Content-Type: multipart/mixed; boundary="-------------0711061840887"

This is a multi-part message in MIME format.

---------------0711061840887
Content-Type: text/plain; name="07-265.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="07-265.keywords"

Operator numerical range; m-sectorial generators; contraction semigroups; quasi-sectorial 
contractions; holomorphic semigroups; semigroup operator-norm approximations.
---------------0711061840887
Content-Type: application/x-tex; name="07-265.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="07-265.comments"

\documentclass{elsart}
---------------0711061840887
Content-Type: application/x-tex; name="JFA-07-395.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="JFA-07-395.tex"

% Template article for preprint document class `elsart'
%SP 2006/04/26

\documentclass{elsart}

% Use the option doublespacing or reviewcopy to obtain double line spacing
% \documentclass[doublespacing]{elsart}

% if you use PostScript figures in your article
% use the graphics package for simple commands
\usepackage{graphics}
% or use the graphicx package for more complicated commands
\usepackage{graphicx}
% or use the epsfig package if you prefer to use the old commands
\usepackage{epsfig}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\authorrunninghead{V.A. Zagrebnov}
%\titlerunninghead{Quasi-Sectorial Contractions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Use the option doublespacing or reviewcopy to obtain double line spacing
%\documentclass[doublespacing]{elsart}

% the natbib package allows both number and author-year (Harvard)
% style referencing;
%\usepackage{natbib}


% The amssymb package provides various useful mathematical symbols

\usepackage{amsfonts,amssymb}

%%%%%%%%%%%%%%%%% LEMME_THEOREME_PROPOSITION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{postulate}{Hypothesis}[section]  %hypoth\`ese
\newcommand{\bhypo}{\begin{postulate} \it}
\newcommand{\ehypo}{\end{postulate}}

\newtheorem{theorem}{Theorem}[section]  %theoreme
\newcommand{\btheo}{\begin{theorem} \it}
\newcommand{\etheo}{\end{theorem}}

\newtheorem{proposition}{Proposition}[section]  %proposition
\newcommand{\bprop}{\begin{proposition}\it}
\newcommand{\eprop}{\end{proposition}}

\newtheorem{corollary}{Corollary}[section]  %corollaire
\newcommand{\bcorol}{\begin{corollary} \it}
\newcommand{\ecorol}{\end{corollary}}

\newtheorem{lemma}{Lemma}[section]  %lemme
\newcommand{\blem}{\begin{lemma}\it}
\newcommand{\elem}{\end{lemma}}

\newtheorem{remark}{Remark}[section]   %remarque
\def\brem{\begin{remark} \rm }
\def\erem{\end{remark}}

\newtheorem{definition}{Definition}[section]   %definition
\newcommand{\bdefi}{\begin{definition} \rm }
\newcommand{\edefi}{\end{definition}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\date{}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\bead}{\begin{eqnarray*}}
\newcommand{\eead}{\end{eqnarray*}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bed}{\begin{displaymath}}
\newcommand{\eed}{\end{displaymath}}
\newcommand{\bl}{\begin{lemmma}}
\newcommand{\el}{\end{lemmma}}
\newcommand{\bp}{\begin{proposition}}
\newcommand{\ep}{\end{proposition}}
\newcommand{\bt}{\begin{theorem}}
\newcommand{\et}{\end{theorem}}
\newcommand{\br}{\begin{remark}}
\newcommand{\er}{\end{remark}}
\newcommand{\bc}{\begin{corollary}}
\newcommand{\ec}{\end{corollary}}
\newcommand{\bd}{\begin{definition}}
\newcommand{\ed}{\end{definition}}

\newcommand{\la}{\Label}
\newcommand{\Label}{\label}
\newcommand{\re}{\Ref}
\newcommand{\Ref}{\ref}
\newenvironment{proof}%
{\begin{sloppypar}\noindent{\bf Proof }}%
{\hspace*{\fill}$\square$\end{sloppypar}}

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



\begin{document}


\begin{frontmatter}
%
% Title, authors and addresses
%
% use the thanksref command within \title, \author or \address for footnotes;
% use the corauthref command within \author for corresponding author footnotes;
% use the ead command for the email address,
% and the form \ead[url] for the home page:
% \title{Title\thanksref{label1}}
% \thanks[label1]{}
% \author{Name\corauthref{cor1}\thanksref{label2}}
% \ead{email address}
% \ead[url]{home page}
% \thanks[label2]{}
% \corauth[cor1]{}
% \address{Address\thanksref{label3}}
% \thanks[label3]{}


\title{Quasi-Sectorial Contractions}
%
% use optional labels to link authors explicitly to addresses:
% \author[label1,label2]{}
% \address[label1]{}
% \address[label2]{}

\author{Valentin A. Zagrebnov}
\address{Universit\'{e} de la M\'{e}diterran\'{e}e (Aix-Marseille II) and
Centre de Physique Th\'{e}orique - UMR 6207,  Luminy-Case 907, 13288 Marseille Cedex 9, France}
\ead{zagrebnov@cpt.univ-mrs.fr}

\begin{abstract}
%
%
We revise the notion of the \textit{quasi-sectorial} contractions.
Our main theorem establishes a relation between semigroups of \textit{quasi-sectorial} contractions and
a class of $m-${sectorial} generators. We discuss a relevance of this kind of contractions to
the theory of operator-norm approximations of strongly continuous semigroups.
%
%
\end{abstract}

\begin{keyword}
%
%
Operator numerical range; $m$-sectorial generators; contraction semigroups; quasi-sectorial
contractions; holomorphic semigroups; semigroup operator-norm approximations.
%
%
\PACS 47A55, 47D03, 81Q10
\end{keyword}

\end{frontmatter}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Sectorial Operators}\label{sec1}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
%\setcounter{theo}{0}
%\renewcommand{\thetheo}{\arabic{section}.\arabic{theo}}



Let $\mathfrak{H}$ be a separable Hilbert space and let $T$ be a densely defined linear
operator with domain ${\rm{dom}}(T)\subset \mathfrak{H}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{def-n-ran}
{\em The set of complex numbers:
$$
{\mathfrak{N}}(T) := \{(u,Tu)\in {{\mathbb{C}}}: u \in {\rm{dom}}(T), \
\|u\| = 1\} ,
$$
is called the \textit{numerical range} of the operator $T$.}
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{rem-1-1}
{\rm{(}}a{\rm{)}} It is known that the set ${\mathfrak{N}}(T)$ is convex
{\rm{(}}the Toeplitz-Hausdorff theorem{\rm{)}}, and in general is
neither open nor closed, even for a closed operator $T$.\\
{\rm{(}}b{\rm{)}} Let $\Delta:= {{\mathbb{C}}} \setminus
\overline{{\mathfrak{N}}(T)}$ be complement of the numerical range
closure in the complex plane. Then $\Delta$ is a connected open set
except the special case, when $\overline{{\mathfrak{N}}(T)}$ is a
strip bounded by two parallel straight lines.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Below we use some important properties of this set, see e.g. \cite[Ch.V]{Kato},
or \cite[Ch.1.6]{Zag}. Recall that ${\rm{dim}} ({\rm{ran}}(T))^{\bot}=:{\rm{def}}(T)$ is called a
\textit{deficiency} (or \textit{defect}) of a closed operator $T$ in $\mathfrak{H}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Proposition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{prop-n-ran}
{\rm{(}}i{\rm{)}} Let $T$ be a closed operator in $\mathfrak{H}$. Then for any complex
number $z\notin \overline{{\mathfrak{N}}(T)}$, the operator $(T-z I)$ is
injective. Moreover, it has a closed range ${\rm{ran}}(T-z I)$ and
a constant deficiency ${\mathrm{def}}(T-z I)$ in each of connected component of
$\ {{\mathbb{C}}}\setminus{\overline{{\mathfrak{N}}(T)}}$.\\
{\rm{(}}ii{\rm{)}} If  ${\mathrm{def}}(T-z I) = 0$
for $z\notin \overline{{\mathfrak{N}}(T)}$, then $\Delta$ is a
subset of the resolvent set $\rho(T)$ of the operator $T$  and
\begin{equation}\label{estres}
\|(T-z I)^{-1}\| \leq {1\over {\rm{dist}}(z,\overline{{\mathfrak{N}}(T)})} \ \ .
\end{equation}
{\rm{(}}iii{\rm{)}} If ${\rm{dom}}(T)$ is dense and ${\mathfrak{N}}(T) \neq {\mathbb{C}}$, then
$T$ is closable, hence the adjoint operator $T^*$ is also densely defined.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{spect-n-ran-bounded} For a bounded operator $T\in \mathcal{L}(\mathfrak{H})$ the spectrum
$\sigma(T)$ is a subset of $\ \overline{{\mathfrak{N}}(T)}$.
\end{corollary}
For unbounded operator $T$ the relation between spectrum and numerical range is more complicated.
For example, it may very well happen that $\sigma(T)$ is not contained
in $\overline{{\mathfrak{N}}(T)}$, but for a closed operator $T$ the essential spectrum
$\sigma_{ess}(T)$ is always a subset of $\overline{{\mathfrak{N}}(T)}$. The condition
${\mathrm{def}}(T-zI) = 0, \ z\notin \overline{{\mathfrak{N}}(T)}$ in Proposition \ref{prop-n-ran} (ii)
serves to ensure that for those unbounded operators one gets
\begin{equation}\label{spect-n-ran}
\sigma(T)\subset \overline{{\mathfrak{N}}(T)} \ ,
\end{equation}
i.e., the same conclusion as in Corollary \ref{spect-n-ran-bounded} for bounded operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{sect-oper}
{\em Operator $T$ is called \textit{sectorial} with {semi-angle}
$\alpha\in(0, \pi/2)$ and a vertex at $z=0$  if
$$
{\mathfrak{N}}(T)  \subseteq S_{\alpha} := \{z\in
{{\mathbb{C}}}: |\arg \ z| \leq \alpha \} \ .
$$
If, in addition, $T$ is closed and there is $z \in
{\mathbb{C}}\setminus S_{\alpha}$ such that it belongs to the resolvent set $\rho (T)$,
then operator $T$ is called  $m$-sectorial.}
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{rem-1-2}
Let $T$ be $m$-sectorial with the {semi-angle} $\alpha\in(0, \pi/2)$ and the vertex at $z=0$.
Then it is obvious that the operators $a T$ and $T_b := T + b$ belong to the same sector $S_{\alpha}$
for any non-negative parameters $a,b \geq 0$. In fact ${\mathfrak{N}}(T_b)\subseteq S_{\alpha} + b$, i.e.
the operator $T_b$ has the vertex at $z=b$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Some of important properties of the $m$-sectorial operators are summarized by the following
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{hol-contr}
If $T$ is {$m$-sectorial} in $\mathfrak{H}$, then the semigroup
$\{U(\zeta):= e^{-\zeta \, T}\}_{\, \zeta }$ generated by the operator $T${\rm{:}} \\
{\rm{(}}i{\rm{)}} is \textit{holomorphic} in the open sector $\{\zeta \in S_{\pi/2 - \alpha}\}${\rm{;}} \\
{\rm{(}}ii{\rm{)}} is a contraction, i.e. ${{\mathfrak{N}}}(U(\zeta))$
is a subset of the unit disc ${\mathfrak{D}}_{r=1}:=\{z\in {{\mathbb{C}}}: |z| \leq 1 \}$ for
$\{\zeta \in S_{\pi/2 - \alpha}\}$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quasi-Sectorial Contractions and Main Theorem}\label{sec2}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

The notion of the \textit{quasi-sectorial} contractions was introduced in \cite{CachZag-Q-S} to study the
operator-norm approximations of semigroups. In paper \cite{CaNeZag} this class of contractions appeared in
analysis of the operator-norm error bound estimate of the exponential Trotter product formula for the
case of accretive perturbations. Further applications of these contractions which, in particular, improve
the rate of convergence estimate of \cite{CachZag-Q-S} for the Euler formula, one can find in \cite{Paul},
\cite{Cach} and \cite{BentPaul}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{alpha-domain}
{\em For $\alpha\in[0, \pi/2)$ we define in the complex plane ${\mathbb{C}}$ a \textit{closed domain}:
% \nonumber to remove numbering (before each equation)
$$ D_{\alpha}:=\{z\in {\mathbb{C}}: |z|\leq \sin \alpha\} \cup
\{z\in {\mathbb{C}}: |\arg (1-z)|\leq \alpha \ {\rm{and}}\ |z-1|\leq
\cos \alpha \} .$$
This is a convex subset of the unit disc ${\mathfrak{D}}_{r=1}$, with \textit{"angle"}
(in contrast to \textit{tangent}) touching of its boundary $\partial {\mathfrak{D}}_{r=1}$
at only one point $z=1$, see Figure 1. It is evident that $D_{\alpha}\subset D_{\beta > \alpha}$.}
\end{definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{Q-S-cont}{\rm{(}}Quasi-Sectorial Contractions {\rm{\cite{CachZag-Q-S})}}
{\em A {contraction} $C$ on the Hilbert space $\mathfrak{H}$ is called
\textit{quasi-sectorial} with  semi-angle $\alpha\in [0, \pi/2)$
with respect to the vertex at $z=1$, if
${\mathfrak{N}}(C) \subseteq D_{\alpha}$.}
\end{definition}

Notice that if operator $C$ is a \textit{quasi-sectorial} contraction, then
$I - C$ is an $m$-\textit{sectorial} operator with
vertex $z=0$ and semi-angle $\alpha$. The limits $\alpha=0$ and $\alpha = \pi/2$ correspond,
respectively, to non-negative (i.e. \textit{self-adjoint}) and to \textit{general} contraction.

The \textit{resolvent} of an $m$-sectorial operator $A$, with {semi-angle}
$\alpha\in(0, \pi/4]$ and vertex at $z=0$, gives the first non-trivial (and for us a \textit{key}) example
of a quasi-sectorial contraction.

\begin{proposition}\label{Th-res}
Let $A$ be $m$-sectorial operator with {semi-angle} $\alpha\in [0, \pi/4]$ and vertex at $z=0$.
Then $\{F(t):= (I + t A)^{-1}\}_{\, t \geq 0}$ is a family
of quasi-sectorial contractions which numerical ranges ${\mathfrak{N}}(F(t)) \subseteq D_{\alpha}$
for all $t \geq 0$.
\end{proposition}
\begin{proof}:
First, by virtue of Proposition \ref{prop-n-ran} (ii) we obtain the estimate:
\begin{equation}\label{F-contr}
\|F(t)\|\leq \frac{1}{t \ {\rm{dist}}(1/t \ , \, - S_{\alpha})} \, = \, 1 \ \ ,
\end{equation}
which implies that operators $\{F(t)\}_{\, t \geq 0}$ are contractions with numerical ranges
${\mathfrak{N}}(F(t)) \subseteq {\mathfrak{D}}_{r=1}$.

Next, by Remark \ref{rem-1-2} for all $u\in \mathfrak{H}$ one gets
$(u,F(t)u)= (v_t,v_t) + t (A v_t,v_t) \in S_{\alpha}$, where $v_t := F (t)u $, i.e. for any $t \geq 0$
the numerical range ${\mathfrak{N}}(F(t)) \subseteq S_{\alpha}$. Similarly, one finds that
$(u,(I-F(t))u) = t (v,A v) + t^2 (A v, A v) \in S_{\alpha}$, i.e.,
${\mathfrak{N}}(I-F(t)) \subseteq S_{\alpha}$. Therefore, for all $t\geq 0$ we obtain:
\begin{equation}\label{N-range-Fb}
{\mathfrak{N}}(F (t)) \subseteq (S_{\alpha} \cap (1 - S_{\alpha})) \subset \mathfrak{D}_{r=1} \ .
\end{equation}
Moreover, since $\alpha \leq \pi/4$, by Definition \ref{alpha-domain} we get
$(S_{\alpha} \cap (1 - S_{\alpha}))\subset D_{\alpha}$, i.e. for these values of $\alpha$ the
operators $\{F(t)\}_{\, t \geq 0}$ are quasi-sectorial contractions with numerical ranges in $D_{\alpha}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we are in position to prove the \textit{main} Theorem establishing a relation between
quasi-sectorial contraction semigroups and a certain class of $m$-sectorial generators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{Th-main}
Let $A$ be an {$m$-sectorial} operator with  semi-angle
$\alpha\in[0, \pi/4]$ and with  vertex at $z=0$. Then $\{e^{-t \, A}\}_{\, t \geq 0}$ is a quasi-sectorial
contraction semigroup with numerical ranges ${\mathfrak{N}}(e^{-t \, A}) \subseteq D_{\alpha}$ for all
$t \geq 0$.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The proof of the theorem is based on a series of lemmata and on the numerical range \textit{mapping}
theorem by Kato \cite{Kato-map} (see also an important comment about this theorem in \cite{Uch}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{Kato-map}{\em \cite{Kato-map}}
Let $f(z)$ be a rational function on the complex plane ${\mathbb{C}}$, with $f(\infty)=\infty$.
Let for some compact and convex set $E'\subset {\mathbb{C}}$ the inverse function
$f^{-1}: E' \mapsto E \supseteq K $, where $K$ is a convex kernel of $E$, i.e., a subset of $E$
such that $E$ is \textit{star-shaped} relative to any $z\in K$.

If $C$ is an operator with numerical range $ {\mathfrak{N}}(C)\subseteq K$, then
${\mathfrak{N}}(f(C))\subseteq E'$.
\end{proposition}
Notice that for a convex set $E$ the corresponding  \textit{convex} kernel $K = E$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{z-n}
Let $f_n(z) = z^n$ be complex functions, for $z\in {\mathbb{C}}$ and $n\in \mathbb{N}$.
Then the sets $f_n(D_\alpha)$ are convex and domains $f_n(D_\alpha)\subseteq D_\alpha$ for
{any $n\in \mathbb{N}$}, if $\alpha\leq \pi/4$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{Euler}{\em(Euler formula)}
Let $A$ be an {$m$-sectorial} operator. Then for $t\geq 0$ one gets the strong limit
\begin{equation}\label{Eul-s-lim}
s-\lim_{n\rightarrow\infty}(F(t/n))^n = e^{-t A} \ .
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The next section is reserved for the proofs. They refine and modify some lines of reasonings
of the paper \cite{CachZag-Q-S}. This concerns, in particular, a corrected proofs of Proposition \ref{Th-res} and
Theorem \ref{Th-main} (cf. Theorem 2.1 of \cite{CachZag-Q-S}), as well as reformulations and proofs of
Propositions \ref{Kato-map} and Lemma \ref{z-n}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs}\label{sec3}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}


\begin{proof}{\rm{(Lemma \ref{z-n})}}:\\
Let $\{z:|z|\leq \sin\alpha\}\subset D_\alpha$, then one gets $|z^n|\leq \sin\alpha$. Therefore, for the
mappings $f_n : z \mapsto z^n$ one obtains  $f_n (z) \in  D_\alpha $ for any $n \geq 1$.

Thus, it rests to check the same property only for images $f_n({\mathcal{G}}_{\alpha}), n\geq1$  of the sub-domain:
\begin{equation}\label{sub-dom}
{\mathcal{G}}_{\alpha}:=\!\{z: \!|\arg(z)|< \!(\pi/2 - \alpha) \}\cap\{z:\!|\arg(z + 1)|>(\pi - \alpha)\}\!
\subset \! D_\alpha ,
\end{equation}
see Definition \ref{alpha-domain} and Figure 1.

For $0\leq t\leq \cos\alpha$, two segments of tangent straight intervals:
$$\{\zeta_{\pm}(t)=1 + t \,e^{i (\pi \mp \alpha)}\}_{0\leq t\leq \cos\alpha} \subset \partial D_\alpha ,$$
are correspondingly \textit{upper} $\zeta_{+}(t)$ and  \textit{lower} $\zeta_{-}(t)= \overline{\zeta_{+}(t)}$
non-arc parts of the total boundary $\partial D_\alpha$; they also coincide with a part of
the boundary $\partial \mathcal{G}_{\alpha}$ connected to the vertex $z=1$.

Now we proceed by induction. Let $n=1$. Then one obviously obtain : $f_{n=1} (D_{\alpha}) = D_{\alpha}$.
For $n=2$ the boundary $\partial f_2 ({\mathcal{G}}_{\alpha})$ of domain $f_2 ({\mathcal{G}}_{\alpha})$ is a
union
$\Gamma_2 (\alpha)\cup \overline{\Gamma_2 (\alpha)}$ of the contour
$$\Gamma_2 (\alpha):= \{f_2 (\zeta_{+}(t))\}_{0\leq t\leq \cos\alpha}\cup \{z: |z|\leq \sin^2 \alpha , \
\arg (z)= (\pi - 2 \alpha)\}$$
and its conjugate $\overline{\Gamma_2 (\alpha)}$.
Since $\arg(\partial_t f_2 (\zeta_{+}(t))\leq (\pi - \alpha)$ for all $0\leq t\leq \cos\alpha$, the contour
$$\{f_2 (\zeta_{+}(t))\}_{0\leq t\leq \cos\alpha} \subseteq  \{z: |\arg(z + 1)|> (\pi - \alpha)\} ,$$
see (\ref{sub-dom}). The same is obviously true for the image of the lower branch $\zeta_{-}(t)$.
If $\alpha \leq \pi/4$,
one gets:
\begin{eqnarray}\label{Im+}
\sup_{0\leq t\leq \cos\alpha} {\rm{Im}} (f_2 (\zeta_{+}(t)))&=& {\rm{Im}} (f_2 (\zeta_{+}(t^* =
(2\cos\alpha)^{-1})))\\ &=&
\frac{1}{2} \tan \alpha < \sin\alpha \cos\alpha \,,\nonumber
\end{eqnarray}
where $t^* = (2\cos\alpha)^{-1} \leq \cos\alpha$, and
$$ 0 \geq {\rm{Re}}(f_2 (\zeta_{+}(t)))\geq -  \sin^2\alpha  \cos2\alpha \geq - \sin\alpha \,.$$
Therefore, $\{f_2 (\zeta_{+}(t))\}_{0\leq t\leq \cos\alpha} \subseteq D_\alpha$. Since the same is
also true for the image of the lower branch $\zeta_{-}(t)$, we obtain
$f_2 ({\mathcal{G}}_{\alpha})\subset D_\alpha$ and by consequence $f_{n=2} (D_\alpha)=
\{w = z\cdot z :  \ z\in D_\alpha,  \ z \in f_{n=1} (D_\alpha)\} \subset D_\alpha ,$ for $\alpha \leq \pi/4$.

Now let $n>2$ and suppose that $f_n (D_\alpha)\subset D_\alpha$. Then the image of the $(n+1)\ -$ order mapping
of domain $D_\alpha$ is:
$$f_{n+1}(D_\alpha)= \{w = z\cdot z^n :  \ z\in D_\alpha,  \ z^n \in f_n (D_\alpha)\}, $$
and since $f_n (D_\alpha)\subset D_\alpha$, we obtain $f_{n+1}(D_\alpha)\subset D_\alpha$ by the same reasoning
as for $n=2$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{max-Im}
Let $\phi(t):= \arg(\zeta_{+}(t))$. Then $\cot(\alpha + \phi(t))= (\cos\alpha - t)/\sin\alpha$ and
\begin{equation}\label{estim-Im}
\sup_{0\leq t\leq \cos\alpha}{\rm{Im}}(f_n (\zeta_{+}(t)))\leq (1 -2t_{n}^*\cos\alpha + (t_{n}^*)^2)^{n/2}
\end{equation}
for $\sin (n\phi(t_{n}^*))=1$. In the limit $n\rightarrow\infty$ this implies that
$\phi(t_{n}^*) = \pi/2n + o(n^{-1})$, $t_{n}^* = \pi/(2n \sin\alpha)  + o(n^{-1})$ and
\begin{equation}\label{Im-lim}
\lim_{n\rightarrow\infty}\sup_{0\leq t\leq \cos\alpha}{\rm{Im}}(f_n (\zeta_{+}(t)))
\leq \exp(- \frac{1}{2}\pi \cot\alpha) < \frac{1}{2} \tan \alpha .
\end{equation}
By the same reasoning one gets the estimates similar to (\ref{estim-Im}) and (\ref{Im-lim}) for $\zeta_{-}(t))$.
Hence, $|{\rm{Im}}(f_n (\zeta_{\pm}(t)))|< {\rm{Im}}(f_{n=1}(\zeta_{+}(t)))< \sin\alpha \cos\alpha$,
cf. (\ref{Im+}).

Notice that in spite of the arc-part of the contour
$\partial{D_\alpha}$ shrinks in the limit $n\rightarrow\infty$ to zero, we obtain
\begin{equation}\label{Re-lim}
\lim_{n\rightarrow\infty}\sup_{0\leq t\leq \cos\alpha}{\rm{Re}}(f_n (\zeta_{+}(t))) =
- \exp(- \pi \cot\alpha) ,
\end{equation}
for the left extreme point of the projection on the real axe ($\sin (n\phi(t_{n}^*))=1$) of the image $f_n (D_\alpha)$.
Since $\exp(- \pi \cot\alpha)< \sin\alpha$, for $\alpha \leq \pi/4$, the arguments (\ref{Im-lim}) and (\ref{Re-lim})
bolster the conclusion of the Lemma \ref{z-n}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}{\rm{(Lemma \ref{Euler})}}:\\
By (\ref{F-contr}) we have for $\lambda > 0$
\begin{equation}\label{H-Y-est}
\|(\lambda I + A)^{-1}\| < \lambda^{-1} \ ,
\end{equation}
and since $A$ is $m$-sectorial, we also get that $(-\infty, 0) \subset \rho(A)$.
Then the \textit{Hille-Yosida} theory ensures the existence of the contraction semigroup
$\{e^{-t \, A}\}_{\, t \geq 0}$,
and the standards arguments (see e.g. \cite[Ch.V]{Kato}, or \cite[Ch.1.1]{Zag})
yield the convergence of the
Euler formula (\ref{Eul-s-lim}) in the strong topology.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}{\rm{(Theorem \ref{Th-main})}}:\\
Take $f(z) = z^2$ and the compact convex set $E':=  f(D_{\alpha})\subseteq D_{\alpha}$, see Lemma \ref{z-n}.
Since the set $E:= f^{-1}(E')= D_{\alpha} \cup (-D_{\alpha})$ is \textit{convex}, its convex kernel $K$
exists and $K = E$. Then by Proposition \ref{Kato-map} we obtain that
${\mathfrak{N}}(f(C))\subseteq E'\subseteq D_\alpha$, if the numerical range ${\mathfrak{N}}(C)\subseteq K$.

Let contraction $C_1 := (I + t \,A /2)^{-1}= F(t/2)$. Since by Proposition \ref{Th-res}
for any $t\geq0$ we have ${\mathfrak{N}}(C_1)\subseteq D_{\alpha}$ and since $D_{\alpha}\subset E$,
we can choose  $K = E$. Then by the Kato numerical range mapping theorem (Proposition \ref{Kato-map}) we get:
\begin{equation}\label{1st-step}
{\mathfrak{N}}(f(C_1)= F (t/2)^2)\subseteq E'\subseteq D_{\alpha}  \ .
\end{equation}
Similarly, take the contraction $C_2 := F (t/4)^2$. Since (\ref{1st-step}) is valid for any $t\geq 0$,
it is true for $t \mapsto t/2$.  Then by definition of $K$ one has
${\mathfrak{N}}(F (t/4)^2)\subseteq D_{\alpha} \subseteq K$. Now
again the Proposition \ref{Kato-map} implies:
\begin{equation}\label{2st-step}
{\mathfrak{N}}(f(C_2)= F(t/4)^4)\subseteq E'\subseteq D_{\alpha} \ .
\end{equation}
Therefore, we obtain ${\mathfrak{N}}(F_b(t/2^n)^{2^n})\subseteq D_{\alpha}$, for any $n \in {\mathbb{N}}$.
By Lemma \ref{Euler} this yields
$$
\lim_{n\rightarrow\infty}(u,(I + t \,A /2^n)^{-2^n} u) = (u,e^{- t \, A} u) \in D_{\alpha} \ ,
$$
for any unit vector $u\in\mathfrak{H}$. Therefore, the numerical ranges of the contraction semigroup
${\mathfrak{N}}(e^{- t \, A})\subseteq D_{\alpha}$ for all $t\geq 0$, if it is generated by {$m$-sectorial}
operator with  the semi-angle $\alpha\in[0, \pi/4]$ and with the vertex at $z=0$.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Corollaries and Applications %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Corollaries and Applications}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

1. Notice that Definition \ref{Q-S-cont} of \textit{quasi-sectorial} contractions $C$ is quite
{restrictive} comparing to the notion of \textit{general} contractions, which demands \textit{only}
${\mathfrak{N}}(C)\subseteq \mathfrak{D}_1$. For the latter case one has a well-known
\textit{Chernoff lemma} \cite{Chern-1}:
\begin{equation}\label{Chern-Lemm}
\|(C^n - e^{n(C-I)})u\| \leq n^{1/2}\|(C-I)u\| \ , \ u\in\mathfrak{H} \ , \ n \in \mathbb{N} \ ,
\end{equation}
which is \textit{not} even a convergent bound.
For quasi-sectorial contractions we can obtain a much stronger estimate \cite{CachZag-Q-S}:
\begin{equation}\label{n-1/3}
\left\|C^n - e^{n(C-I)}\right\| \leq {M \ n^{-1/3}} \ \  , \ \ n \in \mathbb{N} \ ,
\end{equation}
convergent to zero in the uniform topology when $n \rightarrow \infty$.
Notice that the rate of convergence $n^{-1/3}$ obtained in \cite{CachZag-Q-S} with
help of the \textit{Poisson representation} and the
\textit{Tchebychev inequality} is not optimal. In \cite{Paul},
\cite{Cach} and \cite{BentPaul} this estimate was improved up to the
\textit{optimal} rate $O(n^{-1})$, which one can easily verify for a particular
case of self-adjoint contractions (i.e. $\alpha=0$) with help of the  spectral
representation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The inequality (\ref{n-1/3}) and its further improvements are based on the following
important result about the upper bound estimate for the case of \textit{quasi-sectorial} contractions:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{C-n-estim}
If $C$ is a quasi-sectorial contraction on a Hilbert space $\mathfrak{H}$ with semi-angle
$0\leq\alpha < \pi/2$, i.e. the numerical range
${\mathfrak{N}}(C)$ is a subset of the domain $D_\alpha$, then
\begin{equation}\label{C1}
\|C^n (I-C)\|\leq \frac{K}{n+1} \ , \ n\in{\mathbb{N}} \ .
\end{equation}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent For the proof see Lemma 3.1 of \cite{CachZag-Q-S}.

2. Another application of quasi-sectorial contractions generalizes the Chernoff semigroup approximation
theory \cite{Chern-1}, \cite{Chern-2} to the operator-norm approximations \cite{CachZag-Q-S}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{norm-appr}
Let $\{\Phi(s)\}_{s\geq 0}$ be a family of uniformly
quasi-sectorial contractions on a Hilbert space $\mathfrak{H}$, i.e. such that there exists
$0<\alpha<\pi/2$ and  ${\mathfrak{N}}(\Phi(s)) \subseteq D_\alpha$, for all $s\geq 0$. Let
$$
X(s):=(I-\Phi(s))/s  \ ,
$$
and let $X_0$ be a closed operator with non-empty resolvent set, defined in a closed subspace
${\mathfrak{H}}_0 \subseteq \mathfrak{H}$. Then the family $\{X(s)\}_{s>0}$ converges,
when $s\rightarrow +0$, in the
uniform resolvent sense to the operator $X_0$  if and only if
\begin{equation}\label{lim}
\lim_{n\rightarrow \infty} \left\|\Phi(t/n)^n -e^{-tX_0}P_0\right\| = 0 \ , \ \ \ \mbox{for} \ t>0 \ .
\end{equation}
Here $P_0$ denotes the orthogonal projection onto the subspace ${\mathfrak{H}}_0$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3. We conclude by application of Theorem \ref{Th-main} and Proposition \ref{C-n-estim}
to the Euler formula \cite{CachZag-Q-S}, \cite{Paul}, \cite{Cach}.
\begin{proposition}\label{norm-appr-Euler}
If $A$ is an $m$-sectorial operator in a Hilbert space
${\mathfrak{H}}$, with semi-angle $\alpha\in [0,\pi/4]$ and with vertex at $z=0$, then
$$
\lim_{n\rightarrow\infty}\left\|(I+tA/n)^{-n} - e^{-tA}\right\| =
0,\ t\in S_{\pi/2-\alpha}.
$$
Moreover, uniformly in $t\geq t_0 > 0$ one has the error estimate:
$$
\left\|(I+tA/n)^{-n} - e^{-tA}\right\| \leq
O\left(n^{-1}\right) \ , \ n\in{\mathbb{N}} \ .
$$
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Acknowledgements}
I would like to thank Professor Mitsuru Uchiyama for a useful remark indicating a
flaw in our arguments in Section 2 of \cite{CachZag-Q-S} , revision of
this part of the paper \cite{CachZag-Q-S} is done in the present manuscript.
I also thankful to Vincent Cachia and Hagen Neidhardt for a pleasant collaboration.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Biblio %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{999}


\bibitem{BentPaul}
V. Bentkus and V. Paulauskas,  Optimal error estimates in operator-norm
approximations of semigroups.  \emph{Lett. Math. Phys.} {\bf 68}
(2004), 131--138.


\bibitem{Cach}
V. Cachia, Euler's exponential formula for semigroups,
\emph{Semigroup Forum} {\bf 68} (2004), 1--24.


\bibitem{CaNeZag}
V. Cachia, H. Neidhardt, V.A. Zagrebnov,  Comments on the Trotter
product formula error-bound estimates for nonself-adjoint
semigroups,  \emph{Integral Equat. Oper. Theory} {\bf 42}
(2002), 425--448.

\bibitem{CachZag-Q-S}
V. Cachia, V.A. Zagrebnov, Operator-norm approximation of semigroups
by quasi-sectorial contractions,  \emph{J. Funct. Anal.} {\bf 180}
(2001), 176--194.

\bibitem{Chern-1}
P.R. Chernoff, Note on product formulas for operator semigroups.
\emph{J. Funct. Anal.} {\bf 2} (1968)  238--242.

\bibitem{Chern-2}
P.R. Chernoff, \emph{Product formulas, nonlinear semigroups, and addition of unbounded operators.}
Memoirs of the American Mathematical Society, No. 140.
American Mathematical Society, Providence, R. I., 1974.


\bibitem{Kato}
T. Kato,  \emph{Perturbation Theory for Linear Operators},
Springer-Verlag, Berlin, 1966.


\bibitem{Kato-map}
T. Kato, Some mapping theorems for the numerical range, \emph{Proc.
Japan Acad.} {\bf 41} (1966), 652--655.

\bibitem{Paul}
V. Paulauskas, On operator-norm approximation of some semigroups by
quasi-sectorial operators, \emph{J. Funct. Anal.} {\bf 207} (2004),
58--67.

\bibitem{Uch}
M. Uchiyama, Numerical ranges of elements of involutive Banach algebras and commutativity,
\emph{Arch.Math.} {\bf 69} (1997), 313--318.

\bibitem{Zag}
V.A. Zagrebnov,  \emph{Topics in the theory of Gibbs semigroups.}
Leuven Notes in Mathematical and Theoretical Physics. Series A:
Mathematical Physics, 10. Leuven University Press, Leuven, 2003.

\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\epsfysize=10cm

$$\epsfbox{CNZ1.eps}$$
\caption{Illustration of the set $D_{\alpha}$($= \Sigma_{a_*}$ shaded domain) with boundary 
$\partial D_{\alpha} = \Gamma_{a_*}$, where $a_* = \sin \alpha$, as well as of our choice of the contour 
$\Gamma_r$ in the resolvent set $\rho(C)$, where $r=\sin \beta >a_*$. The contour $\Gamma_r$ consists of two 
segments of tangent straight lines $(1,A)$ and $(1,B)$ and the arc $(A,B)$ of radius $r$. 
The dotted circle $\partial \mbox{D}_{r=1/2}$ corresponds to the set of tangent points for different 
values of $\alpha\in[0,\pi/2]$.}

\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
---------------0711061840887
Content-Type: application/postscript; name="CNZ1.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="CNZ1.eps"

%!PS-Adobe-2.0 EPSF-2.0
%%Title: CNZ1.fig
%%Creator: fig2dev Version 3.1 Patchlevel 2
%%CreationDate: Mon Apr 17 18:22:31 2000
%%For: cachia@cptsu6 (Vincent Cachia thesard Zagrebnov)
%Magnification: 1.00
%%Orientation: Portrait
%%BoundingBox: 0 0 362 362
%%Pages: 0
%%BeginSetup
%%IncludeFeature: *PageSize Letter
%%EndSetup
%%EndComments
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def
/col32 {0.733 0.741 0.800 srgb} bind def

end
save
-179.0 469.0 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

$F2psBegin
10 setmiterlimit
n 0 792 m 0 0 l 612 0 l 612 792 l cp clip
 0.06000 0.06000 sc
15.000 slw
% Arc
gs n 6001.3 4800.0 1351.3 -56.4 56.4 arcn
gs col7 0.90 shd ef gr gs col-1 s gr
 gr

% Polyline
n 6750 3675 m 8400 4800 l 6750 5925 l gs col7 0.90 shd ef gr gs col-1 s gr 
7.500 slw
% Arc
gs n 8107.5 4657.5 551.2 -123.9 165.0 arcn
gs col-1 s gr
 gr

/Times-Roman ff 210.00 scf sf
3300 5025 m
gs 1 -1 sc (-1) col-1 sh gr
15.000 slw
% Arc
gs n 8175.0 4725.0 167.7 -116.6 153.4 arcn
gs col-1 s gr
 gr

7.500 slw
% Arc
gs n 5983.7 4800.0 1708.7 -44.6 44.6 arcn
gs col-1 s gr
 gr

% Ellipse
n 6000 4800 2400 2400 0 360 DrawEllipse gs col-1 s gr

 [66.7] 0 sd
% Ellipse
n 7200 4800 1200 1200 0 360 DrawEllipse gs col-1 s gr
 [] 0 sd
% Polyline
gs  clippath
5970 1947 m 6000 1827 l 6030 1947 l 6030 1785 l 5970 1785 l  cp clip
n 6000 7800 m 6000 1800 l gs col-1 s gr gr

% arrowhead
n 5970 1947 m 6000 1827 l 6030 1947 l  col-1 s
% Polyline
 [33.3] 0 sd
n 6000 4800 m 6750 3675 l gs col-1 s gr  [] 0 sd
% Polyline
 [33.3] 0 sd
n 6600 3900 m 6825 4050 l 6975 3825 l gs col-1 s gr  [] 0 sd
% Polyline
n 7200 3600 m 8400 4800 l 7200 6000 l gs col-1 s gr 
% Polyline
gs  clippath
5474 3207 m 5350 3216 l 5455 3150 l 5301 3201 l 5320 3258 l  cp clip
n 5550 3150 m 5325 3225 l gs col-1 s gr gr

% arrowhead
n 5474 3207 m 5350 3216 l 5455 3150 l  col-1 s
% Polyline
 [33.3] 0 sd
gs  clippath
7508 3950 m 7444 3844 l 7550 3908 l 7436 3793 l 7393 3836 l  cp clip
n 8400 4800 m 7425 3825 l gs col-1 s gr gr
 [] 0 sd
% arrowhead
n 7508 3950 m 7444 3844 l 7550 3908 l  col-1 s
% Polyline
gs  clippath
6680 6339 m 6800 6312 l 6707 6393 l 6852 6320 l 6825 6266 l  cp clip
n 6675 6375 m 6825 6300 l gs col-1 s gr gr

% arrowhead
n 6680 6339 m 6800 6312 l 6707 6393 l  col-1 s
% Polyline
 [33.3] 0 sd
n 6000 4800 m 7200 6000 l gs col-1 s gr  [] 0 sd
% Polyline
gs  clippath
4942 6100 m 5005 6205 l 4900 6142 l 5014 6257 l 5057 6214 l  cp clip
n 4950 6150 m 5025 6225 l gs col-1 s gr gr

% arrowhead
n 4942 6100 m 5005 6205 l 4900 6142 l  col-1 s
% Polyline
gs  clippath
8853 4770 m 8973 4800 l 8853 4830 l 9015 4830 l 9015 4770 l  cp clip
n 3000 4800 m 9000 4800 l gs col-1 s gr gr

% arrowhead
n 8853 4770 m 8973 4800 l 8853 4830 l  col-1 s
/Times-Roman ff 210.00 scf sf
8475 5025 m
gs 1 -1 sc (1) col-1 sh gr
/Times-Roman ff 180.00 scf sf
4725 5025 m
gs 1 -1 sc (-sin) col-1 sh gr
% Polyline
 [33.3] 0 sd
n 7050 5850 m 7200 5700 l 7350 5850 l gs col-1 s gr  [] 0 sd
/Symbol ff 240.00 scf sf
5325 3000 m
gs 1 -1 sc (G) col-1 sh gr
/Symbol ff 240.00 scf sf
5250 3450 m
gs 1 -1 sc (G) col-1 sh gr
/Times-Italic ff 180.00 scf sf
5325 3525 m
gs 1 -1 sc (a) col-1 sh gr
/Symbol ff 210.00 scf sf
7425 4500 m
gs 1 -1 sc (b) col-1 sh gr
/Symbol ff 180.00 scf sf
4125 5025 m
gs 1 -1 sc (b) col-1 sh gr
/Times-Italic ff 210.00 scf sf
6600 5325 m
gs 1 -1 sc (r) col-1 sh gr
/Symbol ff 180.00 scf sf
5025 5025 m
gs 1 -1 sc (a) col-1 sh gr
/Symbol ff 210.00 scf sf
7800 4650 m
gs 1 -1 sc (a) col-1 sh gr
/Times-Roman ff 150.00 scf sf
7275 5400 m
gs 1 -1 sc (1/2) col-1 sh gr
/Times-Roman ff 240.00 scf sf
7125 5325 m
gs 1 -1 sc (D) col-1 sh gr
/Times-Italic ff 240.00 scf sf
7200 6225 m
gs 1 -1 sc (B) col-1 sh gr
/Times-Italic ff 180.00 scf sf
5325 4350 m
gs 1 -1 sc (a) col-1 sh gr
/Times-Roman ff 150.00 scf sf
5400 4425 m
gs 1 -1 sc (*) col-1 sh gr
/Times-Roman ff 150.00 scf sf
5400 3600 m
gs 1 -1 sc (*) col-1 sh gr
/Times-Italic ff 210.00 scf sf
6375 4500 m
gs 1 -1 sc (a) col-1 sh gr
/Times-Roman ff 180.00 scf sf
6525 4575 m
gs 1 -1 sc (*) col-1 sh gr
/Times-Roman ff 210.00 scf sf
6150 2250 m
gs 1 -1 sc (i) col-1 sh gr
/Times-Roman ff 210.00 scf sf
6150 7500 m
gs 1 -1 sc (-i) col-1 sh gr
/Times-Roman ff 210.00 scf sf
5850 5025 m
gs 1 -1 sc (0) col-1 sh gr
/Times-Italic ff 240.00 scf sf
7200 3525 m
gs 1 -1 sc (A) col-1 sh gr
/Symbol ff 270.00 scf sf
5175 4275 m
gs 1 -1 sc (S) col-1 sh gr
/Times-Italic ff 180.00 scf sf
5475 3075 m
gs 1 -1 sc (r) col-1 sh gr
/Times-Roman ff 180.00 scf sf
3825 5025 m
gs 1 -1 sc (-sin) col-1 sh gr
$F2psEnd
rs
---------------0711061840887--
