\documentstyle{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ 1997(1997), No.\ 19, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp) 147.26.103.110 or 129.120.3.113}
\thanks{\copyright 1997 Southwest Texas State University  and 
University of North Texas.} 
\vspace{1.5cm}
\title[\hfilneg EJDE--1997/19\hfil  Complex dynamical systems]
{Complex dynamical systems on bounded symmetric domains} 
\author[Victor Khatskevich, Simeon Reich \& David Shoikhet
\hfil EJDE--1997/19\hfilneg]
{Victor Khatskevich\\ Simeon Reich \\ David Shoikhet}
\address{Victor Khatskevich \hfil\break
Department of Applied Mathematics \\
International College of Technology \\
P.O. Box 78,~ 20101 Karmiel, Israel}
\email{}
\address{Simeon Reich\hfil\break
Department of Mathematics \\
Technion -- Israel Institute of Technology \\
32000 Haifa, Israel}
\email{sreich\@tx.technion.ac.il}
\address{David Shoikhet\hfil\break
Department of Applied Mathematics \\
International College of Technology \\
P.O. Box 78,~ 20101 Karmiel, Israel}
\email{davs\@tx.technion.ac.il}
\date{}
\thanks{Submitted August 25, 1997. Published October 31, 1997.}

\subjclass{34G20, 46G20, 47H20, 58C10.}
\keywords{Bounded symmetric domain, complex Banach space, \hfil\break\indent
holomorphic mapping,
 infinitesimal generator, semi-complete vector field.}

\begin{abstract}
We characterize those holomorphic mappings which are 
the infinitesimal generators of semi-flows on bounded 
symmetric domains in complex Banach spaces. 
\end{abstract}
\maketitle

\newtheorem{proposition}{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newcommand{\be}        {\begin{eqnarray}}
\newcommand{\ee}        {\end{eqnarray}}
\newcommand{\RR}        {{\Bbb R}}
\newcommand{\CC}        {{\Bbb C}}

\section{Introduction}
Let $D$ be a bounded domain in a complex Banach space $X$. By 
$\operatorname{Hol}(D,X)$ we denote the set of holomorphic mappings 
from $D$ into $X$.  Let $\operatorname{Hol}(D)$ be the semigroup (with 
respect to composition) of all holomorphic self-mappings of $D$, and 
let $\hbox{Aut}(D)\subset\operatorname{Hol}(D)$ be the subgroup 
consisting of all holomorphic automorphisms of $D$.

A family $S=\{F_t\}\subset\operatorname{Hol}(D),~ t\geq 0 ~(-\infty 
<t<\infty )$, is called a continuous one-parameter semigroup (group) 
if
\be                                                   %(1)
F_{s+t}=F_s\circ F_t,\quad t\geq 0 \quad(-\infty < t<\infty ),
\ee
and
\be                                                   %(2)
\displaystyle\lim_{\stackrel{t\rightarrow 0^+}{(t\rightarrow 0)}} 
F_t(x)=x,\quad x\in D.
\ee

A mapping $f\in\operatorname{Hol}(D,X)$ is said to be an infinitesimal 
generator of a semi-flow (complete flow) if there exists a one-parameter 
semigroup (group) $S_f=\{F_t\}$ such that for each $x\in D$,
\be                                                   %(3)
f(x) = 
\lim_{\stackrel{t\rightarrow 0^+}{(t\rightarrow 0)}}
\frac{x-F_t(x)}{t}\,,
\ee
where once again the limit is taken with respect to the norm of $X$.
We denote by $\operatorname{hol}(D)$ the family of all (infinitesimal) 
holomorphic generators on $D$.

Note that if $f\in\operatorname{hol}(D)$ generates a complete flow 
$S_f=\{F_t\}_{t\in \RR}$, then $F_t\in\hbox{Aut}(D)$ and 
$F_t^{-1}=F_{-t}$ for all $t\in \RR$.  In this case one writes that 
$f\in\hbox{aut}(D)$.

It can be shown (see, for example, \cite{RS1} and \cite{RS2}) that 
since $f\in\operatorname{hol}(D)$ is locally bounded on $D$, the 
Cauchy problem 
\be                                                   %(4)
\begin{cases}
\frac{\partial u(t,x)}{\partial t} + f(u(t,x))= 0\\
u(0,x)=x,\quad x\in D,
\end{cases}
\ee
can be solved on $\RR^+=[0,\infty )$ for each $x\in D$ and 
$u(t,x)=F_t(x)$.  Thus (4) defines an analytic dynamical system and 
$S_f=\{F_t\}_{t\geq 0}$ is a uniquely defined semi-flow on $D$.

Moreover, the convergence in (2) is uniform on each ball strictly 
inside 
$D$.  If, in addition, $f\in\hbox{aut}(D)$, then the Cauchy problem 
(4) can be solved for all $t\in \RR=(-\infty ,\infty )$.

Note also that if $g\in\operatorname{Hol}(D,X)$, then by allowing $f$ 
to operate on $g$ by means of the formula $(fg)(x)=g'(x)\circ f(x)$ we 
can interpret $f$ as a derivation of $\operatorname{Hol}(D,X)$, i.e., 
as a holomorphic vector field.  Using this terminology, 
$f\in\operatorname{hol}(D)$ will be called a semi-complete vector 
field, and $f\in\hbox{aut}(D)$ a complete vector field (see, for 
example, \cite{I-S}, \cite{DS2}, \cite{UH} and \cite{RS1}).  It is known 
that $\hbox{aut}(D)$ is a real Banach Lie algebra, while 
$\operatorname{hol}(D)$ is only a real cone (see \cite{AM}, \cite{RS1} 
and \cite{RS2}).

Our purpose in this paper is to describe the class of semi-complete 
vector fields on a bounded symmetric domain.  To motivate our approach 
we briefly review some previous results.

For the one-dimensional case, namely, $D=\Delta$, the open unit disk 
in the complex plane $\CC$, an implicit condition which characterizes 
$\operatorname{hol}(\Delta)$ was obtained by E. Berkson and H. Porta 
\cite{B-P}.

It was shown by M. Abate \cite{AM} that their condition can be 
rewritten explicitly in the form
\be                                                   %(5)
\hbox{Re}\,f(x)\bar{x}\geq -\frac{1}{2}\hbox{Re}\,f'(x)(1-|x|^2).
\ee
As a matter of fact, this condition is the special case $n=1$ of a 
more general (and more complicated) condition, which is valid for the 
open Euclidean unit ball in $\CC^n$ (see \cite{AM}).

On the other hand, it follows directly from the definition, that if 
$f\in\operatorname{hol}(D)$ has a continuous extension to 
$\bar{\Delta}$, then 
\be                                                   %(6)
\hbox{Re}\,f(x)\bar{x}\geq 0\quad\hbox{ for all }x\in\partial\Delta .
\ee
Unfortunately, it is not clear how to derive (6) from (5) in such a 
situation.  At the same time, by rewriting (6) in the form
\[\hbox{Re}\,[f(x)-f(0)]\bar{x}\geq -\hbox{Re}\,f(0)\bar{x},\]
and dividing the left-hand side by $|x|^2=1$, we get
\[\hbox{Re}\,\left (\frac{f(x)-f(0)}{x}\right )\geq 
-\hbox{Re}\,\overline{f(0)} 
x,\quad x\in\partial\Delta.\]

Now it follows by the maximum principle for harmonic functions that 
the last inequality holds also for $x\in\Delta$.  Multiplying it by 
$|x|^2,~x\in\Delta$, $x\not= 0$, we obtain
\be                                                   %(7)
\hbox{Re}\,f(x)\bar{x}\geq\hbox{Re}\,f(0)\bar{x}(1-|x|^2),~x\in\Delta.
\ee
We claim that even if $f\in\operatorname{Hol}(\Delta,\CC)$ does not 
extend continuously to $\bar{\Delta}$, condition (7) is necessary and 
sufficient for $f$ to be an infinitesimal generator of a semi-flow.

Indeed, for the case of the open unit ball $B$ in a Hilbert space $H$, 
it was shown in \cite{RS2}, by using its hyperbolic metric, that the 
condition
\be                                                   %(8)
\hbox{Re}\,\langle f(x),x\rangle\geq \hbox{Re}\,\langle f(0),x\rangle
(1-\|x\|^2),~x\in B, 
\ee
where $\langle\cdot,\cdot\rangle$ is the inner product in $H$, 
characterizes the class $\operatorname{hol}(B)$.

Note that a crucial point of the approach in \cite{RS2} was the 
smoothness of the boundary of $B$.  It is clear that such a property 
is no longer valid for the finite product $B^n$ equipped with the max 
norm, and all the more so for the open unit ball in ${\cal L}(H,H)$, the 
space of bounded linear operators from $H$ into $H$.

Another technical way to extend (8) to $B^n$, by using a special curve 
defined by a family of M\"{o}bius transformations, was employed in 
\cite{RS3}.

Therefore a natural idea which arises is that this be done for each 
Banach space $X$ the open unit ball $D$ of which is a homogeneous domain 
(i.e., for each pair $x,y\in D$ there is $F\in\hbox{Aut}(D)$ such that 
$F(x)=y$).

Indeed, since every such ball is a bounded symmetric domain (see the 
definition below), one can propose using the more general and 
well-developed theory of such domains to derive an analog of condition 
(8) which will characterize $\operatorname{hol}(D)$.

It will become clear that such an approach does not require difficult 
calculations, and moreover, it establishes new facts concerning the 
description of semi-complete vector fields.

A domain $D$ is called symmetric if for all $a\in D$ there exists 
$F_a \in\hbox{Aut}(D)$ such that $F_a ^2=I_D$ and $a$ is an 
isolated fixed point of $F_a$.

For the case when $D$ is a bounded symetric domain, the class 
$\hbox{aut}(D)$ of all complete vector fields on $D$ has been 
well-described with the help of an algebraic approach (see, for 
example, \cite{I-S}, \cite{UH}, \cite{AJ} and \cite{DS2}).  Namely, 
it is known that $\hbox{aut}(D)$ is a real Banach Lie algebra and each 
$f\in\hbox{aut}(D)$ is a polynomial of degree at most $2$.  Moreover, 
if 
\be                                                   %(9)
p=\{f\in\hbox{aut}(D):~f'(0)=0\}
\ee
and
\be                                                   %(10)
k=\{f\in\hbox{aut}(D):~f(0)=0\},
\ee
then $\hbox{aut}(D)$ is the direct sum decomposition
\[\hbox{aut}(D) = p\oplus k ,\]
and each element of $X$ can be realized as the constant term of a 
unique element of $p$, i.e., for each $y\in X$ there is a unique 
two-homogeneous polynomial $P_y$ such that the mapping 
$g_y\in\operatorname{Hol}(X,X)$ defined by the formula 
\be                                                   %(11)
g_y(x)=y+P_y(x)
\ee
belongs to $p\subset \hbox{aut}(D)$.

Furthermore, by Kaup's theorem \cite{KW}, every bounded symmetric 
domain $D$ can be realized as the open unit ball of a $JB^\ast $-triple 
system, and moreover, it is a homogeneous domain, i.e., for each pair 
$x,y\in D$ there is $F\in\hbox{Aut}(D)$ such that $F(x)=y$.

Note also that an automorphism which moves the origin to $y\in D$ can 
be generated by $g\in p\subset\hbox{aut}(D)$, i.e., $g$ has the form 
(11) (see, for example, \cite{UH} and \cite{DS2}).

So, in the sequel we will always assume that a bounded symmetric 
domain is realized as a convex balanced domain.  At the same time, in 
this case the gauge of $D$ (the Minkowski functional) can be defined 
as $c_D(0,\cdot )$, where $c_D(\cdot ,\cdot )$  is the infinitesimal 
Carath\'{e}odory metric on $D$, and $D$ is the indicatrix of this 
gauge, i.e.,
\[D=\{x\in X: c_D(0,x)<1\}.\]
Thus, since $D$ is bounded, $c_D(0,\cdot )$ is a norm which is 
equivalent to the norm of $X$, and $D$ can be considered the open 
unit ball of $X$ when it is equipped with this norm. So, our problem 
may be formulated as follows.

Let $X$ be a complex Banach space such that the open unit ball $D$ of 
$X$ is a homogeneous domain.  What are the geometric conditions which 
characterize semi-complete vector fields on $D$?

Let $X'$ be the dual space of $X$.  As usual, we use the pairing 
$\langle x,x'\rangle$ to denote the action of a linear functional 
$x'\in X'$ on an element $x\in X$.  In particular, for $X=H$, a 
Hilbert space, $\langle \cdot ,\cdot\rangle$ means the inner product 
in $H$.  Recall also that the normalized duality mapping $J: 
X\rightarrow 2^{X'}$ is defined by 
\[J(x)=\{x'\in X': \langle x,x'\rangle=\|x\|^2=\|x'\|^2\}.\]

\section{Main result}

\begin{theorem} Let $X$ be a complex Banach space such that the 
open unit ball $D$ of 
$X$ is a homogeneous domain.  Then the following assertions hold:
\begin{enumerate}
\item
If $f\in\operatorname{hol}(D)$, then for each $x\in D$ and for each 
$x'\in J(x)$,
\be                                                   %(12)
\hbox{Re}\,\langle f(x),x'\rangle\geq \hbox{Re}\,\langle 
f(0),x'\rangle (1-\|x\|^2).
\ee
\item If $f\in\operatorname{Hol}(D,X)$ is bounded on each subset 
strictly inside $D$ and for each $x\in D$ there exists $x'\in J(x)$ 
such that (12) holds, then $f\in \operatorname{hol}(D)$.
\item If $f\in \operatorname{hol}(D)$ and $S_f=\{F_t\}_{t\geq 0}$ is 
the semi-flow generated by $f$, then $F_t\in\operatorname{Hol}(D)$ 
satisfies the following estimate: 
\be                                                   %(13)
\|F_t(x)\|\leq 
\frac{\|x\|+1-e^{-2\|f(0)\|t}(1-\|x\|)}
{ \|x\|+1+e^{-2\|f(0)\|t}(1-\|x\|) }.
\ee
\end{enumerate}
\end{theorem}

To prove our theorem we need several preliminary assertions.

\begin{proposition}                                   %Proposition 1
\cite{RS1}, \cite{RS2}. Let $D$ be a bounded convex domain in $X$.  
Then $f\in \operatorname{Hol}(D,X)$ is semi-complete (i.e., belongs to 
$\operatorname{hol}(D)$) if and only if for each $\lambda > 0$ the 
nonlinear resolvent $R(\lambda ,f)=(I+\lambda f)^{-1}$ is a 
well-defined holomorphic self-mapping of $D$.

In addition, if $S_f=\{F_t\}_{t\geq 0}$ is the semi-flow generated by 
$f$, then it can be given by the exponential formula
\be                                                   %(14)
F_t=\lim_{n\rightarrow \infty }R^n(\frac{1}{n}t,f),\quad t\geq 0 ,
\ee
where the limit in (14) is taken with respect to the norm of $X$ 
uniformly on each subset strictly inside $D$.
\end{proposition}

\begin{proposition}                                    %Proposition 2
\cite{RS1}, \cite{RS2}.  Let $D$ be as in Proposition 1.  Then 
$\operatorname{hol}(D)$ is a real cone, i.e., for each pair $f$ and 
$g$ from $\operatorname{hol}(D)$ and all $\alpha ,\beta > 0$, the 
mapping $\alpha f+\beta g$ also belongs to $\operatorname{hol}(D)$.
\end{proposition}

Since $\hbox{aut}(D) = \operatorname{hol}(D)\cap 
(-\operatorname{hol}(D))$ is a linear space, Proposition 2 immediately 
implies the following assertion.

\begin{proposition}                                   %Proposition 3
Let $D$ be a bounded balanced convex symmetric domain in $X$.  Then 
each element $f\in\operatorname{hol}(D)$ can be represented as
\be                                                   %(15)
f~=~h+g,
\ee
where $h\in\operatorname{hol}(D)$ with $h(0)=0$ and $g=g_y\in p\subset
\hbox{aut}(D)$ is defined by (11) with $y=f(0)$.  This representation 
is unique.
\end{proposition}

\begin{proposition}                                   %Proposition 4
Let $f\in\operatorname{hol}(D)$ be as above, and let $g_{f(0)}\in 
p\subset \hbox{aut}(D)$ be defined by (11).  Then for each $x\in D$ 
and for each $x'\in J(x)$ the following inequality holds:
\be                                                   %(16)
\hbox{Re}\,\langle f(x),x'\rangle\geq \hbox{Re}\,\langle 
g_{f(0)}(x),x'\rangle.
\ee
\end{proposition}

{\it Proof}. Indeed, it follows by (15) that $h=f-g_{f(0)}$ belongs to 
$\operatorname{hol}(D)$ and 
\be                                                   %(17)
h(0)=0.
\ee
Let $S_h=\{{\cal H}_t\}_{t\geq 0}\subset\operatorname{Hol}(D)$ be the 
semi-flow generated by $h$, i.e., for each $x\in D$,
\[\lim_{t\rightarrow 0^+}\frac{x-{\cal H}_t(x)}{t}=h(x).\]
It follows by the uniqueness of the solution to the Cauchy problem (4) 
and by (17) that the origin is a common fixed point of 
$S_h=\{{\cal H}_t\}_{t\geq 0}$ for all $t\geq 0$.  Since $\|{\cal 
H}_t(x)\|\leq 1$, it follows by the Schwarz Lemma that $\|{\cal 
H}_t(x)\|\leq \|x\|$ for all $x\in D$.  Now using (17), we get 
\be                                                   %(18)
\hbox{Re}\,\langle h(x),x'\rangle\geq 0
\ee
for all $x'\in J(x)$.  By the definition of $h$, (18) is exactly (16), 
and we are done.

Now it is very easy to prove the necessity of (12) for $f$ to be a 
semi-complete vector field.  In fact, for each $u\in\partial D$ and 
each $g\in\hbox{aut}(D)$ we have
\be                                                   %(19)
\hbox{Re}\,\langle g(u),u'\rangle = 0
\ee
whenever $u'\in J(u)$ (note that $g$ is holomorphically extensible to 
$\partial D)$.  In particular, this holds for $g_y=y+P_y(x)\in p$ where 
$P_y$ is a homogeneous polynomial of degree 2.  Therefore, if for 
$x\in D, x\not= 0$, we set $u=\frac{1}{\|x\|}x$, we obtain
\be
\hbox{Re}\,\langle g_y(x),x'\rangle &=& \hbox{Re}\,\langle 
y+P_y(x),x'\rangle ~= ~\hbox{Re}\, \langle y,x'\rangle + 
\hbox{Re}\,\langle P_y(x),x'\rangle
\nonumber\\
&=& \hbox{Re}\,\langle y,x'\rangle + \|x\|^3\hbox{Re}\,\langle 
P_y(u),u'\rangle 
\nonumber\\
&=& \hbox{Re}\,\langle y,x'\rangle + \|x\|^3(\hbox{Re}\,\langle 
P_y(u),u'\rangle +\langle y,u'\rangle ) 
\nonumber\\
&& ~~~ - \|x\|^3\hbox{Re}\,\langle y,u'\rangle
\nonumber\\
&=& \hbox{Re}\,\langle y,x'\rangle
-\|x\|^2\hbox{Re}\,\langle y,\|x\|u'\rangle
\nonumber\\
&=& \hbox{Re}\,\langle y,x'\rangle (1-\|x\|^2).
\nonumber
\ee
Using this equality with $y=f(0)$ and (16) we obtain (12).  Assertion 
1 of our theorem is proved.  To prove assertions 2 and 3 we first 
establish a somewhat more general proposition.
\begin{proposition}                                   %Proposition 5
Let $X$ be an arbitrary complex Banach space, and let $D$ be the open 
unit ball in $X$.  Suppose that $f\in\operatorname{Hol}(D,X)$ is 
bounded on each subset strictly inside $D$ and satisfies the following 
condition:  For each $x\in D$ and some $x'\in J(x)$,
\be                                                   %(20)
\hbox{Re}\,\langle f(x),x'\rangle\geq \alpha (\|x\|)\cdot\|x\|,
\ee
where $\alpha :[0,1]\rightarrow \RR$ is an increasing continuous 
function on $[0,1]$ such that 
\be                                                   %(21)
\alpha (0)\cdot\alpha (1)\leq 0.
\ee
Then 
\begin {enumerate}
\item
$f$ is a semi-complete vector field on $D$.
\item
If $S_f=\{F_t\}$ is the semi-flow generated by $f$, then for all 
$t\geq 0$ and $x\in D$,
\be                                                   %(22)
\|F_t(x)\|\leq \beta_t(\|x\|),
\ee
where $\beta_t$ is the solution of the Cauchy problem
\be                                                   %(23)
\begin{cases}
\frac{d\beta_t(s)}{dt} + \alpha (\beta_t(s))=0,\\
\beta_0(s)=s,\quad s\in[0,1].
\end{cases}
\ee
\end{enumerate}
\end{proposition}
{\it Proof}.  Fix $r\in (0,1)$ and consider the equations
\be                                                   %(24) (25)
x+\lambda f(x) ~ &=& ~ z\\
s+\lambda\alpha (s) ~ &=& ~ \|z\|,
\ee
where $z\in\bar{D}_r=\{x\in X:\|x\|\leq r<1\}, s\in [0,1]$, and 
$\lambda >0$. It follows from (21) that for a fixed 
$z\in\bar{D}_r$, the function \text{$\gamma (s)= s+\lambda\alpha 
(s)-\|z\|$} satisfies the conditions $\gamma (0)\leq 0,~\gamma (1)>0$. 
Hence equation (25) has a unique solution $s_0=s_0(z)\in [0,1)$.
So, for an arbitrary $\delta>0$ we can find $\epsilon >0$ such that 
$\gamma (s_0+\delta )\geq \epsilon$.  Now taking $x\in D$ such that 
$\|x\|=s=s_0+\delta$, we have by (20) for such $x$ and any $x'\in 
J(x)$,
\be
\hbox{Re}\,\langle x+\lambda f(x)-z,x'\rangle &=&
\hbox{Re}\,(\langle x,x'\rangle + \lambda\langle f(x),x'\rangle - 
\langle z,x'\rangle )
\nonumber\\
&\geq & s^2 + \lambda \alpha (s)\cdot s-\|z\|\cdot s
\nonumber\\
&=& s\gamma (s)~ \geq ~ s\cdot\epsilon .
\nonumber
\ee
It follows by the same considerations as in Theorem 3 in \cite{A-R-S} 
that equation (24) has a unique solution $x=x(z)$ such that 
$\|x(z)\|\leq s_0+\delta$.  Since $\delta>0$ is arbitrary, we must 
have
\[\|x(z)\| ~ \leq ~ s_0.\]
In terms of nonlinear resolvents the last inequality can be rewritten 
as
\be
\|R(\lambda ,f)(z)\| &=& \|(I_X+\lambda f)^{-1}(z)\|
~\leq ~R(\lambda ,\alpha )(\|z\|) 
\nonumber\\
&=& (I_\RR+\lambda \alpha )^{-1} (\|z\|).
\nonumber
\ee
Now using Proposition 1 and the exponential formula (14) we deduce our 
assertion.

To prove our theorem we need only observe that the function
\be                                                   %(26)
\alpha (s) ~= ~-\|f(0)\| (1-s^2)
\ee
satisfies all the conditions of Proposition 5, and that the solution 
$\beta_t(s)$ of the Cauchy problem (23) with $\alpha $ defined by (26) 
has the same form as the right-hand side of (13).  The theorem is 
proved.
\begin{remark}                                        %Remark 1
If $X$ is a $J^\ast $-algebra, then condition (16) can be rewritten in the 
form
\be                                                   %(27)
\hbox{Re}\,\langle f(x),x'\rangle\geq \hbox{Re}\,\langle f(0) - 
x[f(0)]^\ast x,x'\rangle,
\ee
which also characterizes those mappings $f\in \operatorname{Hol}(D,X)$ 
which are semi-complete vector fields on the open unit ball of X.
\end{remark}

For example, consider the case of the algebra $X={\cal L}_c(H_1,H_2)$ of 
all linear compact operators ${\cal A}: H_1\rightarrow H_2$ (${\cal A}$ is 
defined on the whole of $H_1$ and maps it compactly into $H_2)$, when 
$H_1$ and $H_2$ are Hilbert spaces.

Let ${\cal D}$ be the open unit operator ball of ${\cal L}_c(H_1,H_2)$, 
that is, ${\cal D} = \{{\cal A}\in{\cal L}_c(H_1,H_2):\|{\cal A}\|<1\}$.  
Suppose that the mapping $f$ belongs to $\operatorname{Hol}(D,X)$.  It 
is easy to see that for any ${\cal A}\in{\cal L}_c(H_1,H_2)$ there exists 
$x_{\cal A}\in H_1$ such that $\|{\cal A}\|=\|{\cal A}x_{\cal A}\|$ and 
$\|x_{\cal A}\|=1$.  Indeed, $\|{\cal A}\|=
\displaystyle\sup_{\stackrel{\|x\|=1}{x\in H_1}}\|{\cal A}x\|$, so 
there exists $\{x_n\}_{n=1}^\infty $ such that $\|x_n\|=1$ and $\|{\cal 
A}x_n\|\rightarrow \|{\cal A}\|$, as $n\rightarrow \infty $.  Since $H_1$ 
is a Hilbert space, there exists a subsequence 
$\{x_{n_k}\}_{k=1}^\infty$ of the sequence $\{x_n\}_{n=1}^\infty $ 
which converges weakly to some $x_A\in H_1$.  Since ${\cal A}$ is 
compact, ${\cal A}x_{n_k}\rightarrow {\cal A}x_A$ as $k\rightarrow \infty 
$.  Hence $\|{\cal A}x_{\cal A}\|=\|{\cal A}\|$ and $\|x_{\cal A}\|= 1.$  

For any ${\cal A}\in{\cal L}_c(H_1,H_2)$ we construct the support
functional 
$g_{\cal A}\in({\cal L}_c(H_1,H_2))^\ast $ in the following way:
\[g_{\cal A}(T):= (Tx_{\cal A},\|{\cal A}\|^{-1}{\cal A}x_A),~T\in{\cal 
L}_c(H_1,H_2).\]
($(x,y)$ is the scalar product in $H_2)$.

We have $|g_{\cal A}(T)|\leq \|Tx_{\cal A}\|\|x_{\cal A}\|\leq \|T\|, 
\quad g_{\cal A}({\cal A})=\|{\cal A}\|$, hence $\|g_{\cal A}\|=1$.  Thus 
$g_{\cal A}$ belongs to $J({\cal A})$.

The following condition is a natural analog of (7) for this algebra:
\be                                                   %(28)
\hbox{Re}\,{\cal A}^\ast f(A)\geq \hbox{Re}\,{\cal A}^\ast 
f(0)({\cal I}-|{\cal A}|^2)
\ee
(here $|{\cal A}|^2={\cal A}^\ast {\cal A})$.

We claim that this simple condition implies (27).  Indeed, (28) is 
equivalent to 
\be
\hbox{Re}\,({\cal A}^\ast f({\cal A})x,x) & \geq & \hbox{Re}\,({\cal
A}^\ast f(0)
({\cal I}-|{\cal A}|^2)x, x)
\nonumber\\
&=& \hbox{Re}\,(({\cal A}^\ast f(0)x,x)-A^\ast f(0){\cal A}^\ast {\cal A}x,x))
\nonumber\\
&=& \hbox{Re}\,(({\cal A}^\ast f(0)x,x)-({\cal A}^\ast 
{\cal A}[f(0)]^\ast {\cal A}x,x)).
\nonumber
\ee
Hence for $x=x_{\cal A}$ we obtain:
\be
\hbox{Re}\,(f({\cal A})x_{\cal A},{\cal A}x_{\cal A})\geq 
\hbox{Re}\,((f(0)x_{\cal A},{\cal A}x_{\cal A})-({\cal A}[f(0)]^\ast
{\cal A}x_{\cal A},{\cal A}x_{\cal A}) ,
\nonumber
\ee
or, setting ${\cal A}'$ to be $g_{\cal A}$, 
\[\hbox{Re}\,\langle f({\cal A}),{\cal A}'\rangle \geq \hbox{Re}\,\langle 
f(0)-{\cal A}[f(0)]^\ast {\cal A},{\cal A}'\rangle ,\]
which is precisely (27).

Note that in the particular case when $\min (\dim H_1,\dim H_2)< 
\infty$, ${\cal L}_c(H_1,H_2)={\cal L}(H_1,H_2)$, the space of all 
bounded linear operators ${\cal A}: H_1\rightarrow H_2$.  So in this 
case all of the above is also true for the open unit ball ${\cal D}$ of 
${\cal L}(H_1,H_2)$.

\begin{remark}                                        %Remark 2
If $f\in \operatorname{hol}(D)$, then it follows from the 
representation (15) (see Proposition 3) that the linear operator 
$A=f'(0)$ is accretive.
\end{remark}
Indeed, if $h=f-g_{f(0)}$, then $h'(0)=f'(0)=A.$  But $h(0)=0$ and 
the origin is a common fixed point of the semi-flow $S_h=\{{\cal 
H}_t\}_{t\geq 0}$.   
Using the Cauchy inequalities, it is easy to check that the family 
$\{B_t=({\cal H}_t)'(0)\}_{t\geq 0}$ is a semigroup of linear 
contractions generated by $A$.  Therefore $A$ is accretive by the 
Lumer-Phillips Theorem.

Thus, if in the $J^\ast $-algebra $X$ we consider the Riccati flow 
equation 
\be
\begin{cases}
\dot{x}_t=a+bx_t-x_ta^\ast x_t,\\
x_0 = x\in D ,
\end{cases}
\nonumber
\ee
then this equation has a solution on $D\times \RR^+$ if and only if the 
element $b\in X$ defines an accretive linear operator by $x\mapsto 
bx$.
\begin{remark}                                        %Remark 3
As a matter of fact, if under the conditions of our Theorem, the  
operator $B=iA$, where $A=f'(0)$, is Hermitian,  
i.e., $\hbox{Re}\,\langle Ax,x'\rangle 
=0$ for all $x\in X$ and $x'\in J(x)$, then $f\in 
\operatorname{hol}(D)$ actually belongs to $\hbox{aut}(D)$.
\end{remark}

Indeed, it is enough to prove that $h$ in the representation (15) has 
the form 
\be                                                   %(29)
h(x)~ = ~ f'(0)x.
\ee

To see this, let us represent $h(x)$ by the Taylor formula
\[h(x)=h'(0)x+k(x),\]
where $k(x)$ contains the terms of order greater or equal to $2$.  
Then, by (18), we have 
\[\hbox{Re}\,\langle h(x),x'\rangle = \hbox{Re}\,\langle 
h'(0)x,x'\rangle + \hbox{Re}\,\langle k(x),x'\rangle \geq 0.\]
Since $h'(0)=f'(0)$ we see that
\[\hbox{Re}\,\langle k(x),x'\rangle\geq 0.\]
Since $k(0)=0$, we get by the theorem that 
$k\in\operatorname{hol}(D)$.  But $k'(0)=0$ and it follows by the 
infinitesimal version of the Cartan Uniqueness Theorem (see 
\cite{RS1}) that $k=0$ and we are done.

Following S. G. Krein \cite{KS} (see also E. Vesentini \cite{VE}), a
linear operator $A:X\rightarrow X$ such that $\hbox{Re}\,\langle 
Ax,x'\rangle = 0$ for all $x\in X$ and $x'\in J(x)$ is called a 
conservative operator.  So we have the following result.

\begin{corollary}                                     %Corollary 1
Let $f\in\operatorname{hol}(D)$.  Then $f$ is a complete vector field 
$(f\in\hbox{aut}(D)$ if and only if the operator $f'(0)$ is conservative. 
\end{corollary}

The following proposition is a direct consequence of assertion 3 of 
the Theorem. It is motivated by Proposition 7 in \cite{DS1}.

\begin{corollary}                                     %Corollary 2
Let $S=\{F_t\}_{t\geq 0}$ be a one-parameter semigroup of holomorphic 
self-mappings of $D$ such that $F_t$ converges to $I$, as $t\rightarrow 
0^+$, locally uniformly on $D$.  Then for each $\rho\in (0,1),~M\in 
\RR^+$ and $\alpha \in\RR^+$, there exists a positive number 
$A=A(\rho ,M,\alpha )<1$ such that 
\[\sup\{\|F_t(x)\|:\|\xi\|\leq M,\quad \|x\|\leq \rho,~ 0\leq t\leq 
\alpha \}\leq A,\]
where $\xi=\frac{d^+F_t(0)}{dt}$.
\end{corollary}
\medskip

{\bf Acknowledgments.} We gratefully acknowledge valuable conversations
with Professors Jonathan Arazy and Wilhelm Kaup. The second author was
partially supported by the Fund for the Promotion of Research at the
Technion and by the Technion VPR Fund - M. and M. L. Bank Mathematics
Research Fund. All the authors thank the referee for several useful
comments.



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\end{document}