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\end{filecontents}{CZ4f3.eps}
\begin{filecontents}{CZ4f2.eps}
%!PS-Adobe-2.0 EPSF-2.0
%%Title: CZ4f2.fig
%%Creator: fig2dev Version 3.1 Patchlevel 2
%%CreationDate: Fri Jan 14 16:51:21 2000
%%For: cachia@cptsu6 (Vincent Cachia thesard Zagrebnov)
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.9 .9 scale % to make patterns same scale as in xfig

% This junk string is used by the show operators
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/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
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    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
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    % Move past the character (charpath modified the
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    % Reposition all characters by rx ry
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  pop pop pop pop pop		% -
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/PATcg {
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    /cm matrix currentmatrix def
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} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index show		% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proe char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
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    % Font dictionary definitions
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    % Create a dummy encoding vector
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    3 string 0 1 255 {
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    /FontBBox BBox def
    /BuildChar {
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	exch begin
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	  PaintType 2 eq Multi 1 ne or
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	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
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	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
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	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
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    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
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    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
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    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
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    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
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} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
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    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
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} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
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\begin{document}


\font\fifteen=cmbx10 at 15pt \font\twelve=cmbx10 at 12pt 

\begin{titlepage} 

\begin{center}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

{\twelve Centre de Physique Th\'eorique\footnote{Unit\'e Propre de
Recherche 7061 }, CNRS Luminy, Case 907}

{\twelve F-13288 Marseille -- Cedex 9}

\vspace{2 cm}

{\fifteen OPERATOR-NORM APPROXIMATION OF SEMIGROUPS BY SECTORIAL CONTRACTIONS}

\vspace{1.5 cm}

\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

{\bf V. CACHIA\footnote{Allocataire-moniteur at Universit\'e de la M\'editerran\'ee (Aix-Marseille II)\\ Email : cachia@cpt.univ-mrs.fr},
and
V.A. ZAGREBNOV\footnote{Universit\'e de la M\'editerran\'ee
(Aix-Marseille II) - Email : zagrebnov@cpt.univ-mrs.fr} }

\vspace{1.5 cm}

{\bf Abstract}

\end{center}

We extend the Chernoff theory for approximation of contraction semigroups \`a la Trotter. We show that the Trotter-Neveu-Kato convergence theorem holds in operator norm for a family of uniformly m-sectorial generators in a Hilbert space. Then we obtain a Chernoff-type approximation theorem for sectorial contractions on a Hilbert space in the operator norm. As examples we give necessary and sufficient conditions for the operator-norm convergence of Trotter-type product formul\ae.


\vspace{1 cm}


\bigskip

\noindent Keywords : Chernoff theory, Trotter product formula, operator-norm convergence, holomorphic semigroups, sectorial operators.

\noindent Mathematics Subject Classification : 47D03, 47B25, 35K22, 41A80



\bigskip
\bigskip


\noindent anonymous ftp :
ftp.cpt.univ-mrs.fr\\
web : www.cpt.univ-mrs.fr\\
May 2000\\
CPT-2000/P.4012

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\end{titlepage}

\setcounter{footnote}{0} \renewcommand{\thefootnote}{\arabic{footnote}}

\newpage




\section{Introduction}

In his article \cite{Chernoff}, Chernoff has proved the following proposition:

\begin{proposition}
Let $F(t)$ be a strongly continuous function from $[0,\infty)$ to the linear contractions on a Banach space $\cal X$ such that $F(0)=I$. Suppose that the closure $C$ of the strong derivative $F'(0)$ is the generator of a contraction semigroup. Then $F(t/n)^n$ converges to $e^{tC}$ in the strong operator topology.
\end{proposition}

This result describes a method for the approximation of a general contraction semigroup. It leads in particular to a simple proof of the strong convergence for the Trotter product formula (see \cite[\S 3.4]{Davies}). For self-adjoint families of operators on a Hilbert space there exists a generalization of the above result in the operator-norm topology \cite{NZ1}, \cite{NZ3}:

\begin{proposition}
Let $\{\Phi(s)\}_{s\geq 0}$ be a family of self-adjoint non-negative contractions on a separable Hilbert space $\cal H$, and let $X_0$ be a self-adjoint operator in the closed subspace ${\cal H}_0\subseteq{\cal H}$. Define $X(s)=s^{-1}(I-\Phi(s))$, $s\geq 0$. Then the family $\{X(s)\}_{s\geq 0}$ converges in the uniform resolvent sense to $X_0$ as $s\rightarrow +0$ if and only if the sequence $\{\Phi(t/r)^r\}_{r\geq 1}$, $t\geq 0$, converges in operator-norm to $e^{-tX_0}P_0$ as $r\rightarrow +\infty$, uniformly on any compact $t$-interval in $(0,\infty)$. Here $P_0$ denotes the orthogonal projection from ${\cal H}$ onto ${\cal H}_0$.
\end{proposition}

The aim of the present paper is to show that the condition of self-adjointness in this result can be replaced by a much weaker condition on the corresponding numerical range. Let $\cal H$ be a separable Hilbert space and let $T$ be a linear operator in $\cal H$. We denote by $\Theta(T)$ the numerical range of $T$ :
%
$$ \Theta(T) = \{ (Tu,u), u\in{\cal D}(T) \mbox{ and } \|u\|=1 \}. $$
%
%The Toeplitz-Hausdorff theorem states that it is a convex set (see e.g. \cite{Halmos} Ch.17). It follows that the complement in $\bb C$ of its closure is connected, except in the special case where $\Theta(T)$ is a strip bounded by two parallel lines.
%
Below we use some important properties of this set (cf \cite[Ch.V Theorem 3.2]{Kato}):

\begin{proposition}\label{propnumrange}
If $T$ is a closed operator in $\cal H$, then for any complex number $z\notin \overline{\Theta(T)}$, $T-z$ has closed range, it is injective and its deficiency ${\mathrm def}(T-z)$ is constant (in each connected component). If ${\mathrm def}(T-z) = 0$, then the spectrum of $T$ is a subset of $\overline{\Theta(T)}$ and
\begin{equation}\label{estres}
\|(T-z)^{-1}\| \leq {1\over dist(z,\Theta(T))}.
\end{equation}
\end{proposition}

For $0 < \alpha \leq \pi$, we denote by $S_\alpha$ the open sector
$$ S_\alpha = \{z\in{\mathbb C}\setminus\{0\}: |\arg z|<\alpha \}.$$

\begin{definition}
Operator $T$ in $\cal H$ is said to be sectorial with a semi-angle $\alpha\in (0,\pi/2)$ and with vertex at $0$ if $\Theta(T)\subseteq S_\alpha$. If, in addition, $T$ is closed and there exists $z\notin S_\alpha$ that belongs to the resolvent set of $T$, then $T$ is said to be m-sectorial.
\end{definition}
%
We recall the following result (see \cite[Ch.IX Theorem 1.24]{Kato}):
%
\begin{proposition}\label{P1}
Let $T$ be an m-sectorial operator in a Hilbert space $\cal H$, with a semi-angle $\alpha \in (0,\pi/2)$ and with vertex at $0$. Then the semigroup $e^{-tT}$ is holomorphic in $S_{\pi/2-\alpha}$ and it is bounded by $\|e^{-tT}\|\leq 1$ for $t\in S_{\pi/2 - \alpha}$.
\end{proposition}

We define for $0 \leq \alpha \leq\pi/2$
\begin{equation}\label{Dalpha}
D_\alpha = \{z\in{\mathbb C}: |z|\leq \sin\alpha\} \cup \{z\in{\mathbb C}: |\arg(1-z)|\leq \alpha \mbox{ and } |z-1|\leq\cos\alpha\}.
\end{equation}

\begin{definition}\label{SC}
We shall say that the contraction $C$ on a Hilbert space $\cal H$ is sectorial with a semi-angle $0 \leq\alpha <\pi/2$ (and with vertex at 1), if its numerical range $\Theta(C)$ is a subset of $D_\alpha$. Notice that the case $\alpha = 0$ corresponds to a non-negative self-adjoint contraction.
\end{definition}

Let $\{\Phi(s)\}_{s\geq 0}$ be a family of sectorial contractions on a Hilbert space $\cal H$ such that $\Theta(\Phi(s))\subseteq D_\alpha$ for some $0\leq \alpha<\pi/2$ and for all $s\geq 0$. Let $X(s)=(I-\Phi(s))/s$, and let $X_0$ be a closed operator in a closed subspace ${\cal H}_0\subseteq{\cal H}$ with non-empty resolvent set. Then the main result of this paper (see Theorem \ref{T})  can be formulated as the equivalence between two limits:
\begin{eqnarray}
\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\| & \longrightarrow & 0 \mbox{ as } s\rightarrow +0,\ \mbox{Re}\,\zeta>0, \label{lim1}\\
\|\Phi(t/n)^n - e^{-tX_0}P_0\| & \longrightarrow & 0 \mbox{ as } n\rightarrow \infty,\ t> 0.\label{lim2}
\end{eqnarray}


Our method is an operator-norm generalization of the Chernoff theory \cite{Chernoff}. Definition \ref{SC} will be illustrated in the next section. Then we generalize Lemma 2 of \cite{Chernoff} to the operator-norm topology. In Section \ref{appth} we prove the operator-norm analogue of the Trotter-Neveu-Kato convergence theorem (see \cite[Theorem 5.1]{Trotter1}, or e.g. \cite[Theorem 3.17]{Davies}, \cite[Ch. 1.7]{Goldstein}) and the main result (\ref{lim1}),(\ref{lim2}) formulated above. Finally, we give some applications concerning Trotter-Kato formul\ae.


\section{Sectorial contractions}\label{sec1}

In this section we give some important examples of operator families corresponding to Definition \ref{SC}, i.e. sectorial contractions. Let $A$ be an m-sectorial operator with a semi-angle $\alpha$ and with vertex at $0$, in a Hilbert space $\cal H$. Then there are two families of sectorial contractions that are generated by $A$ in a natural way.

\subsection{Resolvent of a sectorial operator}\label{cfex1}

The family of resolvents $F(t) = (I+tA)^{-1}$, $t\geq 0$, has the following properties:
\smallskip

$(i)$ Since $F(t=0)=I$ and since for $t>0$
\begin{equation}
\|F(t)\| \leq {1\over t\,\mbox{dist}(1/t,-S_\alpha)} = 1,
\end{equation}
\hspace{1.5cm}$F(t)$ is a family of contractions;

$(ii)$ $\Theta(F(t))\subseteq S_\alpha$, because for any $u\in\cal H$:
\begin{equation}
(u,F(t)u) =  (v,v) + t(Av,v)\in S_\alpha, \end{equation}
\hspace{1.5cm} where $v=(I+tA)^{-1}u$.
\smallskip

$(iii)$ $\Theta(I-F(t))\subseteq S_\alpha$, since for any $u\in\cal H$
\begin{equation}
(u,(I-F(t))u) = ((I+tA)v,tAv) = t(v,Av) + t^2(Av,Av)\in S_\alpha,
\end{equation}
\hspace{1.5cm} where $v=(I+tA)^{-1}u$.
\smallskip

Therefore, we have $\Theta(F(t))\subseteq S_\alpha\cap (1-S_\alpha) \subseteq D_\alpha$, i.e. $F(t)$ are sectorial contractions for $t\geq 0$.


\subsection{Semigroup generated by a sectorial operator}

The semigroup generated by the m-sectorial operator $A$ is holomorphic and contractive in the sector $S_{\pi/2-\alpha}$ (cf Proposition \ref{P1}). We shall show that in fact $\Theta(e^{-tA})\subseteq D_\alpha$ for $t\geq 0$. It is not as simple as for the resolvent. The key statement is the following mapping theorem for the numerical range due to Kato \cite{K1}:

\begin{proposition}\label{mapth}
Let $f(z)$  be a rational function with $f(\infty)=\infty$. Let $E'$ be a compact convex set in the complex plane, let $E=f^{-1}(E')$ and $K$ be the convex kernel of $E$. If $A$ is an operator with $\Theta(A)\subseteq K$, then $\Theta(f(A))\subseteq E'$. In particular if $D$ is a compact convex subset of $\mathbb C$ with $f(D)\subseteq D$ and $\Theta(A)\subseteq D$, then $\Theta(f(A))\subseteq D$.
\end{proposition}

\begin{lemma}\label{z^n}
Let $f(z) = z^n$ where $n\in{\mathbb N}$, then $f(D_\alpha)\subseteq D_\alpha$.
\end{lemma}

\begin{proof}
If $|z|\leq \sin\alpha$, then $|z^n|\leq \sin\alpha$ and $z^n\in D_\alpha$. Thus it remains the case $z\in D_\alpha$, $|z|>\sin\alpha$. Let us consider the family of the straight lines $z(t)=1-te^{i\beta}$ in $D_\alpha$, parametrized by  $0\leq t\leq \cos\alpha$ and $-\alpha\leq \beta \leq\alpha$. We study their images defined by $\zeta(t) = z(t)^n$ to prove that they also lie in $D_\alpha$. They form a regular plane curve, i.e. $\zeta'(t) = -ne^{i\beta}(1-te^{i\beta})^{n-1} \neq 0$. Then for each path $\zeta(t)$ we can define unit tangent vectors $T(t)$:

\begin{equation}\label{tgvector}
T(t) = \frac{\zeta'(t)}{|\zeta'(t)|} = -e^{i\beta} \left(\frac{1-te^{i\beta}}{|1-te^{i\beta}|} \right)^{n-1}.
\end{equation}

The curvature $c(t)$ of a regular plane curve is defined by $T'(t) = ic(t)T(t)$, and we find by (\ref{tgvector}):
\begin{equation}
\hspace{-0.5cm} T'(t) = -e^{i\beta}(n-1)\left(\frac{z}{|z|} \right)^{n-2} \frac{d}{dt}\left(\frac{z}{|z|} \right) = i(n-1) \frac{\sin\beta}{|z|^2} e^{i\beta} \left(\frac{z}{|z|} \right)^{n-1} \!\!\!\!\!\!,
\end{equation}
here $z=z(t)=1-te^{i\beta}$. Hence the expression for the curvature is $c(t)=-(n-1)\sin\beta /|z(t)|^2$. It does not vanish for any $0\leq t\leq \cos\alpha$ and it has the sign opposite to $\beta$. Therefore, the curve $\zeta(t)$ lies in the half-plane defined by any tangent and containing the corresponding curvature center. Let $t=0$, then the tangent is $z(t)$, and the curvature center is at $1+ie^{i\beta}(n-1)\sin\beta$. Finally, all images $\zeta(t)=z(t)^n$, for $0\leq t\leq \cos\alpha$ and $-\alpha\leq \beta\leq \alpha$, are squeezed between two extreme tangent lines corresponding to $\beta = \pm\alpha$. This finishes the proof.

\end{proof}

\begin{theorem}
If $A$ is an m-sectorial operator with a semi-angle $\alpha$ and with vertex at $0$, then $\Theta(e^{-tA})$ is a subset of $D_\alpha$ for any $t\geq 0$.
\end{theorem}

\begin{proof}
By virtue of example from Section \ref{cfex1} we have that $\Theta((I+tA/n)^{-1}) \subseteq D_\alpha$ for any positive real $t$ and any non-zero integer $n$. According to Proposition \ref{mapth} and to Lemma \ref{z^n} for the rational function $f(z) = z^n$ ($f(\infty) = \infty$), the operator $(I+tA/n)^{-1}$, and the compact convex set $D_\alpha$, we obtain that $\Theta((I+tA/n)^{-n}) \subseteq D_\alpha$ for any $t\geq 0$ and $n\in {\mathbb N}\setminus\{0\}$.

On the other hand it is known (see \cite[Ch.IX]{Kato}) that the sequence $(I+tA/n)^{-n}$ converges strongly to the semigroup $e^{-tA}$ as $n\rightarrow \infty$. Therefore, $\lim_{n\rightarrow\infty} ((I+tA/n)^{-n}u,u) = (e^{-tA}u,u) \in D_\alpha$ for any unit vector $u\in {\cal H}$, which proves the assertion.

\end{proof}



\section{Generalization of the Chernoff lemma}

In this section we show that an operator-norm analogue of the main Lemma 2 of \cite{Chernoff} is valid if one supposes that the contraction $C$ is sectorial in the sense of Definition \ref{SC}.

\begin{figure}[h]
\epsfysize=10cm

\setlength{\unitlength}{0.1bp}

$$
\begin{picture}(3000,3000)(0,0)
\put(1200,2300){\makebox(0,0){\small $\partial D_{\alpha'}$}}
\put(1050,2050){\makebox(0,0){\small $\partial D_\alpha$}}
\epsfbox{CZ4f3.eps}
\end{picture}
$$

\caption{Illustration of the set $D_{\alpha}$ (shaded domain) with boundary  $\partial D_{\alpha}$, $a = \sin\alpha$, as well as of our choice of the contour $\partial D_{\alpha'}$ in the resolvent set $\rho(C)$, where $a'=\sin\alpha' >a$. The contour consists of two segments of tangent straight lines $(1,A)$ and $(1,B)$ and the arc $(A,B)$ of radius $a'$. The dotted circle corresponds to the set of tangency points for the different values of $\alpha\in[0,\pi/2]$.}

\end{figure}


\begin{lemma}\label{LC1}
If $C$ is a contraction on a Hilbert space $\cal H$, and there exists $0\leq\alpha < \pi/2$ such that the numerical range $\Theta(C)$ is a subset of the domain $D_\alpha$, then
\begin{equation}\label{C1}
\|C^n (I-C)\|\leq {K\over n+1},\ n\in{\mathbb N}.
\end{equation}
\end{lemma}

\begin{proof}

Since the operator $C$ is bounded, its spectrum is a subset of the closure of the numerical range $\overline{\Theta(C)}$. Then taking $\alpha< \alpha'< \pi/2$ we can choose a contour $\partial D_{\alpha'}$ outside $D_\alpha$, but inside the unit circle (see Figure 1), such that by the Dunford-Taylor formula one has:
\begin{equation}\label{Cauchy}
C^n (I-C) = {1\over 2\pi i} \int_{\partial D_{\alpha'}}{z^n (1-z)\over z-C}dz.
\end{equation}

By Proposition \ref{propnumrange} (see (\ref{estres})) we obtain:
$\|(z-C)^{-1}\|\leq \mbox{dist}(z,D_\alpha)^{-1}$. Thus $\|(z+C)^{-1}\| \leq (\cos\alpha'\sin(\alpha' - \alpha))^{-1}$ for $|\arg z|\geq\pi/2-\alpha'$, and $\|(z+C)^{-1}\| \leq (|1-z|\sin(\alpha'-\alpha))^{-1}$ for $|\arg z|\leq \pi/2-\alpha'$. We consider the following parametrization of $\partial D_{\alpha'}$: for the arc $AB$, we take $z(t)=e^{it}\sin\alpha'$ with $\pi/2-\alpha'\leq t\leq 3\pi/2+\alpha'$, and for the straight lines $(1,A)$, $(1,B)$, we put correspondingly $z_{\mp}(s)=1 - se^{\mp i\alpha'}$ with $0\leq s\leq\cos\alpha'$. Then
\begin{eqnarray}
\|C^n (I-C)\| & \leq & {1\over 2\pi} \int_{{\pi\over 2}-\alpha'}^{{3\pi\over 2}+\alpha'}\!\!{|\sin\alpha'|^{n+1} |1-e^{it}\sin\alpha'|\over \cos\alpha'\sin(\alpha'-\alpha)}dt + \nonumber\\
& & + {1\over\pi} \int_0^{\cos\alpha'}{|(1-e^{i\alpha'}s)^n e^{i\alpha'}s|\over s(\sin(\alpha'-\alpha))}ds \label{para}\\
 &\hspace{-4cm} \leq & \hspace{-2cm} {2(\sin\alpha')^{n+1}\over\cos\alpha'\sin(\alpha'-\alpha)} + \int_0^{\cos\alpha'}{\left((1-s\cos\alpha')^2 + s^2(\sin\alpha')^2\right)^{n/2} \over\pi\sin(\alpha'-\alpha)}ds.\nonumber
\end{eqnarray}

By convexity (for $0\leq s\leq\cos\alpha'$) one gets that
$$
(1-s\cos\alpha')^2 + s^2(\sin\alpha')^2\leq 1-s\cos\alpha',
$$
which leads to the inequality:
\begin{eqnarray*}
\int_0^{\cos\alpha'}\left((1-s\cos\alpha')^2 + s^2(\sin\alpha')^2\right)^{n/2}ds & \leq & \int_0^{\cos\alpha'}(1-s\cos\alpha')^{n/2}ds \\
\int_{(\sin\alpha')^2}^1 u^{n/2} {du\over\cos\alpha'}
& \leq & {1-(\sin\alpha')^{n+2}\over (n/2+1)\cos\alpha'}.
\end{eqnarray*}

Therefore, by (\ref{para}) we get the estimate (\ref{C1})
\begin{equation}
\|C^n (I-C)\| \leq {2(\sin\alpha')^{n+1}\over\cos\alpha'\sin(\alpha'-\alpha)} + 2{1-(\sin\alpha')^{n+2}\over \pi(n+2)\cos\alpha'\sin(\alpha'-\alpha)} \leq {K\over n+1},
\end{equation}
where
\begin{equation}\label{K}
K={2\over\cos\alpha'\sin(\alpha'-\alpha)}\left({1\over\pi} - {1\over e\ln(\sin\alpha')}\right).
\end{equation}
\end{proof}



\begin{lemma}\label{LC2}
If $C$ is a contraction on a Hilbert space $\cal H$ that satisfies the estimate (\ref{C1}), then:
\begin{equation}\label{C2}
\|C^n-e^{n(C-I)}\|\leq {2K+2\over n^{1/3}},\ n\in {\mathbb N}\setminus\{0\}.
\end{equation}
\end{lemma}

\begin{proof}

Since the operator $C$ is bounded, we have the representation:
\begin{equation}\label{somme}
C^n-e^{n(C-I)} = e^{-n}\sum_{m=0}^\infty {n^m\over m!}(C^n-C^m).
\end{equation}
Let $\epsilon_n=n^{2/3}$, $n\geq 1$. We divide the sum (\ref{somme}) into two parts: the central part for $|m-n|\leq\epsilon_n$ and tails for $|m-n|>\epsilon_n$. We estimate tails by using the Tchebychev inequality (see e.g. \cite{Shiryaev}). Let $X$ be a Poisson random variable of parameter $n$, i.e. $\mbox{P}(X=m) = n^m e^{-n}/m!$. Then expectation $\mbox{E}(X)=n$ and variance $\mbox{Var}(X)=n$. Therefore, by the Tchebychev inequality:
\begin{equation}
\forall \epsilon>0,\ \mbox{P}(|X-n|>\epsilon) \leq {n\over\epsilon^2},
\end{equation}
and hence
\begin{equation}\label{part1}
e^{-n}\sum_{|m-n|>\epsilon_n} {n^m\over m!}\|C^n-C^m\| \leq {2n\over\epsilon_n^2} = {2\over n^{1/3}}.
\end{equation}
To estimate the central part of the sum (\ref{somme}), where $|m-n|\leq\epsilon_n$, we use (here $[\cdot]$ denotes the integer part):
\begin{eqnarray*}
\|C^n-C^m\| & = & \left\|C^{n-[\epsilon_n]}\left(C^{[\epsilon_n]}-C^{m-n+[\epsilon_n]}\right) \right\|\\
& \leq & |m-n| \|C^{n-[\epsilon_n]}(I-C)\| \\
& \leq & \epsilon_n\ {K\over n-[\epsilon_n]+1}\leq {2K\over n^{1/3}},
\end{eqnarray*}
and we estimate the sum by $1$. Then, together with (\ref{part1}), we obtain (\ref{C2}):
\begin{equation}
\|C^n-e^{n(C-I)}\| \leq {2K+2\over n^{1/3}}
\end{equation}
for $n\in {\mathbb N}\setminus\{0\}$.

\end{proof}

The next statement is an extension of the famous ``$n^{1/2}$-Lemma'' by Chernoff \cite[Lemma 2]{Chernoff}. It follows directly from Lemmata \ref{LC1}, \ref{LC2} and it gives an operator-norm estimate instead of the well-known strong convergence \cite{Chernoff}.

\begin{theorem}\label{T1}
Let $C$ be a sectorial contraction on $\cal H$ with numerical range $\Theta(C)\subseteq D_\alpha$, $0\leq \alpha <\pi/2$. Then
\begin{equation}\label{eqT1}
\left\|C^n - e^{n(C-I)}\right\| \leq {M\over n^{1/3}},\ n=1,2,3,\dots
\end{equation}
where $M=2K+2$ and $K$ is defined by (\ref{K}).
\end{theorem}


\begin{remark}
If no estimate is required, then the operator-norm convergence of the difference (\ref{eqT1}) to zero follows from the Lebesgue dominated convergence theorem for the Cauchy integral for $C^n - e^{n(C-I)}$ on the contour $\partial D_{\alpha'}$, cf (\ref{Cauchy}).
\end{remark}

\begin{remark}
If $C$ is self-adjoint and non-negative (i.e. $\alpha=0$), then directly by the spectral theorem one obtains (cf \cite{NZ1},\cite{NZ3}):
\begin{equation}
\|C^n (I-C)\|\leq {1\over n+1}\ \mbox{and}\ \|C^n - e^{n(C-I)}\|\leq {1\over n}.
\end{equation}
\end{remark}


\section{Approximation Theorem}\label{appth}

In this section we present the proof of our main Theorem. The first step was the Theorem \ref{T1}, the second step is an operator-norm generalization of the Trotter-Neveu-Kato convergence theorem, see e.g. \cite[Ch. 3.3]{Davies}, \cite[Ch. 1.7]{Goldstein}. In the self-adjoint case, such a generalization in the operator-norm topology is obtained in \cite[Lemma 2.1]{NZ3}. For m-sectorial operators we prove it below.

\begin{lemma}\label{Trgene}
Let $\{X(s)\}_{s>0}$ be a family of m-sectorial operators in a Hilbert space $\cal H$ with $\Theta(X(s))\subseteq S_\alpha$ for some $0< \alpha <\pi/2$ and for all $s>0$. Let $X_0$ be an m-sectorial operator defined in a closed subspace ${\cal H}_0\subseteq{\cal H}$, with $\Theta(X_0)\subseteq S_\alpha$. Then the following conditions are equivalent:
\begin{eqnarray}
(a) & & \lim_{s\rightarrow +0} \left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\| = 0, \mbox{ for } {\mathrm Re}\,\zeta>0; \label{(a)}\\
(b) & & \lim_{s\rightarrow +0} \left\|e^{-tX(s)} - e^{-tX_0}P_0\right\| = 0, \mbox{ for } t> 0.\label{(b)}
\end{eqnarray}
Here $P_0$ denotes the orthogonal projection from $\cal H$ onto ${\cal H}_0$.
\end{lemma}

\begin{proof}

$(a)\Rightarrow(b)$

\begin{figure}[H]
\epsfysize=9cm
$$\epsfbox{CZ4f2.eps}$$
\caption{Illustration of the path $\Gamma$.}
\end{figure}


Since $\{X(s)\}_{s>0}$ are m-sectorial with $\Theta(X(s))\subseteq S_\alpha$, the Dunford-Taylor formula
\begin{equation}
e^{-tX(s)} = {1\over 2\pi i} \int_\Gamma d\zeta {e^{t\zeta}\over\zeta + X(s)}
\end{equation}
defines a family of holomorphic semigroups $\{e^{-tX(s)}\}_{s>0}$ with $t\in S_{\pi/2-\alpha}$. Here $\Gamma\subset S_{\pi-\alpha}$ is a positively-oriented closed (at infinity) path around $-S_\alpha$, i.e. with the spectrum $\sigma(-X(s))\subseteq\overline{\Theta(-X(s))}$ in its interior (see Figure 2). The same is true for the operator $X_0$:
\begin{equation}
e^{-tX_0}P_0 = {1\over 2\pi i}\int_\Gamma d\zeta {e^{t\zeta}\over \zeta + X_0}P_0.
\end{equation}


We set $\Gamma= \Gamma_\epsilon \cup\Gamma_\delta \cup\overline{\Gamma_\epsilon}$, where the arc $\Gamma_\delta = \{z\in{\mathbb C}:  |z|=\delta>0, |\arg z|\leq \pi-\alpha-\epsilon\}$ (for $0<\epsilon<\pi/2-\alpha$) and $\Gamma_\epsilon,\overline{\Gamma_\epsilon}$ are two conjugate straight lines, $\Gamma_\epsilon = \{z\in{\mathbb C},\ \arg z = \pi-\alpha-\epsilon,\ |z|\geq \delta\}$. Then for $t>0$ one gets

\begin{equation}\label{esa1}
\left\| e^{-tX(s)} - e^{-tX_0}P_0\right\| \leq {1\over 2\pi} \int_\Gamma |d\zeta||e^{t\zeta}| \left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\|.
\end{equation}

Since operators $X(s)$ and $X_0$ are m-sectorial, by Proposition \ref{propnumrange} one gets the estimate
\begin{equation}
\left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\| \leq {2\over\mbox{dist}(\zeta, -S_\alpha)},
\end{equation}
which implies
\begin{equation}\label{esa2}
\sup_{\zeta\in\Gamma} \left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\| \leq {2\over \delta \sin\epsilon}.
\end{equation}

Since for the path $\Gamma = \Gamma_\epsilon\cup\Gamma_\delta\cup\overline{\Gamma_\epsilon}$, the integral
$I_t = \int_\Gamma |d\zeta| |e^{t\zeta}|$ converges for any $t>0$, then condition $(a)$, estimates (\ref{esa1}), (\ref{esa2}) and the Lebesgue dominated convergence theorem proves $(b)$.

$(b)\Rightarrow (a)$\\
By the Laplace transform for semigroups $e^{-tX(s)}$ and $e^{-tX_0}$, one gets for $\mbox{Re}\,\zeta>0$:
%
\begin{equation}\label{Laplace}
(\zeta+X(s))^{-1} - (\zeta + X_0)^{-1}P_0 = \int_0^\infty dt e^{-t\zeta}\left(e^{-tX(s)} - e^{-tX_0}P_0\right).
\end{equation}
Since the generators are m-sectorial with vertex at $0$, the corresponding semigroups are contractions: $\|e^{-tX(s)}\|\leq 1$, $\|e^{-tX_0}P_0\|\leq 1$ for $t\in S_{\pi/2-\alpha}$, $s>0$. Then the Lebesgue dominated convergence theorem ensures $(a)$ for $\mbox{Re }\zeta>0$, due to the estimate
\begin{eqnarray}
& & \left\|(\zeta + X(s))^{-1} - (\zeta +X_0)^{-1}P_0\right\| \leq \nonumber\\
& & \hspace{2cm} \int_0^\infty dt e^{-t{\mathrm Re}\,\zeta} \left\|e^{-tX(s)} - e^{-tX_0}P_0\right\|
\end{eqnarray}
and the pointwise $\|.\|$-convergence $(b)$.
\end{proof}


\begin{remark}
The Vitali theorem concerning the convergence of holomorphic (operator-valued) functions (see e.g. \cite[Theorem 3.14.1]{HP}) allows to extend the pointwise convergences in $(a)$ and in $(b)$ because the corresponding functions are holomorphic and uniformly bounded. In particular, we can replace the conditions $(a)$ and $(b)$ respectively by 
\begin{eqnarray*}
& (\tilde{a}) &  \lim_{s\rightarrow +0} \left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\| = 0, \\
& & \mbox{ uniformly on compacts in } S_{\pi-\alpha};\\
& (\tilde{b}) & \lim_{s\rightarrow +0} \left\|e^{-tX(s)} - e^{-tX_0}P_0\right\| = 0, \\
& & \mbox{ uniformly on compacts in } S_{\pi/2-\alpha}.
\end{eqnarray*}
\end{remark}

In fact, we can prove the equivalence in Lemma \ref{Trgene} under weaker conditions:

\begin{corollary}\label{approx}
Let $\{X(s)\}_{s>0}$ be a family of m-sectorial operators with $\Theta(X(s))\subseteq S_\alpha$ for some $0< \alpha <\pi/2$ and for all $s>0$. Let $X_0$ be a closed operator in ${\cal H}_0\subseteq{\cal H}$ with non-empty resolvent set. Then the following conditions are equivalent:
\begin{eqnarray*}
(a') & & \lim_{s\rightarrow +0} \left\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}P_0\right\| = 0, \mbox{ for some } \zeta\in S_{\pi-\alpha};\\
(b') & & \lim_{s\rightarrow +0} \left\|e^{-tX(s)} - e^{-tX_0}P_0\right\| = 0, \mbox{ for t in a subset of } (0,+\infty) \\
\end{eqnarray*}
having a limit point, if $X_0$ is generator of a semigroup, i.e. $e^{-tX_0}$ is well defined at least for $t\geq 0$.
\end{corollary}

\begin{proof}

It is sufficient to check that condition $(a')$ implies: $(a)$ holds and the operator $X_0$ is such that $\Theta(X_0)\subseteq S_\alpha$ for some $\alpha\in (0,\pi/2)$, and that condition $(b')$ implies $(b)$ for the same operator $X_0$.

By \cite[Ch.IV, Theorem 2.25]{Kato}, the condition $(a')$ is sufficient to have a generalized convergence in the sense of Kato. On the other hand, this generalized convergence implies that $(a')$ holds for any $\zeta$ in the resolvent set of $X_0$. Since it is non-empty and open, it has a limit point. Then we can apply the theorem of Vitali, since the norms $\|(\zeta + X(s))^{-1}\|$ are uniformly bounded in $s$ for $\zeta-\epsilon \in S_{\pi-\alpha}$ (for any fixed $\epsilon>0$). Thus we obtain the pointwise convergence for any $\zeta\in S_{\pi-\alpha}$, and in particular, $(a)$ holds for the operator $X_0$.

In order to show that $X_0$ is necessarily m-sectorial with $\Theta(X_0)\subseteq S_\alpha$, we use that the norm-convergence (\ref{(a)}) implies:
\begin{equation}
\|(\zeta+X_0)^{-1}P_0\| = \lim_{s\rightarrow 0}\|(\zeta+X(s))^{-1}\| \leq {1\over\mbox{dist}(\zeta, -S_\alpha)}.
\end{equation}
In the case $\mbox{Re}\,\zeta>0$, we obtain ${\|(\zeta+X_0)^{-1}\|}_0\leq 1/\mbox{Re}\,\zeta$, where ${\|\cdot\|}_0$ is the operator-norm in ${\cal H}_0$. This estimate implies that $X_0$ generates a contraction semigroup, and then is m-accretive. By the same way one checks that $e^{i\varphi}X_0$ is m-accretive for any $\alpha-\pi/2 \leq\varphi\leq \pi/2-\alpha$, which proves that $X_0$ is m-sectorial with semi-angle $\alpha$ and with vertex at $0$.

The implication $(b')\Rightarrow (b)$ is a direct consequence of the Vitali theorem. By $\|e^{-tX(s)}\|\leq 1$ for any $t\in S_{\pi/2-\alpha}$ and $s>0$, and by $(b')$ one gets that the functions $\{e^{-tX(s)}\}_{s>0}$ converge in operator-norm as $s\rightarrow +0$ for any $t\in S_{\pi/2-\alpha}$. Therefore, it follows that the limit $e^{-tX_0}$ is a holomorphic semigroup with $\|e^{-tX_0}\|_0 = \|e^{-tX_0}P_0\|\leq 1$ for $t\in S_{\pi/2-\alpha}$, and then operator $X_0$ is m-sectorial with semi-angle $\alpha$ and with vertex at $0$, in ${\cal H}_0 = P_0{\cal H}$.
\end{proof}

It is well-known that the convergence in the strong resolvent sense is well-adapted to the problem of semigroup approximation in the strong topology (see e.g. \cite[Ch. 3.3]{Davies} or \cite[Ch. 1.7]{Goldstein}). Similarly, the above results show that the convergence in the uniform resolvent sense (i.e. the generalized convergence in the sense of Kato \cite[Ch. IV.6]{Kato}) is well-adapted to the semigroup approximation in the operator-norm topology. This topology is natural for holomorphic contraction semigroups, which are continuous in the operator norm outside $0$. We show that the corresponding class of generators (the m-sectorial operators) is in some sense stable for the convergence we consider.

Now we can prove our main Theorem:

\begin{theorem}\label{T}
Let $\{\Phi(s)\}_{s\geq 0}$ be a family of sectorial contractions on a Hilbert space $\cal H$. Let there exists $0<\alpha<\pi/2$ such that $\Theta(\Phi(s)) \subseteq D_\alpha$, 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 ${\cal H}_0\subseteq{\cal H}$. Then the family $\{X(s)\}_{s>0}$ converges in the uniform resolvent sense to $X_0$ as $s\rightarrow +0$ if and only if
\begin{equation}\label{lim}
\lim_{n\rightarrow \infty} \left\|\Phi(t/n)^n -e^{-tX_0}P_0\right\| = 0,\ t>0.
\end{equation}
Here $P_0$ denotes the orthogonal projection onto ${\cal H}_0$.\end{theorem}

\begin{proof}

Necessity:
Notice that $\Theta(\Phi(s))\subseteq D_\alpha$ means that $\Theta(X(s)) \subseteq S_\alpha$ for any $s>0$. Thus, the convergence of $X(s)$ in the uniform resolvent sense coincides with the condition $(a)$ of Lemma \ref{Trgene}. Since
\begin{eqnarray}
\|\Phi(t/n)^n - e^{-tX_0}P_0\| & \leq & \nonumber\\
& & \hspace{-3cm} \|\Phi(t/n)^n - e^{-n(I-\Phi(t/n))}\| + \|e^{-n(I-\Phi(t/n))} - e^{-tX_0}P_0\|,\label{eqT}
\end{eqnarray}
the assertion follows from Theorem \ref{T1} for $C=\Phi(t/n)$ and part $(a)\Rightarrow(b)$ Lemma \ref{Trgene}.

Sufficiency:
We have to estimate the difference
\begin{eqnarray}
\|e^{-tX(t/n)} - e^{-tX_0}P_0\| & \leq & \nonumber\\
& & \hspace{-3cm} \|e^{-tX(t/n)} - \Phi(t/n)^n\| + \|\Phi(t/n)^n - e^{-tX_0}P_0\|,\ t>0.\label{eq34}
\end{eqnarray}
The first term in the right-hand side of (\ref{eq34}) is estimated by Theorem \ref{T1} for $C=\Phi(t/n)$. The second term is supposed to tend to zero by (\ref{lim}). Then the assertion follows  from the part $(b)\Rightarrow (a)$ of Lemma \ref{Trgene}.

\end{proof}

\begin{remark}
In fact, it is sufficient to have $(a')$, in order to obtain (\ref{lim}). On the other hand, it is sufficient to have (\ref{lim}) for $t$ in a subset of $(0,+\infty)$ having a limit point, in order to obtain the  convergence $(a)$. This, as well as the fact that convergence (\ref{lim}) implies that $e^{-tX_0}$ is a holomorphic contraction semigroup for $t\in S_{\pi/2-\alpha}$, follows from the Corollary \ref{approx}.
\end{remark}


\section{Applications.}

\subsection{Error estimate for the Euler approximation of semigroups}

Let $A$ be an m-sectorial operator with a semi-angle $\alpha$ and with vertex at $0$, $0< \alpha<\pi/2$. Then the operators $F(t) = (I+tA)^{-1}$, $t\geq 0$ are sectorial contractions, i.e. $\Theta(F(t))\subseteq D_\alpha$ (cf Section \ref{cfex1}).
Let $X(s) = (I-F(s))/s$, $s>0$, and $X_0=A$. Then $X(s)$ converges when $s\rightarrow +0$ to $X_0$ in the uniform resolvent sense:
\begin{equation}\label{cond1}
\|(\zeta+X(s))^{-1} - (\zeta+X_0)^{-1}\| = s\left\| {A\over\zeta+A+\zeta sA}\cdot{A\over\zeta+A}\right\| = O(s),\ \zeta\in S_{\pi-\alpha},
\end{equation}
since we have the estimate
\begin{equation}\label{cond1'}
\left\| {A\over\zeta+A+\zeta sA}\cdot{A\over\zeta+A}\right\| \leq \left(1+{|\zeta|\over \mbox{dist}\left({\zeta (1+s\zeta)^{-1}},-S_\alpha\right)}\right) \left(1+{|\zeta|\over \mbox{dist} (\zeta,-S_\alpha)}\right).
\end{equation}

Therefore, the family $\{F(t)\}_{t\geq 0}$ satisfies conditions of Theorem \ref{T}. This gives the operator-norm approximation of the exponential function (semigroup for m-sectorial generator) by powers of resolvent (Euler's approximation):

\begin{theorem}\label{ex1}
If $A$ is an m-sectorial operator in a Hilbert space $\cal H$, with semi-angle $\alpha\in (0,\pi/2)$ and with vertex at $0$, then
\begin{equation}\label{cor1.1}
\lim_{n\rightarrow\infty}\left\|(I+tA/n)^{-n} - e^{-tA}\right\| = 0,\ t\in S_{\pi/2-\alpha}.
\end{equation}
Moreover, if $0$ belongs to the resolvent set of $A$, then uniformly in $t\geq 0$ one has the error estimate:
\begin{equation}\label{cor1.2}
\left\|(I+tA/n)^{-n} - e^{-tA}\right\| \leq O\left(\frac{\ln n}{n}\right).
\end{equation}
\end{theorem}

\begin{proof}
The convergence (\ref{cor1.1}) follows directly from Theorem \ref{T}. To obtain (\ref{cor1.2}), we use the representation:
\begin{eqnarray}
& & (I+tA/n)^{-n} - e^{-tA} = \nonumber\\
& & \hspace{0.5cm} \sum_{m=0}^{n-1} (I+tA/n)^{-(n-m-1)} \left((I+tA/n)^{-1} - e^{-tA/n}\right) e^{-mtA/n}.\label{tele}
\end{eqnarray}
By Lemmata 1-3 of \cite{CZ1}, we have the estimates $\|A^{-1} \left((I+tA/n)^{-1} - \right.$ $\left. e^{-tA/n}\right)\| \leq 2t/n$ and $\|A^{-1} \left((I+tA/n)^{-1} - e^{-tA/n}\right) A^{-1}\|\leq 3/2 (t/n)^2$. Since by Proposition \ref{P1} the operator $A$ generates a holomorphic semigroup, a standard result (see e.g. \cite[Ch. IX, \S 1.6]{Kato}) says that $\|Ae^{-\tau A}\| \leq C_A /\tau$ for some constant $C_A>0$ and any $\tau>0$. Finally, for any integer $k\geq 1$ we have the identity:
\begin{equation}
(I+tA/n)^{-k}A = \frac{n}{t}\left(I- (I+tA/n)^{-1}\right)(I+tA/n)^{-k+1},
\end{equation}
which leads to the estimate $\|(I+tA/n)^{-k}A\| \leq nK/kt$ by Lemma \ref{LC1} for the sectorial contraction $C= (I+tA/n)^{-1}$.

By these estimates and (\ref{tele}) one gets for $n\geq 3$:
\begin{eqnarray}
\|(I+tA/n)^{-n} - e^{-tA}\| & \leq & \frac{nK}{(n-1)t}(2t/n) + (2t/n) \frac{C_A n}{(n-1)t} \nonumber \\
& & + \sum_{m=1}^{n-2} \frac{nK}{(n-m-1)t} \frac{3t^2}{2n^2} \frac{C_A n}{mt}.
\end{eqnarray}
Therefore, uniformly in $t\geq 0$ we obtain:
\begin{equation}
\|(I+tA/n)^{-n} - e^{-tA}\| \leq \frac{4(K+C_A)}{n} + 6 K C_A \frac{\ln n}{n},
\end{equation}
which proves (\ref{cor1.2}).

\end{proof}



\subsection{Arithmetic mean approximation}

Let $A_1, \dots, A_k$ be m-sectorial operators in $\cal H$, such that $\Theta(A_j) \subseteq S_\alpha$ for some $\alpha\in (0, \pi/2)$ and for any $1\leq j \leq k$. By $a_1,\dots, a_k$ we denote the corresponding closed sectorial forms. Similar to the Trotter product formula, there are expressions of the type $F_k(t/n)^n$ that approximate the semigroup generated by the form-sum of $A_1,\dots, A_k$. $F_k(t)$ can be taken as a product $f_1(tA_1)\dots f_k(tA_k)$ or as an arithmetic mean $k^{-1}(f_1(ktA_1) + \dots + f_k(ktA_k))$, where  $\{f_l\}_{1\leq l\leq k}$ are so-called Kato-functions (see e.g. \cite{K}-\cite{NZ1}). Section \ref{sec1} shows that our condition on the numerical range for sectorial contractions (cf Definition \ref{SC}) is easily satisfied if $F_k(t)$ is the arithmetic mean of resolvents or semigroups generated by $\{A_j\}_{1\leq j\leq k}$. Let us consider these families of contractions (cf \cite{Lapidus}):
\begin{eqnarray}
F_k^{res}(t) & = & \frac{1}{k}\sum_{l=1}^k (I+ ktA_l)^{-1}, t\geq 0,\label{Fres}\\
F_k^{sem}(t) & = & \frac{1}{k}\sum_{l=1}^k e^{-ktA_l}, t\geq 0.\label{Fsem}
\end{eqnarray}
Notice that $f(x) = (1+x)^{-1}$ or $f(x) = e^{-x}$ are Kato-functions. Since by conditions on $\{A_j\}_{1\leq j\leq k}$ the operators $\{(I+ktA_l)^{-1}\}_{1\leq j\leq k}$ or $\{e^{-ktA_l}\}_{1\leq j\leq k}$ are sectorial contractions with a semi-angle $\alpha$ (see Section \ref{sec1}), and since $D_\alpha$ is convex, $F_k^{res,sem}(t)$ are families of sectorial contractions with a semi-angle $\alpha$. Moreover, it is shown \cite{Lapidus} that the families $F_k^{res,sem}(t/n)^n$ converge strongly to $e^{-tX_0}P_0$ for any $t\geq 0$ and $k\geq 1$, as $n\rightarrow \infty$, where $X_0$ is the form-sum of $A_1,\dots, A_k$ and $P_0$ is the orthogonal projector on the closure of $\bigcap_{l=1}^k {\cal D}(a_l)$. Therefore, by Theorem \ref{T} we obtain a necessary and sufficient condition for the operator-norm convergence of the Trotter-type formula with arithmetic mean of resolvents (\ref{Fres}) or semigroups (\ref{Fsem}):

\begin{theorem}\label{ex2}
If $A_1,\dots, A_k$ are m-sectorial operators in $\cal H$ such that for $1\leq j\leq k$, $\Theta(A_j) \subseteq S_\alpha$ for some $\alpha\in(0,\pi/2)$, then $X_k(t) = (I-F_k^{res,sem}(t))/t$ converges (as $t\rightarrow +0$) in the uniform resolvent sense to $X_0 = A_1 \dot{+} \dots \dot{+} A_k$ defined in the closed subspace ${\cal H}_0 = \overline{\bigcap_{l=1}^k {\cal D}(a_l)}$, if and only if
\begin{equation}
\lim_{n\rightarrow\infty} \left\| F_k^{res,sem}(t/n)^n - e^{-tX_0}P_0 \right\|  = 0.
\end{equation}
Here $P_0$ is the orthogonal projector on ${\cal H}_0$.
\end{theorem}

If $k=1$, Theorem \ref{ex2} reduces to Theorem \ref{ex1}, where the condition of convergence in the uniform resolvent sense is ensured by (\ref{cond1}).


\subsection{Trotter-Kato product formula}

Let $A$ be a non-negative self-adjoint operator and let $B$ be m-sectorial with $\Theta(B) \subseteq S_\alpha$ for some angle $0< \alpha< \pi/2$. Then our main Theorem \ref{T} gives necessary and sufficient conditions for the operator-norm convergence of a generalized Trotter product formula.

Let us consider the family of operators
\begin{equation}\label{Phi}
\Phi(s) = f(sA)^{1/2}g(sB)f(sA)^{1/2}, s\geq 0,
\end{equation}
where $f(t)$ and $g(t)$ are the Kato-functions $(1+t)^{-1}$ or $e^{-t}$. Since $f(sA)^{1/2}$ is self-adjoint and $\|f(sA)^{1/2}\|\leq 1$, by results of Section \ref{sec1} one has:
\begin{equation}
(u,\Phi(s)u) = (f(sA)^{1/2}u,g(sB)f(sA)^{1/2}u) \in D_\alpha,
\end{equation}
\begin{eqnarray}
& & (u,(I-\Phi(s))u) = (u,(I-f(sA))u) + \nonumber \\
& & \hspace{1cm} + (f(sA)^{1/2}u,(I-g(sB))f(sA)^{1/2}u)\in S_\alpha,\label{48}
\end{eqnarray}
for any $u\in\cal H$, $\|u\|=1$, which proves that $\Theta(\Phi(s))\subseteq D_\alpha$. Therefore, Theorem \ref{T} gives necessary and sufficient conditions for the operator-norm convergence of $\Phi(t/n)^n$ as $n\rightarrow\infty$. At this point we still not identify the limit $X_0$ via operators $A$ and $B$.

But the case is similar to that of \cite[Addendum]{K}. In \cite{K}, Kato proved that for $n\rightarrow \infty$ the generalized Trotter product formula $(f(tA/n)g(Bt/n))^n$ converges strongly to $e^{-tC}P_0$, where $C=A\dot{+}B$ is the form-sum defined in the closure ${\cal H}_0$ of ${\cal D}={\cal D}(A^{1/2})\cap{\cal D}(B^{1/2})$ and $P_0$ is the orthogonal projection onto ${\cal H}_0$, for any non-negative self-adjoint operators $A$ and $B$. One can easily check that this result is also valid for the case of the symmetrized formula $f(tA)^{1/2}g(tB)f(tA)^{1/2}$. Moreover in the Addendum of \cite{K}, it is shown that for $f(t) = g(t) = e^{-t}$ the strong convergence holds also for m-sectorial generators $A$ and $B$. Let $a$ and $b$ be the associated closed m-sectorial forms. Then $A\dot{+}B$ is defined in the closure ${\cal H}_0$ of ${\cal D}(a)\cap{\cal D}(b)$. By the same arguments one checks, that this result is valid for $f(tA)^{1/2}g(tB)f(tA)^{1/2}$, when $f(t)$, $g(t)$ !
are $(1+t)^{-1}$ or $e^{-t}$. Therefore, $\Phi(t/n)^n$ (\ref{Phi}) converges strongly to $e^{-t(A\dot{+}B)}P_0$, where $P_0$ is the orthogonal projector onto ${\cal H}_0$, as $n\rightarrow \infty$. Taking into account these observations and our Theorem \ref{T}, we obtain the following

\begin{theorem}\label{ex3}
Let $A$ be a non-negative self-adjoint operator, and let $B$ be an m-sectorial operator in $\cal H$. Let the functions $f(t)$ and $g(t)$ be $(1+t)^{-1}$ or $e^{-t}$. For $s\geq 0$ we put $X(s)=s^{-1}(I-f(sA)^{1/2}g(sB)f(sA)^{1/2})$. Let $C=A\dot{+}B$ be the form-sum of $A$ and $B$ defined in the closure ${\cal H}_0$ of ${\cal D}(a)\cap{\cal D}(b)$, and let $P_0$ be the orthogonal projection onto ${\cal H}_0$. Then the following assertions are equivalent:
\begin{center}
(i) $X(s)$ converges in the uniform resolvent sense to $C$ as $s\rightarrow +0$,
\end{center}
\begin{equation}\label{sgf}
(ii)\ \lim_{n\rightarrow\infty} \left\| \left(f(tA/n)^{1/2}g(tB/n)f(tA/n)^{1/2}\right)^n - e^{-tC}P_0\right\| = 0.
\end{equation}

\end{theorem}

Notice that in contrast to Theorem \ref{ex1}, see (\ref{cor1.2}), Theorem \ref{ex3} gives no error bound estimate for the rate of convergence in (\ref{sgf}). For self-adjoint semigroups the operator-norm convergence of the Trotter-Kato product formula without error bound estimate is established for a general case of Kato-function $f$ and $g$, see \cite{NZ3}. In particular, there it is shown that compactness of $(I+A)^{-1}$ or $(I+A)^{-1}(I+B)^{-1}$ is a sufficient condition of this convergence for non-negative self-adjoint generators $A$ and $B$. Recently \cite{CZ2} we generalized these results to m-sectorial $A$ and $B$ for the Trotter product formula, i.e. for $f(t) = g(t) = e^{-t}$.

Another strategy to prove the Trotter-Kato product formula for semigroups is based on operator-norm error-bound estimates. After pioneering papers \cite{R}, \cite{IT}, \cite{NZ1} for couples of self-adjoint generators $A$ and $B$, these results have been recently generalized to nonself-adjoint semigroups. In our paper \cite{CZ1} it is done in Banach and Hilbert spaces for holomorphic contraction semigroups under Ichinose-Tamura conditions on generators, cf \cite{IT}. Another result of this kind is established in \cite{CNZ}. Let $A$ be a non-negative self-adjoint operator. If $B$ is m-accretive, then sufficient conditions for the operator-norm convergence of the Trotter product formula with the error bound $O(\ln n/n)$ are the following: $\|Bu\|\leq a\|Au\|$ and $\|B^*u\|\leq a_* \|Au\|$ for $u\in {\cal D}(A)$, $a<1$ , $a_*<1$. Notice that methods of \cite{CNZ} can be adapted without supplementary conditions to the generalized formula $f(tA/n)^{1/2} g(tB/n) f(tA/n)^{1/2}$ for $!
f(t)$ and $g(t)$ be $(1+t)^{-1}$ or $e^{-t}$. Fractional powers conditions on self-adjoint generators ensuring the operator-norm convergence of the Trotter-Kato with error-bound estimates are discussed in \cite{NZ2}. For a generalization to m-sectorial generators see \cite{CZ2}.




\section{Conclusion}

In the present paper we extend the Chernoff theory to approximation of semigroups in the operator-norm topology in a Hilbert space. The condition for this operator-norm convergence is that the operators $\{\Phi(s)\}_{s\geq 0}$ are not only contractions, but sectorial contractions: they have their numerical ranges $\Theta(\Phi(s))$ in a domain $D_\alpha$, $0\leq\alpha<\pi/2$, uniformly in $s\geq 0$, see (\ref{Dalpha}). This allows us to find the $1/n^{1/3}$-estimate in operator-norm for an analogue of the Chernoff lemma (see Theorem \ref{T1}). Notice that the original Chernoff's estimate in this lemma is not convergent: $\|(C^n - e^{n(C-I)})u\|\leq n^{1/2}\|(C-I)u\|$, for any vector $u$. The next step is the analogue of the Trotter-Neveu-Kato approximation theorem in the operator-norm, see Lemma \ref{Trgene}. We consider uniformly m-sectorial families of generators, which leads to uniformly bounded holomorphic semigroups. The Corollary \ref{approx} shows that the convergence in!
 the uniform resolvent sense (i.e. in the generalized convergence in the sense of Kato \cite[Ch. IV]{Kato}) is well-adapted to approximate semigroups in the operator-norm topology. On the other hand, this topology is natural for holomorphic contraction semigroups: they are continuous in this topology for $t>0$, and generators of these semigroups are exactly m-sectorial operators with semi-angles $\alpha<\pi/2$. A first consequence of our main Theorem \ref{T} is that the Euler approximation formula for the exponential function converges in operator-norm for any m-sectorial generator (Theorem \ref{ex1}).

This operator-norm approximation theory leads to necessary and sufficient conditions for convergence of Trotter-type formul{\ae}. We formulate these conditions for the formula with arithmetic mean of resolvents or semigroups (see Theorem \ref{ex2}) and for the symmetrized product formula $f(tA)^{1/2}g(tB)f(tA)^{1/2}$ where $f(t),g(t) = (1+t)^{-1}$ or $e^{-t}$, see Theorem \ref{ex3}. Our condition on the numerical range $\Theta(\Phi(s))\subseteq D_\alpha$ (sectorial contraction) seems to be naturally related to the m-sectorial operators. In fact we have verified it for the resolvent $(I+tA)^{-1}$, for the semigroup $e^{-tA}$ generated by an m-sectorial operator $A$ (see Section \ref{sec1}), and for the family $\Phi(s) = f(sA)^{1/2}g(sB)f(sA)^{1/2}$, see (\ref{Phi})-(\ref{48}).



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\begin{thebibliography}{99}
 
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\end{document}
