\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 $S=\{F_t\}\subset\operatorname{Hol}(D),~ t\geq 0 ~(-\infty
0$ 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}