iff $M(\lambda)$ does. By Theorem \ref{Zexist}
$W(\zeta)$  is holomorphic on $\rho(C)$ and for any
\begin{equation*}
\lambda\in\bigl(\bbC\setminus \spec(M)\bigr)\cup\Delta
\end{equation*}
the operator $W(\lambda)$ has a bounded inverse. Therefore,  using
\eqref{facWW} and \eqref{zvezd} proves that the spectrum of
$\bH$ in the interval $\Delta$ coincides with that of $Z$,
\begin{equation*}
\spec(\bH)\cap\Delta=\spec(Z),
\end{equation*}
and, hence,  it is isolated from the remaining part of the
spectrum of $\bH$.

Using representation \eqref{ResH}, for the spectral projection
$\sE_\bH\bigl(\Delta\bigr)$ of the operator $\bH$ associated
with the interval $\Delta$ the Riesz integration yields
\begin{align*}
\sE_\bH\bigl(\Delta \bigr)=&
\sE_\bH\bigl(\spec(Z)\bigr)=\frac{1}{2\pi\ri}\int_\Gamma
d\zeta(\bH-\zeta I)^{-1}\\
 =&-\frac{1}{2\pi\ri}\int_\Gamma
d\zeta\begin{pmatrix} I \\
-(C-\zeta I)^{-1}B^*
\end{pmatrix}
M^{-1}(\zeta)
\begin{pmatrix}
I & \,\quad -B(C-\zeta I)^{-1}
\end{pmatrix},
\end{align*}
where $\Gamma$ is an arbitrary bounded Jordan contour in $\bbC$ encircling $\spec(Z)$ in the
clockwise direction and having winding number $0$ with respect to the spectrum of $C$. Hence,
\begin{equation}
\label{EH}
\sE_\bH\bigl(\spec(Z)\bigr)=
\begin{pmatrix}
E &\,& G^*\\
G && F\\
\end{pmatrix},
\end{equation}
where
\begin{align}
\label{Eint}
E&=-\frac{1}{2\pi\ri}\int_\Gamma d\zeta M^{-1}(\zeta), \\
\nonumber
F&=-\frac{1}{2\pi\ri}\int_\Gamma d\zeta(C-\zeta I)^{-1}B^*
M^{-1}(\zeta)B(C-\zeta I)^{-1},
\end{align}
and
\begin{align*}
G&=\frac{1}{2\pi\ri}\int_\Gamma d\zeta(C-\zeta I)^{-1}B^* M^{-1}(\zeta)\\
&=\frac{1}{2\pi\ri}\int_\Gamma d\zeta (C-\zeta I)^{-1}B^* (Z-\zeta I)^{-1}
W^{-1}(\zeta),
\end{align*}
using factorization formula \eqref{facWW}. By Theorem
\ref{Zexist} the function $W^{-1}(\zeta)$ is holomorphic in
the domain bounded by $\Gamma$. Then applying the
Daletsky-Krein lemma (\cite[Lemma I.2.1]{DK}) yields
$$
G=\left[\frac{1}{2\pi\ri} \int_\Gamma d\zeta (C-\zeta I)^{-1}B^*(Z-\zeta
I)^{-1}\right]
\left[\frac{1}{2\pi\ri}\int_\Gamma d\zeta
(Z-\zeta I)^{-1}W^{-1}(\zeta)\right].
$$
Hence, combining \eqref{facWW} and \eqref{Eint} proves the
representation
\begin{equation}
\label{XE} G=XE,
\end{equation}
where
\begin{equation}
\label{Xdeff}
X=-\frac{1}{2\pi{\mathrm i}}
\int_\Gamma d\zeta(C-\zeta I)^{-1}B^* (Z-\zeta I)^{-1}.
\end{equation}
In an analogous way one also proves that
\begin{equation}
\label{XEX} F=XEX^*.
\end{equation}

A closer look at the proof of Theorem \ref{RicSol} shows that
the operator $X$ given by \eqref{Xdeff} solves the Riccati
equation \eqref{Ric0}. Then applying Theorem \ref{thHi2} and
using \eqref{EH}, \eqref{XE}, and \eqref{XEX} we obtain by
inspection
\begin{align}
\nonumber I&= \frac{1}{2\pi\ri}\int_\Gamma
d\zeta(Z-\zeta I)^{-1}\\
\nonumber &=\frac{1}{2\pi\ri}\int_\Gamma d\zeta{\sP}_{A}
\begin{pmatrix}
(Z-\zeta I)^{-1} & 0 \\ 0 & (Z_C-\zeta I)^{-1}
\end{pmatrix}
{\sP}_{A}^*\\
\nonumber
&=\frac{1}{2\pi\ri}\int_\Gamma d\zeta {\sP}_{A} \bV^{-1}(\bH-\zeta I)^{-1}\bV
{\sP}_{A}^*\\
\nonumber &={\sP}_{A} \bV^{-1} \sE_\bH\bigl(\spec(Z)\bigr)
\bV{\sP}_{A}^*\\
\label{IXX} &=E(I+X^*X),
\end{align}
where $\widehat{Z}=C-B^*X^*$ with $\Dom(\widehat{Z})=\Dom(C)$,
$$
\bV=\begin{pmatrix}
I &\,& -X^* \\
 X  & & I
\end{pmatrix},
$$
and
${\sP}_A$ is the canonical projection from $\cH$ to
$\cH_A$, i.e., $\sP_A(f_A\oplus f_C)=f_A$ for any
$f_A\in\cH_A$ and $f_C\in\cH_C$. It follows from \eqref{IXX}
that
\begin{equation}
\label{PsiXX} E=(I+X^*X)^{-1}
\end{equation}
since $I+X^*X$ has a bounded inverse. Combining,
\eqref{XE} \eqref{XEX}, and  \eqref{PsiXX} proves that the spectral
projection $\sE_{\bH}(\Delta)$ admits the following
representation
\begin{equation}
\label{sEQ} \sE_{\bH}(\Delta)=
\begin{pmatrix}
(I+X^*X)^{-1}  && (I+X^*X)^{-1}X^* \\
X(I+X^*X)^{-1} && X(I+X^*X)^{-1}X^*
\end{pmatrix}.
\end{equation}
Observing that the r.h.s.\ of \eqref{sEQ} represents the
orthogonal projection in $\cH=\cH_A\oplus\cH_C$ onto the graph
of the operator $X$ completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The proof of the first principal result of the paper is now
straightforward.

\begin{proof}[Proof of Theorem \ref{thm:1}]
By Theorem \ref{RicSol} the Riccati equation \eqref{Ric0} has
a bounded solution $X$ given by \eqref{Xdef} (see the proof of this theorem).
By Theorem \ref{strudom} the graph of the operator $X$ is the
spectral subspace $\Ran \sE_\bH\bigl(\Delta \bigr)$
of the block operator matrix $\bH$ given by \eqref{twochannel}.
Now, applying Theorem \ref{thHi2} shows that $X$ possesses
the properties \eqref{Uniq}. Corollary \ref{Xuniq} then proves
that $X$ is the unique bounded solution to \eqref{Ric0} satisfying
\eqref{Uniq} completing the proof.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{rem}
\label{coptim}
Under hypothesis of Theorem \ref{thm:1} one obviously
concludes that $|\Delta|\geq 2d$ with the equality sign
occurring only if the spectrum of the operator $A$ is a one
point set. Hence, \eqref{korint} implies the solvability of
the Riccati equation \eqref{Ric0} under the condition
$\|B\|<\sqrt{2}d$. Therefore, in the case where the operator
$C$ has a finite spectral gap $\Delta$ containing the spectrum
of $A$ the constant $c_{\mathrm{best}}\geq \sqrt{2}$. On the
other hand in \cite[Lemma 3.11 and Remark 3.12]{AMM} it is
shown that $c_{\mathrm{best}}\leq \sqrt{2}$. Thus,
$c=\sqrt{2}$ is best possible in inequality \eqref{cdest}.
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Norm Estimates of Solutions.
The $\text{tan}{\,\Theta}$ Theorem} \label{SecEst}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}


We start by recalling the concept of the operator angle between two subspaces in a
Hilbert space going back to the works by Friedrichs \cite{Friedrichs}, M.~Krein,
Kransnoselsky, and Milman \cite{Krein:Krasnoselsky},
\cite{Krein:Krasnoselsky:Milman}, Halmos \cite{Halmos:69}, and Davis and Kahan
\cite{Davis:Kahan}. A comprehensive discussion of this notion can be found in
\cite{KMMa}.

Given a subspace $\cQ$ of the Hilbert space
$\cH=\cH_A\oplus\cH_C$, introduce the operator angle $\Theta$
between the subspaces $\cH_A\oplus\{0\}$ and $\cQ$ by
\begin{equation}
\label{sinT}
\Theta=\arcsin(I_{\cH_A}-{\sP}_A\sQ{\sP}_A^*),
\end{equation}
where ${\sP}_A$ is the canonical projection from $\cH$ onto
$\cH_A$ and $\sQ$ the orthogonal projection in $\cH$ onto
$\cQ$. If the subspace $\cQ$ is the graph $\cG(X)$ of a
bounded operator $X$ from $\cH_A$ to $\cH_C$, then (see
\cite{KMMa}; cf.\ \cite{Davis:Kahan} and \cite{Halmos:69})
\begin{equation}
\label{tan-X}
\tan\Theta=\sqrt{X^*X}
\end{equation}
and
\begin{equation}
\label{sinTPQ}
\|\sin\Theta\|=\|\sQ-\sP\|,
\end{equation}
where $\sP={\sP_A}^{\!\!\!*}\sP_A$ denotes the orthogonal
projection in $\cH$ onto the subspace $\cH_A\oplus\{0\}$.

Note that the common definition of the operator
 angle (see, e.g., \cite{KMMa}) slightly differs from \eqref{sinT}. Usually,
the operator angle is defined as the restriction of \eqref{sinT} onto the maximal
subspace of $\cH_A$ where it has a trivial kernel. Obviously, for both definitions
$\|\tan\Theta\|$ is the same.

Now we are ready to prove the second principal result of the paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of Theorem \ref{thm:2}]
By Theorem \ref{thHi2} the operator $Z=A+BX$ is similar to the
bounded self-adjoint operator $\Lambda$ given by \eqref{HA}
and hence the spectrum of $Z$ is a subset of the real axis,
\begin{equation}
\label{spZA} \spec(Z)=\spec(\Lambda)\subset\bbR\,.
\end{equation}
Rewrite the Riccati equation \eqref{Ric0}  in the form
\begin{equation}
\label{SylHelp} X(Z-aI)-(C-aI)X=B^*,
\end{equation}
where
\begin{equation}
\label{lamcen1} a=\frac{1}{2}\bigl(\sup\spec(Z)+\inf\spec(Z)\bigr)
=\frac{1}{2}\bigl(\sup\spec(\Lambda)+\inf\spec(\Lambda)\bigr).
\end{equation}
By hypothesis $\dist\bigl(\spec(\bH|_{\cG(X)}),\spec(C)\bigr)=\delta>0$.
Theorem \ref{thHi2} implies that
$$
\spec(\bH|_{\cG(X)})=\spec(Z)=\spec(\Lambda).
$$
Therefore,
\begin{equation}
\label{spC}
\spec(C)\subset\bigl(-\infty,\inf\spec(\Lambda)-\delta\bigr]\cup
\bigl[\sup\spec(\Lambda)+\delta,\infty\bigr).
\end{equation}
Hence  $C-aI$ has a bounded inverse, and combining
\eqref{spZA}, \eqref{lamcen1}, and \eqref{spC} yields
\begin{equation}
\label{CmA} \|(C-a I)^{-1}\|=\frac{1}{\|\Lambda-aI\|+\delta}.
\end{equation}
Multiplying both sides of \eqref{SylHelp} by $(C-a I)^{-1}$ from the left proves the
representation
\begin{equation}
\label{SylZC} X=(C-a I)^{-1}\bigl(X(Z-aI)-B^*\bigr).
\end{equation}
Using the claim (ii) of Theorem \ref{thHi2} one obtains
\begin{align}
\nonumber \|X(Z-aI)\|=&\|X(I+X^*X)^{-1/2}(\Lambda-aI)
(I+X^*X)^{1/2}\| \\
\label{NX1} &\leq \|X(I+X^*X)^{-1/2}\|\, \|\Lambda-aI\| (1+\|X\|^2)^{1/2}.
\end{align}
Clearly,
\begin{equation*}
\|X(I+X^*X)^{-1/2}\|=\sqrt{\|X^*X(I+X^*X)^{-1}\|} =\frac{\|X\|}{(1+\|X\|^2)^{1/2}}
\end{equation*}
applying the spectral theorem on the last step. Hence
\eqref{NX1} implies the estimate
\begin{equation*}
\|X(Z-aI)\|\leq \|X\|\, \|(\Lambda-aI)\|,
\end{equation*}
which together with \eqref{SylZC} proves the norm inequality
\begin{equation}
\label{XN2} \|X\|\leq \|(C-a I)^{-1}\|\bigl(\|\Lambda-aI\|\|X\|+\|B\|\bigr).
\end{equation}
Solving \eqref{XN2} with respect to $\|X\|$ and taking into
account \eqref{CmA} proves \eqref{NXest}. Finally, since
$\bigl\|\sqrt{X^*X}\bigr\|=\|X\|$, using \eqref{tan-X}  one
concludes that
\begin{equation*}
\|\tan\Theta\|=\|X\|.
\end{equation*}
Hence, \eqref{NXest} is equivalent to
\eqref{TanTheta}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The estimates \eqref{NXest} and \eqref{TanTheta}  depend on the spectral
properties of the perturbed operator $\bH$. Under additional assumptions one can also
get an \emph{a priori} estimate on the norm of  the solution $X$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor}
\label{TanFin} Assume Hypothesis \ref{ABCdiag} and suppose that
\begin{equation*}
\|B\|<\sqrt{d|\Delta|}.
\end{equation*}
If $X$ is the unique solution to the Riccati equation \eqref{Ric0}
referred to in Theorem
\ref{thm:2}, then
\begin{equation}
\label{XestFin}
\|X\|\leq\frac{\|B\|}{\min\{d^--\delta^-_B,d^+-\delta^+_B\}},
\end{equation}
where $d^-=\inf\spec(A)-\alpha $, $d^+=\beta-\sup\spec(A)$,
and $d^\pm_B$ are given by \eqref{dBm} and \eqref{dBp}.
Equivalently,
\begin{equation}
\label{tanThetaTh}
\|\tan\Theta\|\leq
\frac{\|B\|}{\min\{d^--\delta^-_B,d^+-\delta^+_B\}},
\end{equation}
where $\Theta$ is the operator angle between the subspace
$\cH_A\oplus\{0\}$ and spectral subspace of the $2\times2$
operator matrix \eqref{twochannel} associated with the
interval $\Delta$. Moreover, the spectral projections
$\sE_{\bH_0}(\Delta)$ and $\sE_{\bH}(\Delta)$ of the operators
$\bH_0=\begin{pmatrix} A &\,\,& 0 \\ 0 & & C
\end{pmatrix}$ and $\bH=\begin{pmatrix} A &\,\,& B \\ B^* & & C
\end{pmatrix}$, respectively, satisfy the estimate
\begin{equation}
\label{PQ}
\|\sE_{\bH}(\Delta)-\sE_{\bH_0}(\Delta)\|\leq
\sin\arctan\left(\frac{\|B\|}
{\min\{d^--\delta^-_B,d^+-\delta^+_B\}}\right).
\end{equation}
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $X$ be the solution to the Riccati equation \eqref{Ric0}
referred to in Theorem \ref{thm:1}. By Theorem \eqref{RicSol}
\begin{equation}
\label{spek}
\spec(A+BX)=\spec(Z),
\end{equation}
where $Z$ is the operator root of $M$ referred to in Theorem
\ref{Zexist}. Theorem \ref{Zexist} also states that
\begin{equation}
\label{2st} \spec(Z)\subset[\inf\spec(A)-\delta^-_B,\sup\spec(A)+\delta^+_B].
\end{equation}
Recall that by Hypothesis \ref{ABCdiag}
\begin{equation}
\label{3st} \spec(C)\subset\bigl(-\infty,\alpha \bigr]\cup
\bigl[\beta,\infty\bigr).
\end{equation}
Combining \eqref{spek}, \eqref{2st}, and \eqref{3st} yields
\begin{equation*}
\dist\bigl(\spec(A+BX),\spec(C)\bigr)\geq\min\{d^-
-\delta^-_B,d^+-\delta^+_B\},
\end{equation*}
which proves \eqref{XestFin} applying Theorem \ref{thm:2}.

By Theorem \ref{strudom} the spectral projection of the
operator \eqref{twochannel} associated with the interval
$\Delta$ is the orthogonal projection onto the graph  of the
operator $X$. Then it follows from Theorem \ref{thm:2} that
\eqref{XestFin} is equivalent to \eqref{tanThetaTh}.

The last estimate \eqref{PQ} is an immediate consequence of
\eqref{tanThetaTh} by using \eqref{tan-X} and \eqref{sinTPQ}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\ack{V.~Kostrykin in grateful to V.~Enss, A.~Knauf,
H.~Leschke, and R.~Schrader for useful discussions.
K.~A.~Makarov is indebted to Graduiertenkolleg ``Hierarchie
und Symmetrie in mathematischen Modellen" for kind hospitality
during his stay at RWTH Aachen in summer 2002.
A.~K.~Motovilov acknowledges the kind hospitality and support
by the Department of Mathematics, University of Missouri,
Columbia, MO, USA. He was also supported in part by the
Russian Foundation for Basic Research within Project RFBR
01-01-00958.}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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% appendix sections are then done as normal sections
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\begin{thebibliography}{00}

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% \bibitem{label} \note

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% \bibitem{label3} \note
% \end{subbibitems}

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\bibitem{Halmos:69} P.~R.~Halmos, Two subspaces, \textit{Trans.
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\bibitem{KMMa} V.~Kostrykin, K.~A.~Makarov, and A.~K.~Motovilov,
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\bibitem{alpha} V.~Kostrykin, K.~A.~Makarov, and A.~K.~Motovilov,
A generalization of the $\tan 2\Theta$ theorem, in preparation.

\bibitem{KMMgamma} V.~Kostrykin, K.~A.~Makarov, and A.~K.~Motovilov,
Perturbation of spectra and spectral subspaces, in preparation.

\bibitem{Krein:Krasnoselsky} M.~G.~Krein and M.~A.~Krasnoselsky,
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\end{thebibliography}

\end{document}

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  \leftmargin 1em \itemindent-\leftmargin
  \let\makelabel\descriptionlabel}}
\let\enddescription\endlist
\def\@atfmtname{atlplain}
\ifx\fmtname\@atfmtname
 \def\neq{\not\nobreak\mkern -2mu =}%
 \let\ne\neq
\fi
\def\operatorname#1{\mathop{\mathrm{#1}}\nolimits}
\def\lefteqn#1{\hbox to\z@{$\displaystyle {#1}$\hss}}
\newskip\eqnbaselineskip % Standard interline spacing in an {eqnarray}
\jot=2\p@
\newskip\eqnlineskip     % Minimal space between the bottom of
                         % a line and the top of the next line.
\eqnbaselineskip=14\p@  \eqnlineskip=2\p@
\newdimen\mathindent
\if@TwoColumn
  \mathindent 0em
\else
  \mathindent 2em
\fi
\def\[{\relax\ifmmode\@badmath
  \else%\bgroup removed on request from BW (1993-05-17)
  \@beginparpenalty\predisplaypenalty
  \@endparpenalty\postdisplaypenalty
  \begin{trivlist}\@topsep \eqntopsep       % used by first \item
   \@topsepadd \eqntopsep                   % used by \@endparenv
  \item[]\leavevmode
   \hbox to\linewidth\bgroup$ \displaystyle
  \hskip\mathindent\bgroup\fi}
\def\]{\relax\ifmmode \egroup $\hfil \egroup
  \end{trivlist}% \egroup removed on request from BW (1993-05-17)
  \else \@badmath \fi}
\def\equation{\@beginparpenalty\predisplaypenalty
  \@endparpenalty\postdisplaypenalty
  \refstepcounter{equation}\trivlist
  \@topsep \eqntopsep                      % used by first \item
  \@topsepadd \eqntopsep                   % used by \@endparenv
  \item[]\leavevmode
   \hbox to\linewidth\bgroup $ \displaystyle \hskip\mathindent\bgroup}
\def\endequation{\egroup$\hfil \displaywidth\linewidth
  \@eqnnum\egroup \endtrivlist}
\def\eqnarray{%
  \par                                               %BW
  \noindent                                          %BW
  \baselineskip\eqnbaselineskip\lineskip\eqnlineskip %BW
  \lineskiplimit\eqnlineskip                         %BW
  \stepcounter{equation}%
  \let\@currentlabel=\theequation
  \global\@eqnswtrue
  \global\@eqcnt\z@
  \tabskip\mathindent
  \let\\=\@eqncr
  \abovedisplayskip\eqntopsep\ifvmode\advance\abovedisplayskip\partopsep\fi
  \belowdisplayskip\abovedisplayskip
  \belowdisplayshortskip\abovedisplayskip
  \abovedisplayshortskip\abovedisplayskip
  $$\halign to \displaywidth\bgroup\@eqnsel
    \pre@coli$\displaystyle\tabskip\z@{##}$\post@coli
    &\global\@eqcnt\@ne
    \pre@colii$\displaystyle{##}$\post@colii
    &\global\@eqcnt\tw@
    \pre@coliii $\displaystyle\tabskip\z@{##}$\post@coliii
    \tabskip\@centering&\llap{##}\tabskip\z@\cr
}
\def\endeqnarray{\@@eqncr\egroup
 \global\advance\c@equation\m@ne$$\global\@ignoretrue }
\def\pre@coli{\hskip\@centering}              \def\post@coli{}
\def\pre@colii{\hskip 2\eqnarraycolsep \hfil} \def\post@colii{\hfil}
\def\pre@coliii{\hskip 2\eqnarraycolsep}      \def\post@coliii{\hfil}
\arraycolsep 2\p@         % Half the space between columns in array environment.
\tabcolsep 6\p@           % idem in tabular environment.
\def\arraystretch{1.5}   % More vertical space in tables
\arrayrulewidth 0.4\p@    % \@Width of rules and space between adjacent
\doublerulesep 2\p@       % rules in any of these two environments.
\newdimen\rulepreskip \newdimen\rulepostskip
\rulepreskip=4\p@      \rulepostskip=6\p@
\tabbingsep \labelsep   % Space used by the \' command.  (See LaTeX{} manual.)
\skip\@mpfootins = 6\p@ \@plus 2\p@   % Space between last line of text and
                                  % top of first footnote.
\fboxsep = 7\p@    % Space left between box and text by \fbox and \framebox.
\fboxrule = 0.4\p@ % \@Width of rules in box made by \fbox and \framebox.
\newcounter{section}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\newcounter{paragraph}[subsubsection]
\newcounter{subparagraph}[paragraph]
\if@seceqn
 \@addtoreset{equation}{section}
 \def\theequation{\arabic{section}.\arabic{equation}}
\else
  \def\theequation{\arabic{equation}}
\fi
\def\thesection      {\arabic{section}}
\def\thesubsection   {\thesection.\arabic{subsection}}
\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
\def\theparagraph    {\thesubsubsection.\arabic{paragraph}}
\def\thesubparagraph {\theparagraph.\arabic{subparagraph}}
\@addtoreset{section}{part} % reset section numbers at beginning of part

\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
\RequirePackage{latexsym}
\def\qed{\relax\ifmmode\hskip2em \Box\else\unskip\nobreak\hskip1em $\Box$\fi}
\def\proof@headerfont{\upshape\bfseries}
\if@amsthm
  \RequirePackage{amsthm}%
  \def\th@plain{%
    \thm@preskip\parskip
    \thm@postskip\z@
    \itshape % body font
  }
  \def\th@definition{%
    \thm@preskip\parskip
    \thm@postskip\z@
    \normalfont % body font
  }
  \def\th@remark{%
    \thm@headfont{\itshape}%
    \normalfont % body font
    \thm@preskip\parskip \divide\thm@preskip\tw@
    \thm@postskip\z@
  }
\else\let\theoremstyle\@gobble\fi
\newenvironment{pf}%
  {\par\addvspace{\@bls \@plus 0.5\@bls \@minus 0.1\@bls}\noindent
   {\bfseries\Elproofname}\enspace\ignorespaces}%
  {\par\addvspace{\@bls \@plus 0.5\@bls \@minus 0.1\@bls}}
\def\Elproofname{PROOF.}
\@namedef{pf*}#1{\par\begingroup\def\Elproofname{#1}\pf\endgroup\ignorespaces}
\expandafter\let\csname endpf*\endcsname=\endpf
\theoremstyle{plain}
\if@secthm
  \newtheorem{thm}{Theorem}[section]
  \@addtoreset{thm}{section}
\else
  \newtheorem{thm}{Theorem}
\fi
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{claim}[thm]{Claim}
\newtheorem{axiom}[thm]{Axiom}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{fact}[thm]{Fact}
\newtheorem{hypo}[thm]{Hypothesis}
\newtheorem{assum}[thm]{Assumption}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{crit}[thm]{Criterion}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{rem}[thm]{Remark}
\newtheorem{prob}[thm]{Problem}
\newtheorem{prin}[thm]{Principle}
\newtheorem{alg}{Algorithm}
\long\def\@makealgocaption#1#2{\vskip 2ex \small
  \hbox to \hsize{\parbox[t]{\hsize}{{\bfseries #1.} #2}}}
\newcounter{algorithm}
\def\thealgorithm{\@arabic\c@algorithm}
\def\fps@algorithm{tbp}
\def\ftype@algorithm{4}
\def\ext@algorithm{lof}
\def\fnum@algorithm{Algorithm \thealgorithm}
\def\algorithm{\let\@makecaption\@makealgocaption\@float{algorithm}}
\let\endalgorithm\end@float
\newtheorem{note}{Note}
\newtheorem{summ}{Summary}
\newtheorem{case}{Case}
\def\@pnumwidth{2.55em}
\def\@tocrmarg{2.55em \@plus 5em}
\def\@dotsep{-2.5}
\setcounter{tocdepth}{2}
\newcommand\listoffigures{%
    \section*{\listfigurename
      \@mkboth{\MakeUppercase\listfigurename}%
              {\MakeUppercase\listfigurename}}%
    \@starttoc{lof}%
    }
\newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
    \section*{\listtablename
      \@mkboth{%
          \MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
    \@starttoc{lot}%
    }
\let\l@table\l@figure
\def\tableofcontents{%
  \begin{small}
  \leftline {{\bfseries \contentsname\/}}
  \setcounter{secnumdepth}{4}%
  \setcounter{tocdepth}{2}%
  {\@starttoc{toc}}%
\end{small}
}
\newcommand*\l@section{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand*\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\newcommand*\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\newcommand*\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\newcommand*\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\def\@dotsep{2000}
\def\thebibliography{%
  \@startsection{section}{1}{\z@}{20\p@ \@plus 8\p@ \@minus 4pt}
  {\@bls}{\normalsize\bfseries}*{\refname}%
\addcontentsline{toc}{section}{\refname}%
  \@thebibliography}
\let\endthebibliography=\endlist
\def\@thebibliography#1{\@bibliosize
  \list{\@biblabel{\arabic{enumiv}}}{\settowidth\labelwidth{\@biblabel{#1}}
  \if@nameyear
    \labelwidth\z@ \labelsep\z@ \leftmargin\parindent
    \itemindent-\parindent
  \else
    \labelsep 3\p@ \itemindent\z@
    \leftmargin\labelwidth \advance\leftmargin\labelsep
\fi
     \itemsep 0.3\@bls \@plus 0.1\@bls \@minus 0.1\@bls
    \usecounter{enumiv}\let\p@enumiv\@empty
    \def\theenumiv{\arabic{enumiv}}}%
    \tolerance\@M
   \hyphenpenalty\@M
     \hbadness5000 \sfcode`\.=1000\relax}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\if@nameyear
  \def\@biblabel#1{}
\else
 \def\@biblabel#1{[#1]\hskip \z@ \@plus 1filll}
\fi
\let\make@bb@error\relax
\def\@mkbberr{\def\bibitem{\ClassError{elsart}%
    {Bibitem after note}%
    {You are using a bibitem after a note in a subbibitems
      environment;\MessageBreak
      note should the last item in a subbibitems environment}}}
\def\@itemnote{\make@bb@error\item[]}
\def\mk@noitemnote{\ifx\@tempa\note \let\note\@noitemnote \fi}
\def\@noitemnote{\let\note\@itemnote}
\AtBeginDocument{%
  \let\nopeek@bibitem\@bibitem
  \let\nopeek@lbibitem\@lbibitem
  \def\@bibitem#1{\let\note\@itemnote\nopeek@bibitem{#1}%
    \futurelet\@tempa\mk@noitemnote}
  \def\@lbibitem[#1]#2{\let\note\@itemnote\nopeek@lbibitem[#1]{#2}%
    \futurelet\@tempa\mk@noitemnote}
}
\newif\if@natbibloaded\@natbibloadedfalse
\AtBeginDocument{\@ifpackageloaded{natbib}{\@natbibloadedtrue}{}}
\newenvironment{subbibitems}[1]{%
  \if@natbibloaded\def\bib@ctr{NAT@ctr}\else\def\bib@ctr{enumiv}\fi
  \if@filesw {\let \protect \noexpand \immediate \write \@auxout
    {\string \nocollapse@cites}%
    \global\let\nocollapse@cites\relax}\fi
  \def\@itemslabel{#1}%
  \stepcounter{\bib@ctr}%
  \edef\main@bibnum{\the\value{\bib@ctr}}%
  \setcounter{\bib@ctr}{0}%
  \def\thebib@ctr{\main@bibnum\alph{\bib@ctr}}%
  \if@natbibloaded
    \def\bibitem{\@ifnextchar [{\@lbibitem }{\global \NAT@stdbsttrue
        \stepcounter {\bib@ctr}\@lbibitem [\thebib@ctr]}}%
  \else
    \def\@bibitem##1{\stepcounter{\bib@ctr}\@lbibitem[\thebib@ctr]{##1}}%
  \fi
  \let\make@bb@error\@mkbberr
}{%
  \if@filesw {%
    \if@natbibloaded{%
      \let\protect\noexpand
      \immediate\write\@auxout {\string\bibcite{\@itemslabel}%
        {{\main@bibnum a--\alph {\bib@ctr}}{}{{}}{{}}}}%
      \immediate\write\@auxout {\string\bibcite{\@itemslabel :s}%
        {{\main@bibnum}{}{{}}{{}}}}%
    }\else{%
      \let\protect\noexpand
      \immediate\write\@auxout {\string\bibcite{\@itemslabel}%
        {\main@bibnum a--\alph{\bib@ctr}}}%
      \immediate\write\@auxout {\string\bibcite{\@itemslabel :s}%
        {\main@bibnum}}%
    }\fi
  }\fi
  \setcounter{\bib@ctr}{\main@bibnum}%
}
\let\nocollapse@citex\@citex
\newcount\@tempcntc
\def\collapse@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
 \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
  {\@ifundefined
   {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries ?}\@warning
   {Citation `\@citeb' on page \thepage \space undefined}}%
  {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
   \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
    \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
   \else
    \advance\@tempcntb\@ne
    \ifnum\@tempcntb=\@tempcntc
    \else\advance\@tempcntb\m@ne\@citeo
    \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
\let\@citex\collapse@citex
\def\nocollapse@cites{%
  \@ifpackageloaded{hyperref}{}{\global\let\@citex\nocollapse@citex}%
  \global\let\nocollapse@cites\relax}
\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else\@citea\def\@citea{,}%
 \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
  {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else \def\@citea{--}\fi
   \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
\@namedef{cv*}{\section*{Curriculum Vitae}\cv}
 \def\cv{\hangindent=7pc \hangafter=-12 \parskip\bigskipamount \small}
\def\footnote{\@ifnextchar[{\@xfootnote}{\refstepcounter
   {\@mpfn}\protected@xdef\@thefnmark{\thempfn}\@footnotemark\@footnotetext}}
\def\footnotemark{\@ifnextchar[{\@xfootnotemark
    }{\refstepcounter{footnote}\xdef\@thefnmark{\thefootnote}\@footnotemark}}
\def\footnoterule{\kern-3\p@
  \hrule \@width 3pc               % The \hrule has default \@height of 0.4pt.
  \kern 2.6\p@}
\def\thempfootnote{\alph{mpfootnote}}
\def\mpfootnotemark{%
  \@ifnextchar[{\@xmpfootnotemark}{\stepcounter{mpfootnote}%
  \begingroup
    \let\protect\noexpand
    \xdef\@thefnmark{\thempfootnote}%
  \endgroup
  \@footnotemark}}
\def\@xmpfootnotemark[#1]{%
  \begingroup
    \c@mpfootnote #1\relax
    \let\protect\noexpand
    \xdef\@thefnmark{\thempfootnote}%
  \endgroup
  \@footnotemark}
\def\@mpmakefnmark{\,\hbox{$^{\mathrm{\@thefnmark}}$}}
\long\def\@mpmakefntext#1{\noindent
                     \hbox{$^{\mathrm{\@thefnmark}}$} #1}
\def\@iiiminipage#1#2[#3]#4{%
  \leavevmode
  \@pboxswfalse
  \setlength\@tempdima{#4}%
  \def\@mpargs{{#1}{#2}[#3]{#4}}%
  \setbox\@tempboxa\vbox\bgroup
    \color@begingroup
      \hsize\@tempdima
      \textwidth\hsize \columnwidth\hsize
      \@parboxrestore
      \def\@mpfn{mpfootnote}\def\thempfn{\thempfootnote}\c@mpfootnote\z@
      \let\@footnotetext\@mpfootnotetext
      \let\@makefntext\@mpmakefntext
      \let\@makefnmark\@mpmakefnmark
      \let\@listdepth\@mplistdepth \@mplistdepth\z@
      \@minipagerestore\global\@minipagetrue %% \global added 24 May 89
      \everypar{\global\@minipagefalse\everypar{}}}
\def\fn@presym{}
\long\def\@makefntext#1{\noindent\hbox to 1em
  {$^{\fn@presym\mathrm{\@thefnmark}}$\hss}#1}
\def\@makefnmark{\,\hbox{$^{\fn@presym\mathrm{\@thefnmark}}$}\,}
\def\patched@end@dblfloat{%
  \if@twocolumn
      \par\vskip\z@skip      %% \par\vskip\z@ added 15 Dec 87
      \global\@minipagefalse
      \outer@nobreak
    \egroup                  %% end of vbox
    \color@endbox
    \ifnum\@floatpenalty <\z@
      \@largefloatcheck
      \@cons\@dbldeferlist\@currbox
    \fi
    \ifnum \@floatpenalty =-\@Mii \@Esphack\fi
  \else
    \end@float
  \fi
}
\setcounter{topnumber}{5}
\def\topfraction{0.99}
\def\textfraction{0.05}
\def\floatpagefraction{0.9}
\setcounter{bottomnumber}{5}
\def\bottomfraction{0.99}
\setcounter{totalnumber}{10}
\def\dbltopfraction{0.99}
\def\dblfloatpagefraction{0.8}
\setcounter{dbltopnumber}{5}
\long\def\@maketablecaption#1#2{\@tablecaptionsize
    \global \@minipagefalse
    \hbox to \hsize{\parbox[t]{\hsize}{#1 \\ #2}}}
\long\def\@makefigurecaption#1#2{\@figurecaptionsize
  \vskip \@overcaptionskip
  \setbox\@tempboxa\hbox{#1. #2}
  \ifdim \wd\@tempboxa >\hsize              % IF longer than one line THEN
    \unhbox\@tempboxa\par                   %   set as justified paragraph
  \else                                     % ELSE
    \global \@minipagefalse
    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%   center single line.
  \fi}
\def\@makecaption{\@makefigurecaption}
\def\conttablecaption{\par \begingroup \@parboxrestore \normalsize
  \@makecaption{\fnum@table\,---\,continued}{}\par
  \vskip-1pc \endgroup}
\def\contfigurecaption{\vskip-1pc \par \begingroup \@parboxrestore
  \@captionsize
  \@makecaption{\fnum@figure\,---\,continued}{}\par
  \endgroup}
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename~\thefigure}
\def\figure{%
 \let\@makecaption\@makefigurecaption
  \let\contcaption\contfigurecaption \@float{figure}}
\let\endfigure\end@float
\@namedef{figure*}{%
 \let\@makecaption\@makefigurecaption
  \let\contcaption\contfigurecaption \@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
\newcounter{table}
\def\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename~\thetable}
\let\old@floatboxreset\@floatboxreset
\def\table{%
\let\@makecaption\@maketablecaption
\def\@floatboxreset{%
 \old@floatboxreset
 \@tablesize
}%
  \let\footnoterule\relax
  \let\contcaption\conttablecaption \@float{table}}
\let\endtable\end@float
\@namedef{table*}{%
\let\@makecaption\@maketablecaption
\def\@floatboxreset{%
 \old@floatboxreset
 \@tablesize
}%
  \let\footnoterule\relax
  \let\contcaption\conttablecaption \@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}
\newtoks\t@glob@notes             % List of all notes
\newtoks\t@loc@notes              % List of notes for one element
\newcount\note@cnt                % Number of notes per element
\newtoks\corauth@text
\newtoks\email@text
\newtoks\url@text
\newcounter{corauth}
\newcounter{author}               % Author counter
\newcount\n@author                % Total number of authors
\def\n@author@{1}                  % idem, read from .aux file
\newcounter{collab}               % Collaboration counter
\newcount\n@collab                % Total number of collaborations
\def\n@collab@{}                  % idem, read from .aux file
\newcounter{address}              % Address counter
\def\theHaddress{\arabic{address}}% for hyperref
\newdimen\sv@mathsurround         % Dimen register to save \mathsurround
\newcount\sv@hyphenpenalty        % Count register to save \hyphenpenalty
\newcount\prev@elem \prev@elem=0  % Variables to keep track of
\newcount\cur@elem  \cur@elem=0   % types of elements that are processed
\chardef\e@title=1
\chardef\e@subtitle=1
\chardef\e@author=2
\chardef\e@collab=3
\chardef\e@address=4
\newif\if@newelem                 % Switch to new type of element?
\newif\if@firstauthor             % First author or collaboration?
\newif\if@preface                 % If preface: omit history and abstract
\newif\if@hasabstract             % If abstract / keywords: do not omit rules
\newif\if@haskeywords             % If abstract / keywords: do not omit rules
\newbox\fm@box                    % Box for collected front matter
\newdimen\fm@size                 % Total height of \fm@box
\newbox\t@abstract                % Box for abstract
\newbox\t@keyword                 % Box for keyword abstract
 \let\report@elt\@gobble
\def\add@tok#1#2{\global#1\expandafter{\the#1#2}}
\def\add@xtok#1#2{\begingroup
  \no@harm
  \xdef\@act{\global\noexpand#1{\the#1#2}}\@act
\endgroup}
\def\beg@elem{\global\t@loc@notes={}\global\note@cnt\z@}
\def\@xnamedef#1{\expandafter\xdef\csname #1\endcsname}
\def\no@harm{%
  \let\\=\relax  \let\rm\relax
  \let\ss=\relax \let\ae=\relax \let\oe=\relax
  \let\AE=\relax \let\OE=\relax
  \let\o=\relax  \let\O=\relax
  \let\i=\relax  \let\j=\relax
  \let\aa=\relax \let\AA=\relax
  \let\l=\relax  \let\L=\relax
  \let\d=\relax  \let\b=\relax \let\c=\relax
  \let\bar=\relax
  \def\protect{\noexpand\protect\noexpand}}
\def\proc@elem#1#2{\begingroup
    \no@harm                             % make a few instructions harmless
    \let\thanksref\@gobble               % remove \thanksref from element
    \let\corauthref\@gobble
    \@xnamedef{@#1}{#2}%                 % and store as \@#1
    \let\thanksref\add@thanksref
    \let\corauthref\add@thanksref
    \setbox\@tempboxa\hbox{#2}%
  \endgroup
  \prev@elem=\cur@elem                   % keep track of type of previous
  \cur@elem=\csname e@#1\endcsname       % and current element
}
\def\add@thanksref#1{\global\advance\note@cnt\@ne
  \ifnum\note@cnt>\@ne \add@xtok\t@loc@notes{\note@sep}\fi
  \add@tok\t@loc@notes{\ref{#1}}}
\def\note@sep{,}
\def\thanks{\@ifnextchar[{\@tempswatrue
  \thanks@optarg}{\@tempswafalse\thanks@optarg[]}}
\def\thanks@optarg[#1]#2{\refstepcounter{footnote}%
  \if@tempswa\label{#1}\else\relax\fi
  \add@tok\t@glob@notes{\footnotetext}%
  \add@xtok\t@glob@notes{[\the\c@footnote]}%
  \add@tok\t@glob@notes{{#2}}%
  \ignorespaces}
\def\corauth{\@ifnextchar[{\@tempswatrue
  \corauth@optarg}{\@tempswafalse\corauth@optarg[]}}
\def\corauth@optarg[#1]#2{\refstepcounter{corauth}%
  \if@tempswa\label{#1}\else\relax\fi
  \add@tok\corauth@text{\footnotetext}%
  \add@xtok\corauth@text{[\the\c@corauth]}%
  \add@tok\corauth@text{{#2}}\ignorespaces}
\newcommand\ead[1][email]{%
  \add@eadcomma{#1}%
  \expandafter\ifcase\csname has@ead@#1\endcsname
  \expandafter\global\expandafter\chardef
    \csname has@ead@#1\endcsname=1\relax\else
  \expandafter\global\expandafter\chardef
    \csname has@ead@#1\endcsname=2\relax\fi
  \add@ead{#1}%
}
\def\add@ead#1#2{%
  \expandafter\add@tok\csname #1@text\endcsname{\texttt{#2}\ead@au}%
  \expandafter\add@xtok\csname #1@text\endcsname{{\@author}}%
  \ignorespaces
}
\def\add@eadcomma#1{%
  \expandafter\ifcase\csname has@ead@#1\endcsname\else
  \expandafter\add@tok\csname #1@text\endcsname{, }\fi
}
\let\@ead@au\@empty
\def\ead@newau{\ifx\@ead@au\@empty\else\ead@addau\@ead@au
  \let\@ead@au\@empty\fi}
\let\ead@endau\ead@newau
\def\ead@addau#1{ (#1)}
\def\ead@au#1{\edef\@ead@au{#1}}
\def\email@name{Email address}
\def\emails@name{Email addresses}
\chardef\has@ead@email=0
\def\url@name{URL}
\def\urls@name{URLs}
\chardef\has@ead@url=0
\let\real@refstepcounter\refstepcounter
\def\footnote{\@ifnextchar[{\@xfootnote}{\real@refstepcounter
   {\@mpfn}\protected@xdef\@thefnmark{\thempfn}\@footnotemark\@footnotetext}}
\def\footnotemark{\@ifnextchar[{\@xfootnotemark
    }{\real@refstepcounter{footnote}\xdef\@thefnmark{\thefootnote}\@footnotemark}}
\def\footnoterule{\kern-3\p@
  \hrule \@width 3pc               % The \hrule has default \@height of 0.4pt.
  \kern 2.6\p@}
 \let\report@elt\@gobble
\newenvironment{NoHyper}{}{}
\def\frontmatter{%
  \NoHyper
  \let\@corresp@note\relax
  \global\t@glob@notes={}\global\c@author\z@
  \global\c@collab\z@ \global\c@address\z@
  \sv@mathsurround\mathsurround \m@th
  \global\n@author=0\n@author@\relax
  \global\n@collab=0\n@collab@\relax
  \global\advance\n@author\m@ne   % In comparisons later on we need
  \global\advance\n@collab\m@ne   % n@author-1 and n@collab-1
  \global\@firstauthortrue        % set to false by first \author or \collab
  \global\@hasabstractfalse       % Default:  no abstract or keywords
  \global\@haskeywordsfalse       % Default:  no abstract or keywords
  \global\@prefacefalse           %           not preface
  \ifnum\c@firstpage=\c@lastpage
    \gdef\@pagerange{\@pagenumprefix\ESpagenumber{firstpage}}
  \else
    \gdef\@pagerange{\@pagenumprefix
   \ESpagenumber{firstpage}--\@pagenumprefix\ESpagenumber{lastpage}}%
  \fi
 \parskip 4\p@
  \open@fm \ignorespaces}
\def\preface{\@prefacetrue}
\def\endfrontmatter{%
  \ifx\@runauthor\relax
   \global\let\@runauthor\@runningauthor
  \fi
  \global\n@author=\c@author
  \global\n@collab=\c@collab \@writecount
  \global\@topnum\z@
  \thispagestyle{copyright}%            % Format rest of front matter:
  \if@preface \else                     % IF not preface THEN
  \vskip \@overhistoryskip
  \history@fmt                          % print history (received, ...)
  \newcount\c@sv@footnote
  \global\c@sv@footnote=\c@footnote     % save current footnote number
  \if@hasabstract                       % IF abstract/ keywords THEN
   \vskip \@preabstractskip     % Space above rule
  \hrule height 0.4\p@                 % Rule above abstract/keywords
    \vskip 8\p@
    \unvbox\t@abstract                  % print abstract, if any
  \fi
  \if@haskeywords                       % IF keywords THEN
    \vskip \@overkeywordskip
    \unvbox\t@keyword                   % Keyword abstract, if any
  \fi                                   % FI
  \vskip 10\p@
  \hrule height 0.4\p@                 % rule below abstract/keywords
  \dedicated@fmt                        % print dedication
  \vskip \@belowfmskip                  % Vertical space below frontmatter
\fi                                   % FI
  \close@fm                             % Close front matter material.
   \output@glob@notes  % Put notes at bottom of 1st page
  \global\c@footnote=\c@sv@footnote     % restore footnote number
  \global\@prefacefalse
  \global\leftskip\z@                   % Restore the normal values of
  \global\@rightskip\z@                 % \leftskip,
  \global\rightskip\@rightskip          % \rightskip and
  \global\mathsurround\sv@mathsurround  % \mathsurround.
  \let\title\relax       \let\author\relax
  \let\collab\relax      \let\address\relax
  \let\frontmatter\relax \let\endfrontmatter\relax
  \let\@maketitle\relax  \let\@@maketitle\relax
  \normal@text
}
\let\maketitle\relax
\newdimen\t@xtheight
\t@xtheight\textheight \advance\t@xtheight-\splittopskip
\def\open@fm{\global\setbox\fm@box=\vbox\bgroup
  \hsize=\@frontmatterwidth                 % Front matter is page-wide by default
  \centering                                % and centered
  \sv@hyphenpenalty\hyphenpenalty           % (save \hyphenpenalty)
  \hyphenpenalty\@M}                        % and not hyphenated
\def\close@fm{\egroup                       % close \vbox (\fm@box)
  \fm@size=\dp\fm@box \advance\fm@size by \ht\fm@box
  \@whiledim\fm@size>\t@xtheight \do{%
    \global\setbox\@tempboxa=\vsplit\fm@box to \t@xtheight
    \unvbox\@tempboxa \newpage
    \fm@size=\dp\fm@box \advance\fm@size by \ht\fm@box}
  \if@TwoColumn
    \emergencystretch=1pc \twocolumn[\unvbox\fm@box]
  \else
    \unvbox\fm@box
  \fi}
\def\output@glob@notes{\bgroup
  \the\t@glob@notes
  \egroup}
\def\justify@off{\let\\=\@normalcr
  \leftskip\z@ \@rightskip\@flushglue \rightskip\@rightskip}
\def\justify@on{\let\\=\@normalcr
  \leftskip\z@ \@rightskip\z@ \rightskip\@rightskip}
\def\normal@text{\global\let\\=\@normalcr
  \global\leftskip\z@ \global\@rightskip\z@ \global\rightskip\@rightskip
  \global\parfillskip\@flushglue}
\def\@writecount{\write\@mainaux{\string\global
  \string\@namedef{n@author@}{\the\n@author}}%
  \write\@mainaux{\string\global\string
  \@namedef{n@collab@}{\the\n@collab}}}
\def\title#1{%
  \beg@elem
  \title@note@fmt                      % formatting instruction
  \add@tok\t@glob@notes                % for \thanks commands
    {\title@note@fmt}%
  \proc@elem{title}{#1}%
  \def\title@notes{\the\t@loc@notes}%  % store the notes of the title,
  \title@fmt{\@title}{\title@notes}%   % print the title
  \ignorespaces}
\def\subtitle#1{%
  \beg@elem
  \proc@elem{subtitle}{#1}%
  \def\title@notes{\the\t@loc@notes}%  % store the notes of the title,
  \subtitle@fmt{\@subtitle}{\title@notes}% print the title
  \ignorespaces}
\newdimen \@logoheight \@logoheight 5pc
\def\@Lhook{\vrule \@height \@logoheight \@width \z@ \vrule \@height 10\p@ \@width 0.2\p@ \vrule \@height 0.2\p@ \@width 10pt}
\def\@Rhook{\vrule \@height 0.2\p@ \@width 10\p@ \vrule \@height 10\p@ \@width 0.2\p@ \vrule \@height \@logoheight \@width \z@}
\def\title@fmt#1#2{%
\@ifundefined{@runtitle}{\global\def\@runtitle{#1}}{}%
 \vspace*{12pt}             % Vertical space above title
  {\@titlesize #1\,\hbox{$^{#2}$}\par}%
  \vskip\@undertitleskip
\vskip24\p@  % Vertical space below title
  }
\def\subtitle@fmt#1#2{%               % No vertical space above sub-title
  {\@titlesize #1\,\hbox{$^{#2}$}}\par}
\def\title@note@fmt{\def\thefootnote{\fnstar{footnote}}}
\def\author{\@ifnextchar[{\author@optarg}{\author@optarg[]}}
\def\author@optarg[#1]#2{\stepcounter{author}%
  \beg@elem
  \add@tok\email@text{\ead@newau}%
  \add@tok\url@text{\ead@newau}%
  \@for\@tempa:=#1\do{\expandafter\add@thanksref\expandafter{\@tempa}}%
  \report@elt{author}\proc@elem{author}{#2}%
  \ifnum0\n@collab@=\z@ \runningauthor@fmt \fi
  \author@fmt{\the\c@author}{\the\t@loc@notes}{\@author}%
}
\def\runningauthor@fmt{%
  \begingroup\no@harm
    \if@firstauthor
      \ifnum0\n@author@ > 2
        \global\edef\@runningauthor{\@author\ et al.}%
      \else
        \global\let\@runningauthor\@author%
      \fi
    \else % \c@author > 1
      \ifnum0\n@author@ = 2
        \global\edef\@runningauthor{\@runningauthor\ \& \noexpand\@author}%
      \fi
    \fi
  \endgroup
}
\def\author@fmt#1#2#3{\@newelemtrue
  \if@firstauthor
  \first@author \global\@firstauthorfalse \fi
  \ifnum\prev@elem=\e@author \global\@newelemfalse \fi
  \if@newelem \author@fmt@init \fi
  \edef\@tempb{#2}\ifx\@tempb\@empty
    \hbox{{\author@font #3}}\else
    \hbox{{\author@font #3}\,$^{\mathrm{#2}}$}%
  \fi}
\def\first@author{\author@note@fmt
  \corauth@mark@fmt
  \add@tok\t@glob@notes
  {\output@corauth@text
    \output@ead@text{email}%
    \output@ead@text{url}%
    \author@note@fmt}%
}%
\def\author@fmt@init{%
  \par
  \vskip 8\p@ \@plus 4\p@ \@minus 2\p@
  \@authorsize
  \leavevmode}                        % Vertical space above author list
 \def\and{\unskip~and~}
\def\collab{\@ifstar{\collab@arg}{\collab@arg}}
\let\collaboration=\collab
\def\collab@arg#1{\stepcounter{collab}%
  \if@firstauthor \first@collab \global\@firstauthorfalse \fi
  \gdef\@runningauthor{#1}%
  \beg@elem
  \proc@elem{collab}{#1}%
  \collab@fmt{\the\c@collab}{\the\t@loc@notes}{\@collab}%
  \ignorespaces}
\def\collab@fmt#1#2#3{\@newelemtrue
  \ifnum\prev@elem=\e@collab \global\@newelemfalse \fi
  \if@newelem \collab@fmt@init \fi
  \par                                 % Start new paragraph
  {\large #3\,$^{\mathrm{#2}}$}}
\def\first@collab{
  \collab@note@fmt                     % re-define \thefootnote as
  \add@tok\t@glob@notes                % appropriate for collab/address
    {\collab@note@fmt}}%
\def\collab@fmt@init{\vskip 1em}       % Vertical space above list
\def\author@note@fmt{\setcounter{footnote}{0}%
  \def\thefootnote{\xarabic{footnote}}}
\let\collab@note@fmt=\author@note@fmt
\def\corauth@mark@fmt{\def\thecorauth{\astsymbol{corauth}}}
\def\output@corauth@text{\def\thefootnote{\astsymbol{footnote}}%
  \the\corauth@text}
\def\output@ead@text#1{%
  \expandafter\add@tok\csname #1@text\endcsname{\ead@endau}%
  \expandafter\ifcase\csname has@ead@#1\endcsname\else
  {\let\thefootnote\relax
    \footnotetext[0]{\raggedright\textit{%
        \expandafter\ifcase\csname has@ead@#1\endcsname\or
        \csname #1@name\endcsname\else
        \csname #1s@name\endcsname\fi
        : }%
        \expandafter\the\csname #1@text\endcsname.}}%
    \fi}
\def\xarabic#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\arabic{#1} \fi}
\def\xalph#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\alph{#1} \fi}
\def\xfnsymbol#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\fnsymbol{#1} \fi}
\def\address{\@ifstar{\address@star}%
  {\@ifnextchar[{\address@optarg}{\address@noptarg}}}
\def\address@optarg[#1]#2{\real@refstepcounter{address}%
  \beg@elem
  \report@elt{address}\proc@elem{address}{#2}%
  \address@fmt{\c@address}{\the\t@loc@notes}{\@address}{#1}%
  \if@Elproofing\else\label{#1}\fi
  \ignorespaces}
\def\address@noptarg#1{\real@refstepcounter{address}%
  \beg@elem
  \proc@elem{address}{#1}%
  \address@fmt{\z@}{\the\t@loc@notes}{\@address}{\theaddress}%
  \ignorespaces}
\def\address@star#1{%
  \beg@elem
  \proc@elem{address}{#1}%
  \address@fmt{\m@ne}{\the\t@loc@notes}{\@address}{*}%
  \ignorespaces}
\def\theaddress{\alph{address}}
\def\address@fmt#1#2#3#4{\@newelemtrue
  \if@Elproofing\def\@eltag{#4}\else\def\@eltag{\theaddress}\fi
  \ifnum\prev@elem=\e@address \@newelemfalse \fi
  \if@newelem \address@fmt@init \fi
  \noindent \bgroup \@addressstyle
  \ifnum#1=\z@
    #3\,$^{\mathrm{#2}}$\space%
  \else
    \ifnum#1=\m@ne
      $^{\phantom{\mathrm{\@eltag}}}$\space #3\,$^{\mathrm{#2}}$%
    \else
      $^{\mathrm{\@eltag}}\space$#3\,$^{\mathrm{#2}}$%
    \fi
  \fi
  \par \egroup}
\def\address@fmt@init{%
  \par                                % Start new paragraph
   \vskip 6\p@ \@plus 3\p@ \@minus 1.5pt}
\def\abstract{\@ifnextchar[{\@abstract}{\@abstract[]}}
\def\@abstract[#1]{%
  \global\@hasabstracttrue
  \hyphenpenalty\sv@hyphenpenalty     % restore \hyphenpenalty
  \global\setbox\t@abstract=\vbox\bgroup
  \leftskip\z@
  \@rightskip\z@ \rightskip\@rightskip \parfillskip\@flushglue
   \small
  \parindent 1em                      % \parindent in abstract
  \noindent {\bfseries\abstractname}  % caption `Abstract' (bold)
  \vskip 0.5\@bls    % half a line of space below
\noindent\ignorespaces
}
 \def\endabstract{\par \egroup}
\def\keyword{%
  \global\@haskeywordstrue             % Implies rules are to be printed
  \hyphenpenalty\sv@hyphenpenalty      % restore \hyphenpenalty
  \def\sep{\unskip, }                  % separator for multiple keywords
  \def\MSC{\par\leavevmode\hbox {\it 1991 MSC:\ }}%
  \def\PACS{\par\leavevmode\hbox {\it PACS:\ }}%
  \global\setbox\t@keyword=\vbox\bgroup
  \@keywordsize
  \parskip\z@
  \vskip 10\p@ \@plus 2\p@ \@minus 2\p@       % One line of space above keywords.
  \noindent\@keywordheading
  \justify@off                         % Keywords are not justified.
  \ignorespaces}
\def\endkeyword{\par \egroup}
\def\runtitle#1{\gdef\@runtitle{#1}}
\def\runauthor#1{\gdef\@runauthor{#1}}
\let\@runauthor\relax
\let\@runtitle\relax
\let\@runningauthor\relax
\def\RUNDATE{}
\def\RUNJNL{}
\def\RUNART{}
\def\journal#1{\gdef\@journal{#1}}
\def\volume#1{\gdef\@volume{#1}}       \def\@volume{0}
\def\issue#1{\gdef\@issue{#1}}         \def\@issue{0}
\newcount\@pubyear
\@pubyear=\number\year
\def\company#1{\def\@company{#1}}
\def\@copyrightyear{\number\year}
\def\@shortenyear#1#2#3#4\\{\global\def\@shortyear{#3#4}}
\expandafter\@shortenyear\the\@pubyear\\
\def\pubyear#1{\global\@pubyear#1
 \expandafter\@shortenyear\the\@pubyear\\%
  \ignorespaces}
\def\copyear#1{%
  \gdef\@copyrightyear{#1}%
  \ignorespaces}
\let\copyrightyear\copyear
\newcounter{firstpage}
\newcounter{lastpage}
\let\ESpagenumber\arabic
\def\firstpage#1{\def\@tempa{#1}\ifx\@tempa\@empty\else
  \setcounter{firstpage}{#1}%
  \global\c@page=#1 \ignorespaces\fi}
\setcounter{firstpage}{1}
\let\realpageref\pageref
\setcounter{lastpage}{0}
\def\lastpage#1{\def\@tempa{#1}\ifx\@tempa\@empty\else
  \setcounter{lastpage}{#1}\ignorespaces\fi
  }
\AtEndDocument{%
   \clearpage
   \addtocounter{page}{-1}%
   \immediate\write\@auxout{%
   \string\global\string\c@lastpage=\the\c@page}%
   \addtocounter{page}{1}%
}
\def\date#1{\gdef\@date{#1}}                  \def\@date{\today}
\def\aid#1{}
\def\ssdi#1#2{}
\def\received#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@received{#1}\fi}
  \def\@received{\relax}
\def\revised#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@revised{#1}\fi}
  \def\@revised{\relax}
\def\accepted#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@accepted{#1}\fi}
  \def\@accepted{\relax}
\def\communicated#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@communicated{#1}\fi}
  \def\@communicated{\relax}
\def\dedicated#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@dedicated{#1}\fi}
  \def\@dedicated{\relax}
\def\presented#1{\def\@tempa{#1}\ifx\@tempa\@empty\else\gdef\@presented{#1}\fi}
  \def\@presented{\relax}
\def\articletype#1{\gdef\@articletype{#1}}
  \@ifundefined{@articletype}{\def\@articletype{}}{}
\def\received@prefix{Received~}
\def\revised@prefix{; revised~}
\def\accepted@prefix{; accepted~}
\def\communicated@prefix{; communicated~by~}
\def\history@prefix{}
\def\received@postfix{}
\def\revised@postfix{}
\def\accepted@postfix{}
\def\communicated@postfix{}
\def\history@postfix{}
\def\empty@data{\relax}
\def\history@fmt{%
  \bgroup
  \@historysize
  \vskip 6\p@ \@plus 2\p@ \@minus 1\p@         % Vertical space above history
  \ifx\@received\empty@data \else       % If there is no \received,
                                        % do not print anything
    \leavevmode
    \history@prefix
    \received@prefix\@received \received@postfix%
    \ifx\@revised\empty@data \else
      \revised@prefix\@revised \revised@postfix%
    \fi
    \ifx\@accepted\empty@data \else
      \accepted@prefix\@accepted \accepted@postfix%
    \fi
    \ifx\@communicated\empty@data \else
      \communicated@prefix\@communicated \communicated@postfix%
    \fi
    \history@postfix
  \fi
  \par \egroup}
\def\dedicated@fmt{%
  \ifx\@dedicated\empty@data \else
    \vskip 4\p@ \@plus 3\p@
    \normalsize\it\centering \@dedicated
    \fi}
\def\@alph#1{\ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or
k\or \ell\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or
y\or z\or aa\or ab\or ac\or ad\or ae\or af\or ag\or ah\or ai\or aj\or
ak\or a\ell\or am\or an\or ao\or ap\or aq\or ar\or as\or at\or au\or av\or
aw\or ay\or az\or ba\or bb\or bc\or bd\or be\or bf\or bg\or bh\or bi\or
bj\or bk\or b\ell\or bm\or bn\or bo\or bp\or bq\or br\or bs\or bt\or
bu\or bw\or bx\or by\or bz\or ca\or cb\or cc\or cd\or ce\or cf\or cg\or
ch\or ci\or cj\or ck\or c\ell\or cm\or cn\or co\or cp\or cq\or cr\or
cs\or ct\or cu\or cw\or cx\or cy\or cz\or da\or db\or dc\or dd\or de\or df\or dg\or dh\or di\or dj\or dk\or
d\ell\or dm\or dn\or do\or dp\or dq\or dr\or ds\or dt\or du\or dw\or
dx\or dy\or dz\or ea\or eb\or ec\or ed\or ee\or ef\or eg\or eh\or
ei\or ej\or ek\or e\ell\or em\or en\or eo\or ep\or eq\or er\or es\or
et\or eu\or ew\or ex\or ey\or ez\else\@ctrerr\fi}
\def\fnstar#1{\@fnstar{\@nameuse{c@#1}}}
\def\@fnstar#1{\ifcase#1\or
    \hbox{$\star$}\or
    \hbox{$\star\star$}\or
    \hbox{$\star\star\star$}\or
    \hbox{$\star\star\star\star$}\or
    \hbox{$\star\star\star\star\star$}\or
    \hbox{$\star\star\star\star\star\star$}
  \else
    \@ctrerr
  \fi
  \relax}
\def\astsymbol#1{\@astsymbol{\@nameuse{c@#1}}}
\def\@astsymbol#1{\ifcase#1\or
    \hbox{$\ast$}\or
    \hbox{$\ast\ast$}\or
    \hbox{$\ast\ast\ast$}\or
    \hbox{$\ast\ast\ast\ast$}\or
    \hbox{$\ast\ast\ast\ast\ast$}\or
    \hbox{$\ast\ast\ast\ast\ast\ast$}%
  \else
    \@ctrerr
  \fi
  \relax}
\mark{{}{}}   % Initializes TeX's marks
\def\ps@plain{\let\@mkboth\@gobbletwo
 \def\@oddhead{}%
 \def\@evenhead{}%
 \def\@oddfoot{\hfil {\rmfamily\thepage} \hfil}%
 \let\@evenfoot\@oddfoot}
\def\@copyright{\@issn/\@shortyear/\$\@price\ $\copyright$\ \the\@pubyear\
  \@company{} All  rights reserved}
\def\@jou@vol@pag{\@journal\ \@volume\ (\the\@pubyear)\ \@pagerange}
\def\sectionmark#1{}
\def\subsectionmark#1{}
\let\@j@v@p\@jou@vol@pag    % long journal title appears in reprint line
\let\@@j@v@p\@jou@vol@pag   % long journal title appears in running headline
\def\sectionmark#1{}
\def\subsectionmark#1{}
\def\ps@copyright{\let\@mkboth\@gobbletwo
  \def\@oddhead{}%
  \let\@evenhead\@oddhead
  \def\@oddfoot{\small\slshape
    \def\@tempa{0}
    \ifx\@volume\@tempa
                                                  %
%      Preprint submitted to \@journal\hfil\@date\/%
    \else
      Article published in \@jou@vol@pag\hfil\hbox{}\fi}%
  \let\@evenfoot\@oddfoot
}
\let\ps@noissn\ps@empty
\let\ps@headings\ps@plain
\def\today{\number\day\space\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\year}
\def\nuc#1#2{\relax\ifmmode{}^{#1}{\protect\text{#2}}\else${}^{#1}$#2\fi}
\def\itnuc#1#2{\setbox\@tempboxa=\hbox{\scriptsize\it #1}
  \def\@tempa{{}^{\box\@tempboxa}\!\protect\text{\it #2}}\relax
  \ifmmode \@tempa \else $\@tempa$\fi}
\let\old@vec\vec % save old definition of \vec
\def\pol#1{\old@vec{#1}}
\def\half{{\textstyle\frac{1}{2}}}
\def\threehalf{{\textstyle\frac{3}{2}}}
\def\quart{{\textstyle\frac{1}{4}}}
\if@symbold\else\def\d{\,\mathrm{d}}\fi
\def\e{\mathop{\mathrm{e}}\nolimits}
\def\int{\intop}
\def\oint{\ointop}
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\newbox\Slashbox \setbox\Slashbox=\hbox{\large$/$}
\def\pFMslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\pFMSlash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\Slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\Slashbox \kern-\@tempdima \box\@tempboxa}
\def\FMslash{\protect\pFMslash}
\def\FMSlash{\protect\pFMSlash}
  \def\Cset{\mathbb{C}}
  \def\Hset{\mathbb{H}}
  \def\Nset{\mathbb{N}}
  \def\Qset{\mathbb{Q}}
  \def\Rset{\mathbb{R}}
  \def\Zset{\mathbb{Z}}
\if@TwoColumn
  \adjdemerits=100
  \linepenalty=100
  \doublehyphendemerits=5000        % experimental (1993-12-14)
  \emergencystretch=1.6pc
  \spaceskip=0.3em \@plus 0.17em \@minus 0.12em
\fi
\@frontmatterwidth\textwidth
\ps@headings                                % 'headings' page style
\pagenumbering{arabic}                      % Arabic page numbers
\def\thepage{\@pagenumprefix\ESpagenumber{page}}  % preceded by \@pagenumprefix
\let\baselinestretch\@blstr
\InputIfFileExists{\@shortjid.cfg}{}{}
\endinput
%%
%% End of file `elsart.cls'.
---------------0210032148468--
