\magnification=\magstep1
\parindent=0pt


\long\def\DEL#1{{}}

\newcount\SEC
\newcount\THM
\def\ADS{\advance \SEC by 1 \THM=0}
\def\ADT{\advance \THM by 1}
\def\NUM{{\the\SEC.\the\THM}}


\font\tenopen=msbm10 \font\sevenopen=msbm7 \font\fiveopen=msbm5
\newfam\openfam
\def\openo{\fam\openfam\tenopen}
\textfont\openfam=\tenopen \scriptfont\openfam=\sevenopen
\scriptscriptfont\openfam=\fiveopen

\font\tenujsym=msam10 \font\sevenujsym=msam7 \font\fiveujsym=msam5
\newfam\ujsymfam
\def\ujsym{\fam\ujsymfam\tenujsym}
\textfont\ujsymfam=\tenujsym \scriptfont\ujsymfam=\sevenujsym
\scriptscriptfont\ujsymfam=\fiveujsym

\font\Fr=eufm10

\def\RR{{\openo{R}}}
\def\CC{{\openo{C}}}
\def\QQ{{\openo{Q}}}
\def\ZZ{{\openo{Z}}}

\def\eps{\varepsilon}

\def\qed{\hskip10mm Q.e.d.}

%References
\def\ABATE{1}    %Abate
\def\BKU{2}      %Brau-Kaup-Upmeier
\def\DINEEN{3}   %Dineen
\def\NAG{4}      %Engel-Nagel
\def\FRVE{5}     %Franzoni-Vesentini
\def\GOEBEL{6}   %Goebel
\def\HS{7}       %Hayden-Suffridge
\def\IST{8}     %Isidro-Stacho
\def\KHAS{9}    %Kashkevits-Reich-Shoikhet
\def\KAUP{10}     %Kaup1981
\def\KAUPR{11}   %Kaup1983
\def\NACHBIN{12} %Nachbin
\def\RSZ{13}      %Riesz-Sz-Nagy
\def\FSZ{14}     %SzNagy-Foias
\def\VESi{15}    %Vesentini87
\def\VESii{16}   %Vesentini94
\def\VIG{17}     %Vigue



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


%\centerline{\bf A Stone type theorem for non-linear automorphisms in Hilbert space}
\centerline{\bf A fixed point approach to unitary $C_0$-groups in Hilbert space}

\bigskip

\centerline{L.L. STACH\'O}

\ADS
\bigskip\bigskip
\noindent{\bf \the\SEC. Introduction, main results}

\bigskip





Throughout this work ${\bf H}$ denotes an arbitrarily fixed
infinite dimensional
complex Hilbert space
with the scalar product $\langle x \vert y \rangle$ which is linear in $x$ and conjugate linear in $y$,
giving rise to the norm $\Vert x\Vert = \langle x \vert x\rangle^{1/2}$.
We denote the open unit ball
$\{ e\in{\bf H}:\ \Vert e\Vert<1\}$ with ${\bf B}$ and
for any vector $a\in{\bf H}$ we shall write $a^*:= [x\mapsto \langle x\vert a\rangle]$
for its dual functional.

Recall that the group ${\rm Aut}({\bf B})$ of all holomorphic automorphisms of ${\bf B}$
consists the biholomorphic maps
${\bf B}\leftrightarrow{\bf B}$, and the  ${\bf H}$-unitary operators restricted to ${\bf B}$
form the isotropy subgroup of the origin of ${\rm Aut}({\bf B})$.
Stone's classical theorem on strongly continuous one-parameter groups of unitary operators
can be reformulated in terms of ${\rm Aut}({\bf B})$ as a statement that the infinitesimal
generator of a strongly continuous one-parameter subgroup of ${\rm Aut}({\bf B})$
leaving fixed the origin can be identified canonically with the restriction $iA\vert{\bf B}$
where $A$ is a possibly unbounded self-adjoint linear operator with dense domain in ${\bf H}$.
The first attempt to reach
an analogous description for the strongly continuous
one-parameter subgroups of ${\rm Aut}({\bf B})$ formed by possibly non-linear maps
can be found in Vesentini's celebrated paper [\VESi] in 1987 based on a linear model
generalizing naturally a well-known analogous concept for finite dimensional M\"obius groups.
Later on [\VESii] he returned to the theme with the aim of extending the results
to strongly continuous one-parameter semigroups holomorphic automorphisms of the unit ball of a Cartan factor of type 1
that is a space of the form ${\cal L}({\bf H}_1,{\bf H}_2)$ with Hilbert spaces ${\bf H}_k$
where a linear model is still available. Katshkevich-Reich-Shoiket [\KHAS] extended these investigations to general
strongly continuous one-parameter semigroups of holomorphic fractional linear transformations.
Nevertheless a simple explicit algebraic description for these semigroups seems not yet being appeared in the literature.

Our purpose
in this paper will be to develop an alternative shorter approach to the description of vector fields
arising as infinitesimal generators of strongly continuous one-parameter subsemigroups of ${\rm Aut}({\bf B})$.
Though several details presented here seem to be contained implicitly in [\VESi,\VESii,\KHAS],
our treatment treatment based on the existence of joint fixed points uses essentially different ideas
which may be of independent geometric interest concerning the structure of the Banach-Lie group of the surjective isometries of a hyperbolic space.
We try give a self-contained presentation staring only from the familiar form (2.3) %(\the\MobshS.\the\MobshT)
for the M\"obius shifts establish first in [\FRVE,Ch.X] in infinite dimensions.
At the beginning we also provide some simple general results concerning the existence of joint fixed points and continuity of boundary extensions
in the setting of reflexive spaces.
We pay particular attention (Section 4) to the characterize the cases where a Kaup type formula ($x\mapsto b - \langle x\vert b\rangle + iAx$
given first in [\KAUP] for the {\it uniformly} continuous case) is available for the vector fields of the infinitesimal generators.
%
%including approximations with norm-continuous one-parameter subgroups.
%
%Actually
We focus to one-parameter groups, establishing the following main results.

\bigskip
\ADT
\newcount\MainS \MainS=\SEC
\newcount\MainT \MainT=\THM
\bf Theorem \NUM. \it
Assume $\big[ \Psi^t:\ t\in\RR]$ is a strongly continuous one-parameter
group of holomorphic automorphisms of \ ${\bf B}$.\footnote{$^1$}{\rm
That is $\Psi^{t+s}=\Psi^t\circ\Psi^s \in{\rm Aut}({\bf B})$ for all couples $t,s\in\RR$ and the functions
$[t\mapsto\Psi^t(x)]$ are continuous $\RR\to{\bf H}$ for any fixed vector $x\in{\bf B}$.}
Then there exists a vector $\overline{x}$ with $\Vert \overline{x}\Vert \le 1$
along with a
constant $\lambda\in\RR$ and a
densely defined possibly unbounded self-adjoint operator $A:{\bf Z}\to{\bf H}$ with
dense domain such that
%the section
\ADT
\newcount\NipiiS \NipiiS=\SEC
\newcount\NipiiT \NipiiT=\THM
$${\bf B} \cap(\overline{x} + {\bf Z}) ={\bf D} \quad \hbox{where} \quad
{\bf D}:= \big\{ x\in{\bf D}:\ t\mapsto \Psi^t(x) \ \hbox{is differentiable on} \ \RR \big\} ,
\leqno(\NUM)$$
%and we have
\vskip-6mm
\ADT
\newcount\NipiiiS \NipiiiS=\SEC
\newcount\NipiiiT \NipiiiT=\THM
$${d\over dt}\Big\vert_{t=0} \Psi^t(x) =
-\big\langle (iA-\lambda)(x-\overline{x}) \big\vert \overline{x} \big\rangle x +
(iA+\lambda) (x-\overline{x}) \qquad (x\in{\bf D}).
\leqno(\NUM)$$
%In case $\Vert \overline{x}\Vert <1$ we have necessarily $\lambda=0$
Given any tuple $(A,\overline{x},\lambda)$ consisting of a densely defined self-adjoint operator $A:{\bf Z}\to{\bf H}$,
a vector $\overline{x}$ with $\Vert\overline{x}\Vert\le 1$ and a real number $\lambda$,
there exists (a necessarily unique) strongly continuous one-parameter group $[\Psi^t:\ t\in\RR]$
satisfying $(\the\NipiiS.\the\NipiiT)$ and $(\the\NipiiiS.\the\NipiiiT)$ if and only if one of the following alternatives holds: \quad
%$(1)$ $\overline{x}\!\in\!{\bf Z}$; \
%$(2)$ $\overline{x}\!\not\in\!{\bf Z},\,$
$(1)$ $\Vert \overline{x}\Vert \!=\! 1$; \
$(2)$
%$\overline{x}\!\not\in\!{\bf Z},\,$
$ \Vert \overline{x}\Vert \!<\! 1,\ \lambda \!=\! 0$.

\bigskip
\ADT
\newcount\CorIS \CorIS=\SEC
\newcount\CorIT \CorIT=\THM
\bf Corollary \NUM. \it If $\Vert\overline{x}\Vert =1$ above and $\overline{x}$ is an eigenvector of the operator $A$ with eigenvalue $\rho(\in\RR)$ then the following alternatives hold: either for some $0\ne\lambda\in\RR$ we have 
$$\Psi^t(x_0 + \xi \overline{x}) =
{e^{\lambda t} \exp(it A) x_0 \over 2\lambda - \varphi(\rho,\lambda,t)(\xi-1)} +
\Big( 1 + {2\lambda e^{2\lambda t} (\xi-1) \over
2\lambda - \varphi(\rho,\lambda,t) (\xi-1) } \Big) \overline{x} $$
where $\lambda\ne0$ and $\varphi(\rho,\lambda,t):=(i\rho - \lambda) e^{2\lambda t}-(\lambda+i\rho)$, or
$$\Psi^t(x_0 + \xi \overline{x}) =
{ \exp(it A) x_0 \over 1 + i\rho t (\xi-1) } + \Big( 1 + {\xi-1 \over 1 + i\rho t (\xi-1)} \Big) \overline{x}$$
for all $t\in\RR$, and $x_0+\xi\overline{x} \in{\bf B}$ with $x_0 \perp \overline{x}$.

\bigskip
\ADT
\newcount\CorIIS \CorIIS=\SEC
\newcount\CorIIT \CorIIT=\THM
\bf Corollary \NUM. \it
If $\overline{x}\ne\overline{x}^\prime \in\partial{\bf B}$ are the only common fixed poits of
$[\overline{\Psi^t} : t\in\RR]$,
%$\bigcap_t {\rm Fix}(\overline{\Psi^t})=\{ p,q \} \subset \partial{\bf B}$ with $p\ne q$  then
there exists $\Theta\in{\rm Aut}({\bf B})$ along with a constant $0\ne \lambda\in\RR$ such that
$\overline{\Theta}(\overline{x})=\overline{x}$, 
$\overline{\Theta}(\overline{x}^\prime)=-\overline{x}$ and
$$\Theta\circ\Psi^t\circ\Theta^{-1} (x+\xi \overline{x}) =
{e^{\lambda t} \exp(it A) x_0 \over 2\lambda - \varphi(0,\lambda,t)(\xi-1)} +
\Big( 1 + {2\lambda e^{2\lambda t} (\xi-1) \over
2\lambda - \varphi(0,\lambda,t) (\xi-1) } \Big) \overline{x} $$
for all $t\in\RR$, and $x_0+\xi\overline{x} \in{\bf B}$ with $x_0 \perp \overline{x}$.


\DEL{
\bigskip
\ADT
\bf Corollary \NUM. \it By setting $A_n:= \!\int_{\vert\xi\vert\!\le\! n} \xi\, dE(\xi)$,
$\overline{x}_n:=\!\int_{\vert\xi\vert\!\le\! n} dE(\xi) \overline{x}$
%$(n \!=\!1,2,\ldots)$
in terms of the spectral resolution $A=\!\int_{\xi\!\in\!\RR} \xi\, dE(\xi)$ for the self-adjoint operator in
$(\the\NipiiS.\the\NipiiT\!-\!3)$, the norm-continuous one-parameter subgroups $[\Psi^t_n:\ t\in\RR] \subset{\rm Aut}({\bf B})$
defined by the generator equations
$${d\over dt}\Big\vert_{t=0} \Psi_n^t(x) =
-\big\langle (iA_n-\lambda)(x-\overline{x}_n) \big\vert \overline{x} \big\rangle x +
(iA_n+\lambda) (x-\overline{x}_n) \qquad (x\in{\bf B}).
\leqno(1.3)_n $$
approximate $[\Psi^t:\ t\in\RR]$ pointwise, that is \
$\Psi_t(x)=\lim\limits_{n\to\infty} \Psi_n^t(x)$ \ $(x\in{\bf B})$.

\rm

%\eject
}%ENDDEL

\DEL{
\bigskip
\rm
It is tempting to conjecture that all strongly continuous one-parameter groups of holomorphic automorphisms of bounded circular domains
in Banach spaces can be reconstructed similarly from a polynomial of second degree defined on some dense algebraic submanifold.
However, the techniques we use here depend heavily on Hilbert space structure.
}%ENDDEL


\bigskip\bigskip
\ADS
{\bf \the\SEC. Preliminaries: linear model with joint fixed points} % for one-parameter groups}
% in ${\rm Aut}({\bf B})$}







%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


\bigskip
\ADT
\bf Lemma \NUM. \it Assume ${\cal K}$ is a compact topological space and let
$[f_t :\ t\in\RR_+]$ be a one-parameter semigroup of continuous maps
${\cal K}\to{\cal K}$ admitting fixed points
such that all the functions $t\mapsto f_t(x)$ are continuous. Then
also $\bigcap_{t\in\RR_+} {\rm Fix}(f_t) \ne \emptyset$.
%
\bigskip
\bf Proof. \rm
Consider any parameter $t>0$ and a point $x\in{\rm Fix}(f_t)$.
For $n=1,2,\ldots$ recursively we have
$f_{nt}(x) =f_t\big( f_{(n-1)t} (x)\big) = x$.
Thus
${\rm Fix}(f_{nt}) \supset {\rm Fix}(f_t) \ne \emptyset$ $(t\in\RR_+, \ n=1,2,\ldots)$.
From the continuity of the maps $f_t$ it follows that
${\rm Fix}(f_{1/n!})$ \ $(n=1,2,\ldots)$ is a decreasing sequence of non-empty compact sets
with non-empty intersection ${\cal X}:= \bigcap_n {\rm Fix}(f_{1/n!})$.
Since any rational number $0\ne q\in\QQ_+$ can be written in the form $q=m/n!$
for suitable integers $m,n>0$, it follows even that
$\bigcap_{q\in \QQ_+} {\rm Fix}(f_q) = {\cal X} \ne \emptyset$.
Consider any parameter $t>0$ and any point $x\in{\cal X}$.
Given any sequence $q_1,q_2,\ldots\in\QQ_+$ converging to $t$, the continuity of
the orbit $t\mapsto f_t(x)$
ensures that
$x=\lim_n f_{q_n}(x) = f_t(x)$.

\DEL{

\bigskip
%\ADT
\bf Lemma \NUM a. \it Assume $[ f_q :\ t\in\QQ_+]$ is a one-parameter semigroup of
w*-continuous maps $f_t: \overline{{\bf K}}\to\overline{{\bf K}}$.
Then $\bigcap_{q\in\QQ_+}  {\rm Fix}(f_q) \ne\emptyset$.

\bigskip
\bf Proof. \rm
By the Schauder-Tychonoff fixed point theorem, ${\rm Fix}(f_q)\ne \emptyset$ for any $q\in\QQ_+$.
Given any $q\in\QQ_+$ and $n\in\NN$ along with a fixed point $x\in{\rm Fix}(f_{q/n})$,
for $k=1,2,\ldots,n$ consecutively we have
$x=f_{q/n}\big( f_{q/n}^{k-1}(x)\big) = f_{q/n}^k(x)$ showing that
${\rm Fix}(f_{q/n}) \subset {\rm Fix}(f_q)$ for all $q\in\QQ_+$ and $n\in\NN$.
Hence
${\rm Fix}(f_{1/n!})$ \ $(n=1,2,\ldots)$ is a decreasing sequence of non-empty compact sets
with non-empty intersection ${\bf X}:= \bigcap_n {\rm Fix}(f_{1/n!})$.
Given any $0\ne q\in \QQ_+$ we can write $q=m/n!$ for suitable $m,n\in\NN$ entailing that
${\bf X} \subset {\rm Fix}(f_{1/n!}) \subset {\rm Fix}(f_{m/n!}) = {\rm Fix}(f_{q})$.


}%ENDDEL


\bigskip
\ADT
\bf Lemma \NUM. \it
Let ${\bf K}$ be a domain in a Banach space ${\bf E}$ and
let $f_t : {\bf D}_t \to {\bf E}$ $(t\in\RR_+]$ be a family of holomorphic maps
defined on open neighborhoods of $\overline{{\bf K}}$
such that the restrictions $[ f_t\vert {\bf K} : t\in\RR_+]$ form
a strongly continuous one-parameter semigroup.
Assume that for every boundary point $x\in\partial {\bf K}$ there exists
a 1-dimensional complex disc $\Delta_x$ centered in $x$
and intersecting ${\bf K}$
such that
$\Delta_x \subset \bigcap_{t\in[0,\delta_x]} {\bf D}_t$
and $\bigcup_{t\in[0,\delta_x]} f_t(\Delta_x)$ is a bounded set
for some $\delta_x>0$.
Then
$[ f_t\vert \overline{{\bf K}} : t\in\RR]$ is also a strongly continuous one-parameter semigroup.
%the functions $t\mapsto f_t(x)$ are continuous
%for every point $x\in\partial {\bf K}$.


\bigskip
\bf Proof. \rm By assumption $f_s(f_t(x)) =f_{s+t}(x)$ $(x\in {\bf K}, \ s,t\in\RR_+)$.
Since the maps $f_t\vert \overline{{\bf K}}$ are all continuous,
and it follows $f_s(f_t(x)) =f_{s+t}(x)$ $(x\in \overline{{\bf K}}, \ s,t\in\RR_+)$
that is $[f_t\vert\overline{{\bf K}}: t\in\RR_+]$ is a one-parameter semigroup of continuous maps on
the closure $\overline{{\bf K}}$.
Hence, to complete the proof, it suffices to see only that for any $x\in\partial{\bf K}$, the function
$t\mapsto f_t(x)$ is continuous on some neighborhood the origin, namely on $[0,\delta_x)$.
Fix any $x\in\partial{\bf K}$ and consider a convergent sequence $t_n\to t$ within $[0,\delta_x]$.
We show the convergence $f_{t_n}(x)\to f_t(x)$ as follows.
We can write $\Delta_x = \{ x+\zeta v: \vert \zeta\vert < 1\}$
with a suitable vector $v\in{\bf E}$.
By assumption, the functions $g_n(\zeta):= f_{t_n}(x+\zeta v)$ $(n=1,2,\ldots)$ are uniformly bounded
and holomorphic on the unit disc. Furthermore they are assumed to converge pointwise to
$g(\zeta):= f_t(x+\zeta v)$ on the non-empty open complex domain $\{ \zeta\in\CC:\ x+\zeta v \in{\bf K} \}$.
In Banach space setting,
pointwise convergence implies uniform convergence on compact sets for holomorphic maps [\NACHBIN].
In particular, we have uniform convergence for $[g_n: n=1,2,\ldots]$ on some compact disc with positive radius.
By a theorem of Vigu\'e [\VIG, \IST],
for a uniformly bounded sequence of holomorphic maps,
the uniform convergence on some subdomain entails locally uniform (and hence pointwise) convergence on the whole domain.
In particular, $g_n\to g$ pointwise and  hence $f_{t_n}(x) = g_n(0) \to g(0) =f_t(x)$.



\bigskip
\ADT
\bf Corollary \NUM. \it If ${\bf E}$ is a JB*-triple, ${\bf K}$ is its open unit ball and $[f_t: t\in\RR]$ is a strongly continuous one-parameter subgroup of ${\rm Aut}({\bf K})$
then the maps $\overline{f}_t$ obtained with graph closure from the respective $f_t$, form a
strongly continuous one-parameter group of maps $\overline{{\bf K}}\to\overline{{\bf K}}$.

\bigskip
\bf Proof. \rm
It is well-known [\KAUPR] that
we can write $f_t=M_{f_t(0)} \circ U_t$ with some invertible linear operator $U_t\in{\cal L}({\bf E})$ and a
so-called M\"obius transformation with the fractional linear form $x\mapsto f_t(0) + B_t [I+L(x,f_t(0)]^{-1}x$
where $B_t, L(x,f_t(0)) \in{\cal L}({\bf E})$ and $\Vert L(x,f_t(0))\Vert\le \Vert f_t(0)\Vert \Vert x\Vert$.
In particular each $f_t$ extends holomorphically to the ball of radius $1/\Vert f_t(0)\Vert$.
Thus the conditions required by the lemma are fulfilled since $\lim\limits_{t\to 0} f_(0) =0$ by assumption.

\DEL{
\bigskip
\ADT
\bf Lemma \NUM. \it Assume $[U^t: t\!\in\!\RR_+] \!\subset\! {\cal L}({\bf E})$ resp. $[u^t : t\!\in\!\RR_+] \!\subset\! {\bf E}^*$
are families of bounded linear operators resp. linear functionals
of a Banach space ${\bf E}$ such that the operators
$${\cal U}^t := \left[ {U^t \ 0 \atop u^t \ 1}\right] \qquad (t\in \RR_+)$$
form a strongly continuous one-parameter semigroup of ${\cal L}({\bf E}\oplus\CC)$.
Then $[U^t: t\in\RR_+]$ is a strongly continuous one-parameter semigroup of ${\cal L}({\bf E})$.
By writing $A:=dU^t/dt\vert_{t=0}$ resp. ${\cal A}:=d{\cal U}^t/dt\vert_{t=0}$ for the generators
and by setting
$$D_k :=\big\{ x\in{\rm dom}(A): [t\mapsto {\cal U}^t (x\oplus 0)]
\ \hbox{is $k$ times differentiable on $[0,\infty)$}\big\}
,$$
we have ${\rm dom}({\cal A}) \!=\! D_1\!\oplus\!\CC$.
The linear functional
$a x  \!:=\! \lim\limits_{s\downarrow 0} {1\over s} u^s x$  $(x\!\!\in\!\! D_1)$ is well-defined and
$${\cal A} = \Big[ { A \ 0 \atop a \ 0}\Big] ,  \qquad
{\cal U}^t (x\oplus \xi) = \big[U^t x\big] \oplus \big[ {\scriptstyle{\int}}_0^t \ a U^s x \ ds + \xi\big]
\quad (x\in D_2,\ \xi\in\CC).$$
\bf Proof. \rm Using the matrix form of the identity ${\cal U}^t {\cal U}^s \!=\! {\cal U}^{t+s}$, for any $x \!\in\!{\bf E}$ and $\xi\!\in\!\CC$  we get
\ADT
$$\big[ U^t U^s x \big] \oplus \big[ u^t(U^s x) + u^s x + \xi \big] = \big[ U^{t+s} x \big] \oplus \big[ u^{t+s} x + \xi \big] .
\leqno(\NUM)$$
Regarding the first components in (\NUM), we deduce the fact that $[U^t: t\in\RR_+]$ is necessarily a strongly continuous semigroup.
In particular ${\cal U}^t (0\oplus 1) = 0\oplus 1$ $(t\in\RR_+)$, whence we see that \
$0\oplus 1\in {\rm dom}({\cal A})=\{ x\oplus\xi: t\mapsto {\cal U}^t (x\oplus\xi)$ is differentiable$\}$  \ entailing also that
${\rm dom}({\cal A}) = {\rm dom}({\cal A}) + \CC (0\oplus 1) = D_1\oplus \CC$.
From (\NUM) it is also immediate that
$d{\cal U}^t(x\oplus \xi)= \big[ A U^t x\big] \oplus \big[ a U^t x\big]$ $(x\in D_1)$. In particular, if $x\in D_2$
(or only $[t\mapsto u^t x] \in {\cal C}^2[0,\infty)$), we have
$u^t x = u^0 x + \int_0^t [ d u^s x / ds] ds = 0 + \int_0^t [a U^s x] ds$ \ which completes the proof.

\bigskip
\ADT
\bf Remark \NUM. \rm
It is well-known that the linear submanifolds $D_k$ above are dense in ${\bf E}$.
Actually,
given any function $\phi\in{\cal C}^k_0(\RR)$ with compact support within $(0,\infty)$ and with $\int \phi =1$,
for any $\Goth{z} \!\in\!{\bf E} \!\oplus\! \CC$ and $h \!>\! 0$ the vectors
$\Goth{z}_h:=h^{-1} \int_0^\infty \phi(s/h) {\cal U}^s \Goth{z} \ ds$
satisfy $\Goth{z}=\lim\limits_{h\downarrow 0} \Goth{z}_h$ and the functions
$t\mapsto {\cal U}^t \Goth{z}_h =
h^{-1} \int_0^\infty  \phi\big( (s-t)/h \big) {\cal U}^{t} \Goth{z} \ ds$ belong to  ${\cal C}^k ([0,\infty),{\bf E}\oplus\CC)$.
}%ENDDEL

\DEL{ 
\bigskip
\ADT
%\newcount\TriS \TriS=\SEC 
%\newcount\TriT \TriT=\THM
\bf Lemma \NUM. \it Let ${\bf E}_1,{\bf E}_2$ be Banach spaces and suppose $[W^t: t\in\RR_+]$ is a
strongly continuous one-parameter subsemigroup of ${\cal L}({\bf E}_1)$ with the infinitesimal generator $G$.
Furthermore let $G_1\in{\cal L}({\bf E}_2,{\bf E}_1), \ G_2\in{\cal L}({\bf E}_1,{\bf E}_2)$ with
${\cal G}_1:=\left[ \matrix{G &0 \cr G_1& 0}\right]$ and ${\cal G}_2:=\left[ \matrix{G &G_2 \cr 0& 0}\right]$.
Then
$$\left[\left[ \matrix{W^t &0 \cr G_1\int_0^t W^\tau d\tau & I\cr}\right] : t\in\RR_+\right] , \quad
\left[\left[ \matrix{W^t &\int_0^t W^\tau d\tau G_2 \cr 0 &I\cr}\right] : t\in\RR_+\right] $$
are strongly continuous one-parameter subsemigroups of ${\cal L}({\bf E}_1\oplus{\bf E}_2)$ with the infinitesimal generators
%$\left[ \matrix{G &0 \cr G_1& 0}\right]$
${\cal G}_1, {\cal G}_2$, respectively.
\DEL{
{\rm (i)} $\left[\left[ \matrix{W^t &0 \cr G_1\int_0^t W^\tau d\tau & I\cr}\right] : t\in\RR_+\right]$
%$ \left[\big[ { W^t \qquad 0  \atop  G_1\int_0^t W^\tau d\tau \ I } \big] : t\in\RR_+ \ \right]$
is a
strongly continuous one-parameter subsemigroup of ${\cal L}({\bf E}_1\oplus{\bf E}_2)$ with the infinitesimal generator
$\left[ \matrix{G &0 \cr G_1& 0}\right]$;

\smallskip
{\rm (ii)} $\left[\left[ \matrix{W^t &\int_0^t W^\tau d\tau G_2 \cr 0 &I\cr}\right] : t\in\RR_+\right]$ is a
strongly continuous one-parameter subsemigroup of ${\cal L}({\bf E}_1\oplus{\bf E}_2)$ with the infinitesimal generator
$\left[ \matrix{G &G_1 \cr 0& 0}\right]$.

\smallskip
If the generator $G$ is invertible then above we have $\int_0^t W^\tau d\tau=G^{-1}[W^t-I]=[W^t-I]G^{-1}$.
}%ENDDEL

\bigskip
\bf Proof. \rm
Since ${\cal G}_1$ is a bounded perturbation of the generator ${\cal G}:= G\oplus 0$ of the semigroup $[W^t \oplus 0 : t\in\RR_+]$ with ${\rm dom}({\cal G}) = {\rm dom}(G)\oplus  E_2$,
it suffices to see only that
$${d\over dt}\Big\vert_{t=0+} \left[ \matrix{W^t &0 \cr G_1\int_0^t W^\tau d\tau & I\cr}\right] \left[ \matrix{ x \cr y} \right] =
{\cal G}_1 \left[ \matrix{ x \cr y} \right] \qquad (x\in{\rm dom}(G),\ y\in E_2)$$
which is immediate. The case of ${\cal G}_2$ is analogous.

}%ENDDEL

\DEL{
\bigskip
\ADT
%\newcount\DilS \DilS=\SEC
%\newcount\DilT \DilT=\THM
\bf Lemma \NUM. \it Assume $[U^t: t\in\RR_+]$ is a strongly continuous one-parameter semigroup formed by isometries of a Hilbert space ${\bf H}$.
Then there is Hilbert space $\widehat{{\bf H}}$ containing ${\bf H}$ as a subspace along with a strongly continuous one-parameter group
$[ \widehat{U}^t : t\in\RR]$ formed by isometries of $\widehat{{\bf H}}$ such that $U^t = \widehat{U}^t \big\vert {\bf H}$ $(t\ge 0)$.

\bigskip
\bf Proof. \rm
This is implicit in [\FSZ, III. Thm. 9.3].
%Below we give a short elementary proof.

\bigskip
\ADT
%\newcount\DILS \DILS=\SEC
%\newcount\DILT \DILT=\THM
\bf Lemma \NUM. \it Assume $[U^t: t\in\RR_+]$ is a strongly continuous one-parameter semigroup of (necessarily linear) isometries of a Banach space ${\bf E}$,
${\bf E}$ is a subspace of another Banach space ${\bf F}$ and $U^{t_0}$ is
the restriction of a surjective isometry of $\widehat{{\bf E}}$ for some $t_0>0$. Then
there is subspace $\widehat{{\bf E}}$ of ${\bf F}$ along with a strongly continuous one-parameter group $[\widehat{U}^t: t\in\RR]$ of
surjective isometries of $\widehat{{\bf E}}$
such that $U^t = \widehat{U}^t \big\vert {\bf E}$ $(t\ge 0)$.

\bigskip
\bf Proof. \rm
We may assume $t_0=1$ without loss of generality and let $V$ be a surjective isometric continuation of $U^1$ to ${\bf F}$.
Notice that $V^{-1}$ a surjective ${\bf F}$-isometry with ${\bf E}\subset V^{-1}{\bf E}$ and hence, in general,
$V^n {\bf E} \supset V^{n+1}{\bf E}$ $(n\in\ZZ)$.
Observe also that we have
\ADT
$$V^n U^t x = U^s V^m x \quad \hbox{whenever $V^m x\in {\bf E}$, $n+t=m+s$ with $t,s\ge 0$ and $n,m\in\ZZ$}.
\leqno(\NUM)$$
Hence the definitions
$$\eqalign{
&\widehat{{\bf E}}_0 := \bigcup_{n=0}^\infty V^{-n}{\bf E} = \big\{ x \in{\bf F}: \exists\, n\in\ZZ \quad V^n x \in {\bf E} \big\}, \cr
&\widehat{U}^t_0 x:= \Big[ U^s V^m x: \ m\in\ZZ, \ s\ge 0, \ m+s=t, \ V^m x \in{\bf E} \Big] \qquad (x\in\widehat{{\bf E}}_0)\cr}$$
make sense. It is also immediate that each operator $\widehat{U}^t_0$ is an isometric extension of $U^t$ to the not necessarily closed subspace $\widehat{{\bf E}}_0$
and, as direct a consequence of (\NUM), they have
the one-parameter group property $\widehat{U}_0^t  \widehat{U}_0^t = \widehat{U}_0^{t+s}$ $(s,t\in\RR)$.
Thus taking the closure of $\widehat{{\bf E}}_0$ in ${\bf F}$ for $\widehat{{\bf E}}$ and setting the graph closure of $\widehat{U}_0^t$ for
$\widehat{U}^t$ suit the requirements of the lemma.

}%ENDDEL


%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!






\rm
\bigskip
Henceforth we focus to the case of the unit ball ${\bf B}$ of an infinite dimensional Hilbert space.




\rm
\bigskip
Recall [\FRVE, Ch. VI] that the group of
${\rm Aut}({\bf B})$
all holomorphic automorphisms of ${\bf B}$ admits a matrix
representation. % in the following sense:
Namely each element $\Psi$ of ${\rm Aut}({\bf B})$ has the fractional linear form
\ADT
\newcount\NiipiS \NiipiS=\SEC
\newcount\NiipiT \NiipiT=\THM
$$\Psi(x) = {{Ax + b}\over \langle x \vert c\rangle + d} \ , \qquad
A\in{\cal L}({\bf H}), \ \ b,c\in {\bf H}, \ \ d\in\CC
\leqno(\NUM)$$
and we have
$$\Psi_1 \!\circ\! \Psi_2(x) = \Psi_1\big(\Psi_2(x)\big) =
{Ax + b\over \langle x \vert c\rangle + d} \quad
\hbox{whenever} \quad
\left[ \matrix{A & b\cr c^* &d \cr} \right] =
\left[ \matrix{A_1 & b_1\cr c_1^* &d_1 \cr} \right]
\left[ \matrix{A_2 & b_2\cr c_2^* &d_2 \cr} \right].$$
This representation is unique up to a constant, since in (\the\NiipiS.\the\NiipiT) we necessarily have
$$\left[ \matrix{ A &b \cr c^* & d\cr}\right] =
d \!\left[ \matrix{ {\rm Q}_a & \!\!\!a \cr
a^* &\!\!\! 1 \cr}\right] \! \left[ \matrix{U & 0\cr 0 &1 \cr} \right]
\qquad \hbox{where} \quad a \!:=\! \Psi(0), \
U \!=(\beta_a^2 {\rm P}_a \!+\beta_a\overline{\rm  P}_a)^{-1}\Psi^\prime(0)$$
in terms of the the Fr\'echet derivative $\Psi^\prime$ and the standard notations
$${\rm P}_a \!:=\! \big[ \hbox{orthogonal projection ${\bf H}\!\to\!{\CC}a$} \big], \ \
%\overline{\rm P}_a:=1-{\rm P}_a , \ \
\beta_a \!=\!\sqrt{1 \!-\!\Vert a\Vert^2}, \ \
{\rm Q}_a:= {\rm P}_a+\beta_a (I-{\rm P}_a) .$$
We call the matrix
$$\widetilde{\Psi} :=
\left[ { {\rm Q}_a  \ a \atop a^* \ 1 }\right] \! \left[ { U \ 0 \atop 0 \ \ 1} \right] =
\left[ { {\rm Q}_a U  \ \ \ a \atop (U^*a)^* \ 1 }\right]$$
corresponding to the case with constant $d=1$ the
{\it canonical %matrix
representation} of $\Psi$.
In the sequel we shall write
$$\hbox{
${\cal H}:=\! {\bf H} \!\oplus\!\CC \!=\!
\big\{ \! \big[ {x\atop \xi}\big] \! : x\!\in\!{\bf H},\, \xi\!\in\!\CC\big\}$ }$$
and
%conveniently we
identify the matrix $M:=\big[ m_{ij} \big]_{i,j=1}^2$ where
$m_{11}\!\in\!{\cal L}({\bf H}), \, m_{12}\!\in\! {\bf H}, \, m_{2,1}\!\in\! {\bf H}^*$
and $m_{22}\!\in\!\CC$
with the linear operator $\big[ {x\atop \xi}\big] \mapsto M \big[ {x\atop \xi}\big]$ on ${\cal H}$.
Notice that,
by (\the\NiipiS.\the\NiipiT) we have
\ADT
\newcount\NiipiiS \NiipiiS=\SEC
\newcount\NiipiiT \NiipiiT=\THM
$$\Psi(x) = \Big[ \widetilde{\Psi} \left[{x\atop 1}\right] \Big]_{\CC}^{-1}
\Big[ \widetilde{\Psi} \left[{x\atop 1}\right] \Big]_{{\bf H}}
%= \Pi \widetilde{\Psi}\left[{x\atop 1}\right]
\qquad (x\in{\bf B})
\leqno(\NUM)$$
where $[ \,\cdot\, ]_{\CC}$ %\big[{x\atop \xi}\big] \big]_{\CC} :=\xi$
resp. $[ \,\cdot\, ]_{{\bf H}}$ %\big[{x\atop \xi}\big] \big]_{{\bf H}} :=x$
are the standard notations for the canonical projections ${\cal H}\to\CC$ resp. ${\cal H}\to{\bf H}$.
%
It is immediate that any $\Psi \in{\rm Aut}({\bf B})$
extends holomorphically to the ball $(1-\Vert \Psi(0)\Vert)^{-1}{\bf B}$.
Hence we can define the group of all automorphisms of the closed unit ball
$\overline{{\bf B}} :=\{ x\in{\bf H}:\ \Vert x\Vert\le 1\}$
as
$${\rm Aut}(\overline{{\bf B}}) :=
\big\{ \overline{\Psi}:\ \Psi\in{\rm Aut}\,({\bf B}) \big\}  \quad \hbox{where} \quad
\overline{\Psi}:= \big[
\hbox{continuous extension of $\Psi$ to $\overline{{\bf B}} \big]$}.$$
It is also well-known [\FRVE, Ch.VI] that any mapping
$\overline{\Psi}\in{\rm Aut}(\overline{{\bf B}})$ is weakly continuous
and
preserves the Grassmann family ${\rm Aff}(\overline{{\bf B}})$ of all
complex affine
closed subspaces intersected with $\overline{{\bf B}}$.\footnote{$^1$}{If
$x \!=\!\sum_{k\!=\!1}^2 \!\lambda_k x_k$ with $\lambda_1,\lambda_2\!\!\in\!\CC$
and $\sum_{k\!=\!1}^2 \!\lambda_k \!=\!1$ then
$\overline{\Psi}(x) \!=\! \sum_{k\!=\! 1}^2 \!\alpha_k \overline{\Psi}(x_k)$ \
for some $\alpha_1,\alpha_2\!\!\in\!\CC$ with
$\sum_{k\!=\!1}^2 \!\alpha_k \!=\! 1$ \ $\big($namely
$\alpha_k\! =\! \lambda_k [1 \!+\!\langle x_k\vert U^* a\rangle]/
[1 \!+\!\langle \lambda_1 x_1 \!+\! \lambda_2 x_2 \vert U^* a\rangle]\big)$.}
By Schauder's fixed point theorem, ${\rm Fix}(\overline{\Psi})\ne\emptyset$ since
$\overline{{\bf B}}$ is weakly compact.
Moreover we have the following alternatives:

\smallskip
\hskip10mm (1) ${\rm Fix}(\overline{\Psi}) \in {\rm Aff}(\overline{{\bf B}})$,\qquad
(2) ${\rm Fix}(\overline{\Psi})$ consists of two boundary points.
%$\{ p,q \}\subset\partial {\bf B}$.
\smallskip

In case (2) from the proof of [\FRVE, Thm.VI.4.8] we see even that
$\overline{\Psi} =
\overline{\Phi}\circ \overline{\Theta}_a \circ\overline{\Phi}\,\strut^{-1}$
with a suitable automorphism $\Phi\in{\rm Aut}(\overline{{\bf B}})$ and a
{\it M\"obius shift}
\newcount\MobshS \MobshS=\SEC
\newcount\MobshT \MobshT=\THM
\ADT
$$\Theta_a: x\mapsto {{\rm Q}_a x+a \over 1+\langle x \vert a \rangle}
\leqno(\NUM)$$
for some $0\ne a\in{\bf B}$ such that
${\rm Fix}(\overline{\Theta}_a)=\{-e,e\}$ where $e :=a/\Vert a\Vert$.



\DEL{

\medskip

The next result is an infinite dimensional extension for a simple special case
of a far reaching theorem of Abate [\ABATE]
established for finite dimensional uniformly convex domains.
It seems that Dineen's bidual embedding method [\DINEEN] is suitable in proving a complete
infinite dimensional analogy with uniformly convex domains in Banach spaces.
Instead, below we give a short direct proof on the basis of the special algebraic form $(2.1)$.

\ADT
\bigskip
\bf Lemma \NUM. \it  Abelian subsets of ${\rm Aut}(\overline{{\bf B}})$,
in particular one-parameter subgroups,
admit common fixed points.

\bigskip
\bf Proof. \rm
Assume $\{ \overline{\Psi}_j :\, j\!\in\!{\cal J}\} \!\subset\! {\rm Aut}(\overline{{\bf B}})$
with $\overline{\Psi}_j \!\circ\! \overline{\Psi}_k = \overline{\Psi}_k \!\circ\! \overline{\Psi}_j$
$(j,k \!\in\! {\cal J})$.
By weak continuity, for any index family ${\cal K}\subset{\cal J}$, the set
$\bigcap_{k\in{\cal K}} {\rm Fix}(\overline{\Psi}_k)$ of common fixed points
is weakly compact.
Thus, according to Riesz' intersection theorem, it suffices to see that
$\bigcap_{k\in{\cal K}} {\rm Fix}(\overline{\Psi}_k) \ne \emptyset$
for finite index families ${\cal K}$.
By proceeding to contradiction,
let ${\cal K}=\{ k_1,\ldots,k_N\}$ be a subset of ${\cal J}$ with minimal cardinality such that
$\bigcap_{k\in{\cal K}} {\rm Fix}(\overline{\Psi}_k) = \emptyset$.
Necessarily $N>1$ and ${\bf S}:=\bigcap_{n=1}^{N-1} {\rm Fix}(\overline{\Psi}_{k_n}) \ne \emptyset$
is either a weakly compact convex subset of $\overline{{\bf B}}$ or it consists of two boundary points.
Since $\overline{\Psi}_{k_N}$ commutes with all the maps $\overline{\Psi}_{k_n}$ $(n<N)$,
we have $\overline{\Psi}_{k_N}({\bf S}) \subset {\bf S}$.
Indeed, if $x\in{\bf S}$ then
$\overline{\Psi}_{k_n}\big(\overline{\Psi}_{k_N}(x)\big) =
\overline{\Psi}_{k_N}\big(\overline{\Psi}_{k_n}(x)\big) =
\overline{\Psi}_{k_N}(x) \Rightarrow \overline{\Psi}_{k_N}(x)\in{\rm Fix}(\overline{\Psi}_{k_n})$ $(n<N)$.
Hence Schauder's fixed point theorem excludes the case of ${\bf S}$ being convex.
Suppose ${\bf S}=\{p,q\}\subset\partial{\bf B}$.
Then necessarily ${\rm Fix}(\overline{\Psi}_{k_M}) =\{ p,q\}$
for some index $M<N$ and $\overline{\Psi}_{k_N}: p\leftrightarrow q$.
However, in this case we can write
$\overline{\Psi}_{k_N} =
\overline{\Phi}\circ \overline{\Theta}_a \circ\overline{\Phi}\,\strut^{-1}$
with a suitable automorphism $\overline{\Phi}\in{\rm Aut}(\overline{{\bf B}})$ and a
M\"obius shift
$\overline{\Theta}_a$
of the form $(2.3)$
where $0 \!\ne\! a\!\in\!{\bf B}$.
Then, by setting
$\overline{\Omega} \!:=\!\overline{\Phi}\,\strut^{-1}\!\circ\overline{\Psi}_{k_M}\!\circ\overline{\Phi}$
and $e \!:=\! a/\vert a\Vert$, we have
$\{ \pm e\} \!=\! {\rm Fix}(\overline{\Theta}_a)$ and $\overline{\Omega}: e\!\leftrightarrow\! -e$.
On the other hand, it is immediate that
$e=\lim_{n\to\infty}\overline{\Theta}_a^n (x)$ for every point $x\!\in\!\overline{{\bf B}}\!\setminus\!\{ e\}$.
Taking any point $f\in{\rm Fix}(\overline{\Omega})$ we get the contradiction
$e=\lim_{n\to\infty} \overline{\Theta}_a^n(f) =
\lim_{n\to\infty} \overline{\Theta}_a^n \!\!\circ \overline{\Omega}(f) =
\lim_{n\to\infty} \overline{\Omega} \!\circ \overline{\Theta}_a^n(f) = \overline{\Omega} (e) =-e$.
%\qed

}%ENDDEL

\ADT
\bigskip
\bf Remark \NUM. \rm In finite dimensions, it is customary to normalize (\the\NiipiS.\the\NiipiT) by requiring
${\rm det} \left[ {A \ b\atop c^* d} \right] \!=\! 1$.
Thus, in case of ${\rm dim}({\bf H})=N$, in this manner one can establish a canonical identification
of ${\rm Aut}({\bf B})$
with a subgroup of the classical matrix group ${\rm SL}(N+1)$.
Though in infinite dimensions such a normalization is not available,
for one-parameter groups with common fixed point there is an alternative way as follows.


\ADT
\bigskip
\bf Definition \NUM. \rm
Let
$\big( [\Psi^t :\ t\in\RR], \overline{x} \big)$
be a couple of a
one-parameter subgroup of ${\rm Aut}({\bf B})$ with common fixed point $\overline{x}$
for the continuous extensions of its members to $\overline{{\bf B}}$:
$\overline{x}\in\overline{{\bf B}},\
\overline{\Psi}^t(\overline{x}) =\overline{x} \ \ (t\in\RR).$
In terms of the canonical representations
\DEL{
$\widetilde{\Psi^t} = \left[ {Q_t \ a_t \atop a_t^* \ 1 }\right]
\left[ { U_t \ 0 \atop 0 \ 1} \right]$
}%ENDDEL
define
$$\widehat{\Psi}^t_{\overline{x}} :=
%{1\over 1+\langle U_t \overline{x}\vert a_t\rangle}
\Big[ \widetilde{\Psi^t} \Big[ {\overline{x} \atop 1} \Big]\Big]_{\CC}^{-1}
\widetilde{\Psi^t}  =
{1\over 1+\langle U_t \overline{x}\vert a_t\rangle}
\left[ \matrix{ Q_t &a_t \cr a_t^* &1 \cr}\right]
\left[ \matrix{ U_t  &0 \cr 0 &1 \cr}\right] \qquad (t\in\RR).$$
where \ \
$a_t \!=\!\Psi_t(0)\!\in\! {\bf B}$, $U_t \!\in\!{\cal L}({\bf H})$
is a suitable unitary operator
and
$Q_t:={\rm Q}_{a_t} = P_t + \beta_t (I-P_t)$ \ with \
$P_t x :={\rm P}_{\! a_t} x \!= \Vert a_t\Vert^{- \!2}\langle x \vert a_t\rangle a_t, \
%\overline{P}_t =I \!-\! P_t, \
\beta_t \!=\! \sqrt{1 \!-\! \Vert a_t\Vert^2}$.

Later on, conveniently we shall simply write $\widehat{\Psi}^t$ instead of $\widehat{\Psi}^t_{\overline{x}}$
without danger of confusion.

\ADT
\bigskip
\bf Remark \NUM. \rm
As we have seen
$\Psi^t(x) = (1+\langle U_t x \vert a_t \rangle)^{-1} [Q_{t} U^t x + a_t]
= \big[\Theta_{a_t} \!\circ U_t \big] (x)$.

Thus, by construction we have
$$\overline{\Psi^t}(x) = \Big[ \widehat{\Psi}^t \left[{x\atop 1}\right] \Big]_{\CC}^{-1}
\Big[ \widehat{\Psi}^t \left[{x\atop 1}\right] \Big]_{{\bf H}}
%\Pi \widehat{\Psi}^t \left[{x\atop 1}\right]
\quad (x\in\overline{{\bf B}}) , \qquad
\widehat{\Psi}^t \Big[ { \overline{x} \atop 1} \Big] = \Big[ { \overline{x} \atop 1} \Big] .$$
It is worth to notice that the term $\langle U_t\overline{x}\vert a_t\rangle$ is
actually independent of $U_t$ as
\ADT
\newcount\NiipviiiS \NiipviiiS=\SEC
\newcount\NiipviiiT \NiipviiiT=\THM
$$\langle U_t\overline{x}\vert a_t\rangle =
{\langle \overline{x}-a_t\vert a_t\rangle  \over 1 - \langle \overline{x}\vert a_t \rangle} , \qquad
\widehat{\Psi}^t  =
{ 1-\langle \overline{x}\vert a_t\rangle \over 1 -\langle a_t\vert a_t\rangle}
\left[ \matrix{ Q_t %P_t + \beta_t \overline{P}_t
&a_t \cr a_t^* &1 \cr}\right]
\left[ \matrix{ U_t  &0 \cr 0 &1 \cr}\right] .
\leqno(\NUM)$$
Proof: In general we have $P_t y = \langle y\vert a_t\rangle \langle a_t \vert a_t\rangle^{-2} a_t$
$(0\ne a_t,y\in{\bf H})$. It follows
$\langle P_t U_t \overline{x} \vert a_t\rangle = \langle U_t\overline{x}\vert a_t\rangle$
with $\langle \overline{P}_t U_t \overline{x} \vert a_t\rangle =0$ for any $t\in\RR$.
Thus multiplying the fixed point equation
$\overline{x} = \overline{\Psi}^t(\overline{x}) =
(1+\langle U_t\overline{x}\vert a_t\rangle)^{-1} (P_t+\beta_t\overline{P}_t)U_t\overline{x}$
with $\vert a_t\rangle$, we get
$(1+\langle U_t\overline{x}\vert a_t\rangle)^{-1} \langle U_t\overline{x} +a_t\vert a_t\rangle =
\langle \overline{x} \vert a_t\rangle$
whence the relations $(\NUM)$ are immediate.

\medskip
The power style indexing of $\widehat{\Psi}^t$ in $t$ is justified by the proposition below.

\ADT
\newcount\NiipixS \NiipixS=\SEC
\newcount\NiipixT \NiipixT=\THM
\bigskip
\bf Proposition \NUM. \it Given a strongly continuous one-parameter group
$[\Psi^t:\! t\!\in\!\RR]$ in ${\rm Aut}({\bf B})$ with common fixed point $\overline{x}\in\overline{{\bf B}}$,
the family $[\widehat{\Psi}^{\, t}_{\overline{x}}:\ t\in\RR ]$
\DEL{
$$\widehat{\Psi}^t := {1\over 1+\langle U_t \overline{x}\vert a_t\rangle}
\left[ \matrix{ (P_t + \beta_t \overline{P}_t)U_t &a_t \cr [U_t^* a_t]^* &1 \cr}\right] =
{1\over 1+\langle U_t \overline{x}\vert a_t\rangle}
\left[ \matrix{ P_t + \beta_t \overline{P}_t &a_t \cr a_t^* &1 \cr}\right]
\left[ \matrix{ U_t  &0 \cr 0 &1 \cr}\right]$$
}%ENDDEL
is
a strongly continuous one-parameter group of operators in ${\cal H}$.

\bigskip
\bf Proof. \rm
Since
 $\Psi^t\!\circ\!\Psi^s \!=\! \Psi^{t+s}$ $(t,s\!\in\!\RR)$,
for the representation matrices we have
$\widehat{\Psi}^t \widehat{\Psi}^s \!=\! d_{t,s} \widehat{\Psi}^{t+s}$
with suitable constants $d_{t,s}\in\CC_*$.
The fixed point property
$\overline{\Psi}^t (\overline{x})=\overline{x}$
implies
$$\widehat{\Psi}^t \left[ {\overline{x} \atop 1}\right] =
\left[ {\overline{x} \atop 1}\right] \qquad (t\in\RR).$$
Hence necessarily $d_{t,s}\!=\!1$ $(t,s\!\in\!\RR)$, thus
$\big[ \widehat{\Psi}^t: t\!\in\!\RR\big]$ is a one-parameter matrix group.

By assumption, the function $t\mapsto a_t=\Psi^t(0)$ is norm-continuous $\RR\to{\bf B}$.
Hence we can deduce the strong continuity of the ${\bf H}$-unitary operator valued
%$\RR\to{\cal L}({\f H})$
function $t\mapsto U_t$. Namely consider any vector $x\in{\bf H}$.
To establish the norm-continuity of the function $t\mapsto U_t$,
we may assume without loss of generality that $x\in{\bf B}$.
Then, by the aid of the M\"obius shifts $(2.3)$
we can write
$$U_t x = \big[\Theta_{a_t}^{-1} \circ \Psi_t \big](x) = \Theta_{-a_t} \big( \Psi(x)\big)
\qquad (t\!\in\!\RR)
%\quad \hbox{where} \ \
%\Theta_a : z\!\mapsto\! { [{\rm P}_a \!+\! \beta_a \overline{{\rm P}}_a]z \over 1 \!+\!\langle z\vert a\rangle}
%\ \ \hbox{with} \ \ \beta_a=\sqrt{1 \!-\!\Vert a \Vert^2}
.$$
Observe that the norm continuity of $t\mapsto a_t$ implies the
continuity of $t\mapsto \langle x \vert a_t\rangle$ and $t\mapsto\beta_t \in [0,1)$
entailing the norm-continuity of
$t\mapsto P_t +\beta_t \overline{P}_t \in\overline{{\rm Ball}}\big({\cal L}({\bf H})\big)$.
Hence the required norm-continuity of
$t\mapsto U_t x = (1-\langle x\vert a_t\rangle)^{-1} [(P_t x - a_t + \beta_t \overline{P}_t x]$
is immediate.
In general, the product of two bounded strongly continuous linear operator valued functions $\RR\to{\cal L}({\bf X})$
over a normed space ${\bf X}$
is strongly continuous. Hence we conclude that the entries $(1,1),(1,2),(2,1)$ resp. $(2,2)$
of the matrices $\widehat{\Psi}^t$ are strongly continuous functions
$\RR\to{\cal L}({\bf H})$, $\RR\to{\bf H}$, $\RR\to {\bf H}^* \!\!\simeq\!{\bf H}$ resp. $\RR\to\RR$
which completes the proof.
%\qed


\ADT
\newcount\NiipxS \NiipxS=\SEC
\newcount\NiipxT \NiipxT=\THM
\bigskip
\bf Corollary \NUM. \it Given a strongly continuous one-parameter group $[{\cal T}^t: \ t\in\RR]$ in ${\cal L}({\cal H})$,
the following statements are equivalent

\hskip6mm {\rm (i)} for all $t\in\RR$, the maps $x\mapsto \big[{\cal T}^t\big[ {x\atop 1} \big]\big]_{\CC}^{-1} \big[ {\cal T}^t\big[ {x\atop 1} \big]\big]_{{\bf H}}$
belong to ${\rm Aut}({\bf B})$;

\hskip5.5mm {\rm (ii)} \vtop{
\hbox{we have ${\cal T}^t = e^{\mu t} \widehat{\Psi}^t$ $(t\in\RR)$ for some strongly continuous one-parameter subgroup}
\hbox{$[\Psi^t:\ t\in\RR]$ of ${\rm Aut}({\bf B})$
and a constant $\mu\in\CC$;}}

\hskip5mm {\rm (iii)} each operator ${\cal T}^t$ maps the cone ${\cal K}:=\big\{ \big[{x\atop\xi}\big] :\, \vert\xi\vert^2 > \Vert x\Vert^2\big\}$ onto itself;

\hskip5mm {\rm (iv)} each operator ${\cal T}^t$ maps $\partial{\cal K}:=\big\{ \big[{x\atop\xi}\big] :\, \vert\xi\vert^2 = \Vert x\Vert^2\big\}$ onto itself.

\bigskip
\bf Proof. \rm
The implication (ii)$\Rightarrow$(i) is trivial by (\the\NiipiiS.\the\NiipiiT).

\medskip
Proof of (i)$\Rightarrow$(ii): By assumption the maps
$\Psi^t(x) := \big[ {\cal T}^t\big[ {x\atop 1}\big] \big]_{\CC}^{-1} \big[ {\cal T}^t\big[ {x\atop 1}\big] \big]_{{\bf H}}$
$(t\in\RR,\, x\in{\bf B})$.
are well-defined holomorphic automorphisms of the unit ball ${\bf B}$.
By (\the\NiipiiS.\the\NiipiiT) we have ${\cal T}^t = d_t \widehat{\Psi}^t$ $(t\in\RR)$
with suitable constants $d_t\in\CC_*$.
Fixing any point $x\in{\bf B}$, the strong continuity of the group $[{\cal T}^t:\ t\in\RR]$ implies the
continuity of the function ${\cal T}^t \big[ {x\atop 1}\big]$ whence we deduce also the continuity of
$t\mapsto \Psi^t(x)$ which entails the continuity of
$t\mapsto \widehat{\Psi}^t \big[ {x\atop 1}\big] =d_t^{-1}{\cal T}^t \big[ {x\atop 1}\big]$
and hence the continuity of $t\mapsto d_t$.
%On the other hand,
By the one-parameter group property,
all the relations ${\cal T}^{t+s}={\cal T}^t {\cal T}^s$, $\widehat{\Psi}^{t+s}=\widehat{\Psi}^t \widehat{\Psi}^s$
$(t,s\in\RR)$ hold.  Therefore $d_{t+s}=d_t d_s$ $(t,s\!\in\!\RR)$ and the continuity of $t\mapsto d_t$ establishes the
existence of a constant $\mu\in\CC$ with $d_t=e^{\mu t}$ $(t\in\RR)$.

\medskip
Proof of (i)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv):
Consider the projective Hilbert space ${\cal H}_* \! / \!\!\approx$
associated with ${\cal H}$
regarded as
the set of all nontrivial punctured complex rays $\CC_* \big[{x\atop\xi}\big]$
with the factor topology.\footnote{$^2$}{As usually,
${\cal H}_*:= {\cal H}\!\setminus\!\{\! 0 \!\}$
with the equivalence relation
$\big[{x\atop \xi}\big] \!\sim\! \big[{y\atop \eta}\big] :\!
\iff \CC_*\big[{y\atop \eta}\big] \!=\! \CC_*\big[{x\atop \xi}\big]$
where $\CC_*:=\CC\!\setminus\!\{\! 0\!\}$.
A subset of ${\cal H}\!/\!\approx$ is open if the union of it members (rays in ${\cal H}_*$) is open in ${\cal H}$.}
By homogeneity, any injective linear operator ${\cal T}\in{\cal L}({\cal H})$ acts holomorphically
on ${\cal H}_*\!/\!\approx$ by its factorization
${\cal T}_\approx: \CC_*\big[{x\atop\xi}\big]\mapsto \CC_* {\cal T}\big[ {x\atop \xi}\big]$.
In particular, as admitting a continuous inverse, each map ${\cal T}^t_\approx$ is a holomorphic automorphism of ${\cal H}/\approx$.
Hence the equivalences (i)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv) are straightforward consequences of the facts that,
with the embedding $\Pi: x\mapsto \CC\big[{x\atop 1}\big]$ and its inverse $\pi\big( \CC\big[{x\atop \xi}\big] \big) := x/\xi$ $(\xi\ne0)$,
we have $\Pi {\bf B} := {\cal K}$, $\pi{\cal K}={\bf B}$ and
$\big[{\cal T}^t \big[{x\atop 1}\big]\big]_{\CC}^{-1} \big[{\cal T}^t \big[{x\atop 1}\big]\big]_{{\bf H}} =
\pi\circ {\cal T}^t_\approx \circ \Pi \big[{x\atop 1}\big]$ whenever $\big[ {\cal T}^t \big[{x\atop 1}\big] \big]_{\CC} \ne0$.

\bigskip
\ADT
\bf Corollary \NUM. \it Given any $\Theta \in  {\rm Aut}(\overline{{\bf B}})$,
the $\Theta$-shifted automorphisms $\Phi^t := \Theta\circ\Psi^t\circ\Theta^{-1}$
form strongly continuous one-parameter group with common fixed point $\overline{y}:=\Theta(\overline{x})$
when extended continuously to $\overline{{\bf B}}$ and \ \
$\widehat{\Phi}^t_{\overline{y}} =
e^{\mu t} \widetilde{\Theta}^{-1} \widehat{\Psi}^t_{\overline{x}} \widetilde{\Theta}$ \
$(t\!\in\!\RR)$ \ \ for some $\mu\!\in\!\CC$.



\bigskip
\bigskip

\ADS
{\bf \the\SEC. Infinitesimal generators}
\rm

\bigskip
Throughout this section, let
$\big( [\Psi^t\!\! : t \!\in\!\RR], \overline{x}\big)$
be an arbitrarily fixed couple of a strongly
continuous one-parameter group in ${\rm Aut}({\bf B})$ with a common
fixed point for the continuous extensions in $\overline{{\bf B}}$.
Recalling the Hille--Yosida theorem [\RSZ, Kap.10], Proposition \the\NiipixS.\the\NiipixT
ensures that the differential \ADT
$${\cal A} :  \hbox{{\Fr h}} \mapsto
{d\over dt} \widehat{\Psi}^t \hbox{{\Fr h}}
\quad \hbox{with} \ \ {\cal D} :={\rm dom}({\cal A}) =
\big\{ \hbox{{\Fr h}}\in{\cal H} : t\mapsto \widehat{\Psi}^t \hbox{{\Fr h}} \
\hbox{is differentable on} \ \RR\big\}
\leqno(\NUM)$$
(called the {\it infinitesimal generator} of the linear model
$\big[ \widehat{\Psi}^t \!\! : t\!\in\!\RR\big]$
where $\widehat{\Psi}^t \!\equiv\! \widehat{\Psi}^t_{\overline{x}}$ for short)
is a not necessarily bounded linear map
with closed graph and
$[\widehat{\Psi}^t \!: t\!\in\!\RR]$-invariant domain being dense in ${\cal H}$.
%
Instead of the differential ${\cal A}={d\over dt}\vert_{t=0} \widehat{\Psi}^t$ of the representations
we are primarily interested in the differential
$$\Omega :={d\over dt}\Big\vert_{t=0} \Psi^t:  {\bf D} \to{\bf H}
\quad\hbox{where} \ \ {\bf D}={\rm dom}(\Omega)=\Big\{ x\in{\bf B} : \
{d\over dt}\Big\vert_{t=0} \Psi^t(x) \ \hbox{exists}\Big\} .$$
In order that we could regard the vector field $\Omega$ as a non-linear infinitesimal generator
for $[\Psi^t :\ t\in\RR]$, we should see the density of ${\bf D}$ in $\overline{{\bf B}}$.
In order to establish a non-linear Stone-type theorem, we should determine precise links to
%unbounded
self-adjoint linear operators.


\bigskip
\ADT
\bf Lemma \NUM. \it
${\bf D}$ is
$\big[ \overline{\Psi}^t: t\!\in\!\RR\big]$-invariant.
We have $\big[ {x \atop 1}\big] \!\in\! {\cal D} \!\iff\!  x\!\in\!{\bf D}$ whenever $x\!\in\!{\bf B}$.

\bigskip
\bf Proof. \rm The $\big[ \Psi^t: t\!\in\!\RR\big]$-invariance of ${\bf D}$ is clear
from the group property $\Psi^{t+s}=\Psi^t \circ \Psi^s$ $(t,s\in\!\RR)$.
Moreover even
${\rm dom}\big({d\over dt}\big\vert_{t=0} \Theta \circ \Psi^t \circ \Theta^{-1} \big) = \Theta({\bf D})$
whenever $\Theta $ is any holomorphic automorphism of ${\bf B}$.
Hence, given any point $x\in{\bf B}$, we have
$x\!\in\!{\bf D}$ if and only if $0=\Theta_{-x}(x) \!\in\!
{\rm dom}\big({d\over dt}\big\vert_{t=0} \Phi^t\big)$
with the one-parameter group of the maps $\Phi^t := \Theta_{-x} \circ \Psi^t \circ \Theta_x$
in terms of the M\"obius transformations (2.3).
That is, without loss of generality, it suffices only to see the equivalence \
$0\!\in\!{\rm dom}\big({d\over dt}\big\vert_{t=0} \Phi^t\big) \!\iff\!  \big[{0\atop 1}\big] \!\in \!{\cal D}$.
According to (\the\NiipviiiS.\the\NiipviiiT), by setting
$a_t:=\Psi^t(0)$ and $\hbox{\Fr a}_t:= \widehat{\Psi}^t \big[ {0\atop 1}\big]$
we have
$\hbox{\Fr a}_t=(1-\langle a_t\vert \overline{x} \rangle (1-\Vert a_t\Vert^2)^{-1} \big[ {a_t \atop 1} \big]$
$(t\in\RR)$. Hence the curves $t\mapsto a_t$ resp. $t\mapsto\hbox{\Fr a}_t$ are differentiable in the same time,
which completes the proof.

\bigskip
For later use we also introduce the notations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                             %
%  {\cal D}_0 helyett {\bf Z} %
%  {\cal B}   \Psi^thelyett {\bf D} %
%                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\ADT
$${\bf Z} := \Big\{ z\in{\bf H}:\ \left[ {z \atop 0}\right] \in {\cal D} \Big\}; \qquad
Bz \!:=\!\Big[ {\cal A}\Big[{z \atop 0}\Big]\Big]_{\!{\bf H}}, \
\Lambda z \!:=\!\Big[ {\cal A}\Big[{z \atop 0}\Big]\Big]_{\!\CC} \quad
(z\in{\bf Z}) .
%\leqno(\NUM)
$$

\bigskip
\ADT
\newcount\NiiipiiiS \NiiipiiiS=\SEC
\newcount\NiiipiiiT \NiiipiiiT=\THM
\bf Lemma \NUM. \it ${\bf Z}$ is a dense linear submanifold in ${\bf H}$ \
with \
${\bf D} = \big(\overline{x}+{\bf Z}\big) \cap {\bf B}$ and
${\cal D}= \big[ { {\bf Z} \atop 0}\big] + \CC \big[ {\overline{x} \atop 1}\big]$.
The set ${\bf D}$ is dense in ${\bf B}$
and
$${d\over dt}\Big\vert_{t=0} \overline{\Psi}^t(x) =
%-\Big\{ {\cal A} \left[ {x\atop 1}\right] \Big\}_{\CC} x +
%\Big\{ {\cal A} \left[ {x\atop 1}\right] \Big\}_{{\bf H}}
[\Lambda(\overline{x} -x)]x + B(x-\overline{x})
\qquad (x\in{\bf D}).$$

\bf Proof. \rm
By definition,
$\big[ {\overline{x} \atop 1}\big] \!\in\!{\cal D}$ with ${\cal A} \big[ {\overline{x} \atop 1}\big]\!=\! 0$
since $\widehat{\Psi}^t \!\big[ {\overline{x} \atop 1}\big] \!=\!\big[ {\overline{x} \atop 1}\big]$ $(t\!\in\!\RR)$.
Since ${\cal D}$ is closed for linear combinations, it follows that
$\big[ {{\bf Z} \atop 0}\big] + \CC \big[ {\overline{x} \atop 1}\big] = {\cal D}$
and that ${\bf Z}$ is the image of ${\cal D}$ by the bounded linear operator
$\Pi\big[ {x\atop\xi}\big] := x-\xi\overline{x}$.
Since $\Pi {\cal H}={\bf H}$ and since ${\cal D}$ is dense in ${\cal H}$,
${\bf Z}=\Pi {\cal D}$ is also dense in ${\bf H} =\Pi{\cal H}$.
From Lemma 3.2 we know that ${\bf D}={\bf B} \cap \big\{ x: \ \big[{x\atop 1}\big] \in {\cal D} \big\}$.
Hence the relation ${\bf D} = \big(\overline{x}+{\bf Z}\big) \cap {\bf B}$
along with the density of ${\bf D}$ in ${\bf B}$ is immediate.
%
Given any $x\in{\cal D}$, the relation
$\big[ {x\atop 1} \big] \in{\cal D}$  implies
${\cal A}\big[ {x\atop 1} \big] = {d\over dt}\big\vert_{t=0} \widehat{\Psi}^t \big[ {x\atop 1} \big]$.
Since
$\overline{\Psi}^t (x) =
\big\{ \widehat{\Psi}^t \big[ {x\atop 1} \big] \big\}_{\!\CC}^{\!\!-1}
\big\{ \widehat{\Psi}^t \big[ {x\atop 1} \big] \big\}_{\!{\bf H}}$
along with
$\widehat{\Psi}^0={\rm Id}$
and
${\cal A} \big[{\overline{x}\atop 1}\big] = 0$,
%{d\over dt}\big\vert_{t=0} \widehat{\Psi}^t  \big[{\overline{x}\atop 1}\big] =
%{d\over dt}\big\vert_{t=0} \big[{\overline{x}\atop 1}\big] =0 $,
we get
$$\eqalign{
{d\over dt}\Big\vert_{t=0} \!\!\!\!\overline{\Psi}^t(x) &=
-\Big[ \!\widehat{\Psi}^0 \Big[ {x\atop 1} \Big] \!\Big]_{\!\CC}^{\!\!-2} \!
\left( {d\over dt}\Big\vert_{t=0} \! \Big\{ \!\widehat{\Psi}^0 \Big[ {x\atop 1} \Big] \!\Big\}_{\!\CC} \right) \!
\Big[ \!\widehat{\Psi}^0 \Big[ {x\atop 1} \Big] \!\Big]_{\!{\bf H}} \!\!+
\Big[ \!\widehat{\Psi}^0 \Big[ {x\atop 1} \Big] \!\Big]_{\!\CC}^{\!\!-1}
{d\over dt}\Big\vert_{t=0} \!\Big\{ \!\widehat{\Psi}^0 \Big[ {x\atop 1} \Big] \!\Big\}_{\!{\bf H}} \!\!=\cr
&= - \Big[ {\cal A} \Big[ {x\atop 1} \Big] \Big]_{\CC} x +
\Big[ {\cal A} \Big[ {x\atop 1} \Big] \Big]_{\!{\bf H}} =
 - \Big[ {\cal A} \Big[ {x-\overline{x}\atop 0} \Big] \Big]_{\CC} x +
\Big[ {\cal A} \Big[ {x-\overline{x}\atop 0} \Big] \Big]_{\!{\bf H}}. \ \
%\hbox{Q.e.d.}
\cr}$$

\ADT
\bf Lemma \NUM. \it
Suppose a Hilbert space ${\bf W}$ is the orthogonal sum
of the subspaces ${\bf W}_1,{\bf W}_2$
%${\bf W_k}$ $(k\!=\!1,2)$
and ${\cal C}$ is the infinitesimal generator of a
strongly continuous one-parameter subgroup
\break
$[T^t: t\!\in\!\RR ]$ of ${\cal L}({\bf W})$.
Then,
for the cone
${\bf K} \!:=\!\big\{ w_1\!\oplus\!w_2: %{\bf w}_k\!\in\!{\bf W}_k, \,
\Vert{w}_1\Vert \!>\! \Vert{w}_2\Vert \big\}$,
we have $T^t {\bf K} \!=\!{\bf K}$ $(t\!\in\!\RR)$
if and only if ${\cal C}$ is tangent to the boundary of ${\bf K}$
that is if
\ADT
$${\rm Re}\big\langle {\cal C}({w}_1\!\oplus\!{w}_2) \big\vert {w}_1 \big\rangle =
{\rm Re}\big\langle {\cal C}({w}_1\!\oplus{w}_2) \big\vert {w}_2 \big\rangle
\quad
\big( {w}_1\!\oplus\!{w}_2 \!\in\!{\rm dom}({\cal C}) ,\
\Vert{w}_1\Vert \!=\! \Vert{w}_2\Vert\big) .
\leqno(\NUM)$$

\bf Proof. \rm
It is immediate that $T^t{\bf K\!}\!\subset\! {\bf K}$ $(t\!\in\!\RR)$
$\Rightarrow T^t\partial{\bf K}\!\subset\! \partial{\bf K}$ $(t\!\in\!\RR)$
$\Rightarrow {d\over dt}\vert_{t=0} T^t({w}_1\oplus{w}_2) \in
{\rm Tan}_{{w}_1\oplus{w}_2}({\bf K})$ for ${w}_1\oplus{w}_2\in{\rm dom}({\cal C})$
$\Rightarrow (\NUM)$.
Assume $(\NUM)$ and
let $P$ denote the canonical projection of ${\bf W}$ onto ${\bf W}_1$
and define
$T^{t,s}:=\exp(t{\cal C}+sP)$ $(s,t\in\RR)$.
By the theorem of bounded perturbations [\NAG, p.158]
the operators $T^{t,s}$ are all well-defined.
Moreover, by [\NAG, Corollary 1.7 p. 161]
(applied with $B:=sP$ and $A:={\cal C}$ there)
we have $\lim_{s\to 0} T^{t,s}{w} = T^t{w}$ $({w}\in{\rm dom}({\cal C}), t\in\RR)$.
Therefore, to establish that $T^t{\bf K}\subset {\bf K}$ $(t\in\RR)$,
it suffices to see only that \ %$(*)$
$T^{t,s}{w} \!\in\!{\bf K}$ whenever
${w}\!\in\!{\rm dom}({\cal C})\!\cap\! {\bf K}$ and $t,s\!>\! 0$.
To proceed to contradiction, let
$s,t>0$ and ${w}:={w}_1\oplus {w}_2 \in{\rm dom}({\cal C})$ with
$\Vert{w}_1\Vert > \Vert{w}_2\Vert$ but
$\Vert [T^{t,s}{w}]_1\Vert \le \Vert [T^{t,s}{w}]_2\Vert$.
The function
$\delta(\tau):=\Vert [T^{\tau,s}{w}]_1\Vert^2 - \Vert [T^{\tau,s}{w}]_2\Vert^2$
is differentiable in $\tau$ on the whole $\RR$ and
$\delta(0)> 0 \ge \delta(t)$. Thus there exist a point $t_*\in(0,t]$
such that $\delta(\tau)>0=\delta(t_*)$ $(0\le\tau<t_*)$.
Since $\delta(t_*)=0$, the vector
${w}_*:=T^{t_*,s}{w}$ belongs to $\partial{\bf K}$
and hence
${\rm Re}\langle {\cal C}{w}_* \vert [{w}_*]_1\rangle=
{\rm Re}\langle {\cal C}{w}_* \vert [{w}_*]_2\rangle$.
We get the contradiction $0\ge \delta^\prime(t_*) =
2{\rm Re}\langle ({\cal C}+sP){w}_* \vert [{w}_*]_1\rangle -
2{\rm Re}\langle ({\cal C}+sP){w}_* \vert [{w}_*]_2\rangle =
2s \Vert[{w}_*]_1\Vert^2 >0$.



\ADT
\newcount\NiiipviS \NiiipviS=\SEC
\newcount\NiiipviT \NiiipviT=\THM
\bigskip
\bf Corollary \NUM. \it %Given any vector
%$v\in{\bf Z}$ we have
${\rm Re} \Big( -\Lambda v +
\big\langle B v \big\vert \overline{x} \big\rangle +
\big\langle B v \big\vert v \big\rangle \Big)=0$
whenever $v\in{\bf Z}$ with $\big\Vert \overline{x}+v \big\Vert =1$.


\bigskip
\bf Proof. \rm
By Corollary \the\NiipxS.\the\NiipxT, we have
$\widehat{\Psi}^t {\cal K} = {\cal K}$ %$\widehat{\Psi}^t \partial{\cal K} = \partial{\cal K}$
$(t\!\in\!\RR)$
where
${\cal K}:=\big\{ \big[ {x\atop\xi}\big]:\ \vert\xi\vert >\Vert x\Vert\big\} \subset{\cal H}$.
An application of Lemma 3.4 with
${\bf W}_1:=\CC$, ${\bf W}_2:={\bf H}$,
${\bf K}:={\cal K}$, $T^t:=\widehat{\Psi}^t$, ${\cal C}:={\cal A}$
establishes that \
${\rm Re}\, \big[(\Lambda x) \overline{\xi} \big] = {\rm Re}\, \big\langle B x \big\vert x \big\rangle$
whenever $\big[ {x\atop \xi}\big]  \in{\cal D}$ and $\Vert x\Vert = \vert\xi\vert$.
We obtain the statement with the choice  $x:= v+\overline{x}$ and $\xi:=1$
if $v\in{\bf Z}$ with $\Vert v+\overline{x}\Vert=1$ because then, by Lemma \the\NiiipiiiS.\the\NiiipiiiT,
we have $\big[{x \atop 1}\big] \in{\cal D}$.

\DEL{
\bigskip
\bf Proof. \rm
Let $v\!\in{\bf Z}$, $u := \overline{x} \!+\! v$ and assume $\Vert u\Vert \!=\! 1$.
Since
$\overline{\Psi}\big( \partial {\bf B} \big) =
\partial {\bf B}$ for any
$\Psi \in {\rm Aut}({\bf B})$ (cf. [\FRVE]), we have
$\big\Vert \overline{\Psi}^t(u) \big\Vert \!=\! 1$ \ $(t\!\in\!\RR)$.
It follows
$$0 \ = \ {d\over dt}\Big\vert_{t=0}
\Big\langle \overline{\Psi}^t(u) \,\Big\vert\, \overline{\Psi}^t(u) \Big\rangle \ = \
2 \,{\rm Re} \, \Big\langle {d\over dt}\Big\vert_{t=0} \!\!
\overline{\Psi}^t(u) \ \Big\vert \ u \Big\rangle .$$
Applying Lemma \the\NiiipiiiS.\the\NiiipiiiT, we see that
$\big\langle [-\Lambda v] u \!+\! Bv \,\big\vert\, u \big\rangle \!=\!0$.
Taking into account that $\langle u\vert u\rangle =1$,
the statement is immediate.
%\qed
}%ENDDEL

\bigskip
\ADT
\newcount\NiiipviiS \NiiipviiS=\SEC
\newcount\NiiipviiT \NiiipviiT=\THM
\bf Proposition \NUM. \it For some symmetric linear operator $A:{\bf Z}\to{\bf H}$ and a suitable constant $\lambda\in\RR$
which is necessarily $=0$ if $\Vert \overline{x}\Vert \ne 1$, we have $(\the\NipiiiS.\the\NipiiiT)$ as
$$B=iA+\lambda I, \qquad \Lambda z= \big\langle (iA-\lambda I)z \big\vert \overline{x} \big\rangle \quad (z\in{\bf Z}). $$

\DEL{
\it For some constant $\lambda\in\RR$
we have
$$\Lambda z = \big\langle (B \!-\! 2\lambda I) z \,\big\vert\, \overline{x} \big\rangle , \quad
{\rm Re} \big\langle \big(B \!-\!\lambda I \big)z \,\big\vert\, z \big\rangle =0 \qquad (z\in{\bf Z}) .$$
In particular, if \ $\Vert \overline{x}\Vert <1$ then necessarily $\lambda=0$ above.
}%ENDDEL


%\bigskip
\bf Proof. \rm
Consider any vector $0\ne z\in{\bf Z}$.
Let $\zeta\in \CC$ be the (unique) constant such that
$\overline{x}+\zeta z \perp z$ and define
$\varrho :=\sqrt{1-\Vert \overline{x}+\zeta z\Vert^2}$.
Actually we have
$\zeta = -\langle \overline{x} \vert z\rangle / \langle z\vert z\rangle$ and
$1\ge \Vert\overline{x} \Vert^2 = \Vert \overline{x}+\zeta z\Vert^2 + \Vert -\zeta z\Vert^2$
showing that both $\zeta$ and $\varrho$ are well-defined.
Consider the unit vectors
$$v_\varphi:= \overline{x}+\zeta z + e^{i\varphi} \varrho z \qquad (\varphi\!\in\!\RR) .$$
According to Corollary \the\NiiipviS.\the\NiiipviT,
$${\rm Re} \Big( (\zeta \!+\! e^{i\varphi} \varrho) \big[- \Lambda z +
\big\langle Bz \,\big\vert\, \overline{x}\big\rangle \big] +
\vert \zeta \!+\! e^{i\varphi}\varrho\vert^2 \big\langle Bz\,\big\vert\, z \big\rangle \Big) =0
\qquad (\varphi\in\RR).$$
Thus the identity \ \
${\rm Re} \big( \alpha + \beta e^{i\varphi} +\gamma  e^{-i\varphi}\big)=0$
for all $\varphi\in\RR$ \ \
holds with the constants
\break
$\alpha:=
\zeta\big[ -\Lambda z + \big\langle Bz \,\big\vert\, \overline{x}\big\rangle \big] +
(\vert\zeta\vert^2 + \varrho^2) \big\langle Bz \,\big\vert\, z\big\rangle$,
$\beta:= \varrho\, \big[ -\Lambda z + \big\langle Bz \,\big\vert\, \overline{x} \big\rangle +
\overline{\zeta} \big\langle Bz \,\big\vert\, z \big\rangle\big]$
and
$ \gamma:= \varrho\zeta \big\langle Bz \,\big\vert\, z \big\rangle$.
\DEL{
$$\eqalign{
&{\rm Re} \big( \alpha + \beta e^{i\varphi} +\gamma  e^{-i\varphi}\big)=0
\quad  (\varphi\in\RR) \cr
&{\rm where} \ \ \alpha:=
\zeta\big[ -\Lambda z + \big\langle Bz \,\big\vert\, \overline{x}\big\rangle \big] +
(\vert\zeta\vert^2 + \varrho^2) \big\langle Bz \,\big\vert\, z\big\rangle,\cr
&\hskip11mm
\beta:= \varrho\, \big[ -\Lambda z + \big\langle Bz \,\big\vert\, \overline{x} \big\rangle +
\overline{\zeta} \big\langle Bz \,\big\vert\, z \big\rangle\big],
\quad \gamma:= \varrho\zeta \big\langle Bz \,\big\vert\, z \big\rangle .\cr}$$
}%ENDDEL
Since \
$2\, {\rm Re} \big( \alpha + \beta e^{i\varphi} +\gamma  e^{-i\varphi} \big) =
2\, {\rm Re} (\alpha) + (\beta+\overline{\gamma})
e^{i\varphi} + (\overline{\beta}+\gamma)e^{-i\varphi}$,
\break
we have necessarily
${\rm Re}(\alpha) = \beta+\overline{\gamma}=0$.
%The identity $\beta+\overline{\gamma}=0$ corresponds to our statement.
From the relation \ $\beta+\overline{\gamma}=0,$ \ it follows
\ADT
\newcount\NiiipviiiS \NiiipviiiS=\SEC
\newcount\NiiipviiiT \NiiipviiiT=\THM
$$\Lambda z - \langle Bz \,\vert\, \overline{x} \rangle =
2\, \overline{\zeta}\, {\rm Re}\, \langle Bz \,\vert\, z \rangle =
-2\, \langle z \,\vert\, \overline{x} \rangle \,
{{\rm Re}\, \langle Bz \,\vert\, z \rangle \over \langle z \,\vert\, z \rangle} \ ,
\leqno(\NUM)$$
and substituting this into the relation
$0={\rm Re} (\alpha)$, we get
$$0=(\varrho^2 - \vert\zeta\vert^2) \, {\rm Re}\, \big\langle Bz \,\big\vert\, z\big\rangle =
\big(1-\Vert \overline{x}\Vert^2 \big) \, {\rm Re}\, \big\langle Bz \,\big\vert\, z\big\rangle \ .
\leqno(\the\NiiipviiiS.\the\NiiipviiiT^\prime)$$
From $(\the\NiiipviiiS.\the\NiiipviiiT)$ we see also that
$$z\mapsto{ {\rm Re} \big\langle Bz \,\big\vert\, z\big\rangle
\over  \big\langle z \,\big\vert\, z\big\rangle } =
-{1\over 2}\,
{\Lambda z - \big\langle Bz \,\big\vert\, \overline{x} \big\rangle
\over\big\langle z \,\big\vert\, \overline{x} \big\rangle }$$
is a
real valued G\^ateaux holomorphic function on the algebraically open
and in ${\bf Z}$ algebraically dense
domain $\{ z\!\in\!{\bf Z} \! :  z\not\perp \overline{x} \}$ which
is possible only if being constant on ${\bf Z}$.
\break
By writing $\lambda$ for this constant value,
%the statement is immediate
from $(\the\NiiipviiiS.\the\NiiipviiiT)$ and $(\the\NiiipviiiS.\the\NiiipviiiT^\prime)$
we conclude that
$$\Lambda z = \big\langle (B \!-\! 2\lambda I) z \,\big\vert\, \overline{x} \big\rangle , \quad
{\rm Re} \big\langle \big(B \!-\!\lambda I \big)z \,\big\vert\, z \big\rangle =0 \qquad (z\in{\bf Z}) ,$$
and, in particular, if $\Vert \overline{x}\Vert \!<\! 1$ then necessarily $\lambda \!=\! 0$ above.
By setting $A \!:=\! -i(B \!+\!\lambda I)$, hence the statement including the symmetry of $A$ is immediate.

%\qed


\DEL{
\bigskip
Taking int account the obvious reversibility properties of the steps of the proof,
we conclude immediately the following technical statement.\footnote{$^3$}{Direct proof.
Given any $\lambda\in\RR$ and a vector $z\in{\bf H}$ with $\Vert \overline{x}+z \Vert=1$,
we have the following chain of equivalences:
$0= {\rm Re} \big\langle -\langle (iA-\lambda)z \vert \overline{x} \rangle (\overline{x}+z) +
(iA+\lambda I)z \big\vert \overline{x}+z \big\rangle$
iff
$0= {\rm Re} \big[ -\langle (iA-\lambda)z \vert \overline{x} \rangle +
\langle (iA+\lambda I)z \vert \overline{x}+z \rangle \big]$
iff
$0= {\rm Re} \big[ 2\lambda \langle z \vert \overline{x} \rangle +
\langle iA z \vert z \rangle + \lambda \langle z \vert z\rangle \big]$
iff
$0= {\rm Re} \big[ \lambda \big( \langle \overline{x}+z \vert \overline{x} +z \rangle
- \langle \overline{x} \vert \overline{x}\rangle
- \langle \overline{x} \vert z\rangle + \langle z \vert \overline{x} \rangle \big) +
\langle iA z \vert z \rangle \big]$
iff
$0= {\rm Re} \big[ \lambda ( 1 - \Vert \overline{x} \Vert^2 ) +
\langle iA z \vert z \rangle \big]$.
}




\bigskip
\ADT
\bf Corollary \NUM. \it
Given a possibly unbounded linear functional
$\Lambda: {\bf Z}\to\CC$ along with a possibly unbounded linear operator
$B:{\bf Z}\to {\bf H}$, we have
$\big[\Lambda(\overline{x}-x)\big]x + B(x-\overline{x}) \perp x$ for any
$x\in [\overline{x}+{\bf Z}]\cap\partial{\bf B}$
if and only if
$\Lambda z \!=\! \big\langle (B \!-\! 2\lambda I) z \,\big\vert\, \overline{x} \big\rangle , \
{\rm Re} \big\langle \big(B \!-\!\lambda I \big)z \,\big\vert\, z \big\rangle \!=\!0$ $(z\!\in\!{\bf Z})$
with some real constant $\lambda$ if $\Vert\overline{x}\Vert=1$ and
with $\lambda=0$ in case of $\Vert\overline{x}\Vert<1$.

}%ENDDEL

\bigskip
The following geometric converse of Proposition {\the\NiiipviiS.\the\NiiipviiT} is elementary:

\bigskip
\ADT
\bf Remark \NUM. \rm\ Given a dense linear submanifold ${\bf Z}$ in ${\bf H}$, a
symmetric linear operator $A:{\bf Z}\to{\bf H}$, a vector $\overline{x}\in\overline{{\bf B}}$,
the vector field $\Omega_\lambda(x) :=
-\langle (iA-\lambda I)(x-\overline{x}\vert\overline{x}\rangle x + (iA+\lambda I)(x-\overline{x})$
define for $\overline{x}+{\bf Z} $
is tangent to the unit sphere $\partial{\bf B}$ at the points $x\in \overline{x}+{\bf Z}$ with $\Vert x\Vert =1$
whenever either $\Vert\overline{x}\Vert=1$ and $\lambda\in\RR$ or $\Vert\overline{x}\Vert<1$ and $\lambda=0$.

\bigskip
In the sequel we proceed to the problem if the operator $A$ in 
Proposition {\the\NiiipviiS.\the\NiiipviiT} arising from the differential
${d\over dt}\big\vert_{t=0}\Psi^t$ of a strongly continuous one-parameter subgroup of ${\rm Aut}({\bf B})$
is necessarily self-adjoint
and conversely if every self-adjoint operator may appear there. %in \the\NiiipviiS.\the\NiiipviiT.




\DEL{
\bigskip
\bf Lemma. \it
The operator $B:{\bf Z}\to{\bf Z}$ is closed.

\bigskip
\bf Proof. \rm
Consider a convergent sequence
${\bf Z} \ni z_n \to z$ with limit point in ${\bf H}$
such that
$B z_n \to w$ for some $w\in{\bf H}$.
We have to show that \ $(*)$ $z\!\in\!{\bf Z}$ with $Bz \!=\! w$,
For $(*)$, it suffices to
observe that
$$\Lambda z_n = \big\langle (B-2\lambda I) z_n \big\vert \overline{x} \big\rangle
\to \big\langle w -2\lambda z \big\vert \overline{x} \big\rangle ,$$
recalling that
%, by the Hille-Yosida theorem,
the not necessarily bounded linear operator
${\cal A} \big[ { x \atop \xi} \big] =
\big[ { B(x-\xi\overline{x}) \atop \Lambda(x-\xi\overline{x})} \big]$ mapping
the linear manifold
${\cal D} := \big[ { {\bf Z} \atop 0} \big] \!\!+\! \CC \big[ { \overline{x} \atop 1} \big]$
into itself is closed.
Indeed we have
$$\Big[ { {\bf Z} \atop 0} \Big] \ni \Big[ { z_n \atop 0} \Big] \to \Big[ { z \atop 0} \Big] , \quad
{\cal A} \Big[ { z_n \atop 0} \Big] = \Big[ { B z_n \atop \Lambda z_n} \Big] \to
\Big[ { w \atop \langle w -2\lambda z \vert \overline{x} \rangle} \Big]$$
implying $(*)$ along with the relation
$\Lambda z= \langle w -2\lambda z \vert \overline{x} \rangle =
\langle (B-2\lambda I) z \vert \overline{x} \rangle$.
%\qed


\ADT
\bf Definition \NUM. \rm
We shall write
$A:= -i(B+\lambda I)$. Thus henceforth
\ADT
$$Bz=(iA +\lambda I)z \, , \quad
\Lambda z = \big\langle (iA-\lambda I)z \big\vert \overline{x} \big\rangle
\qquad (z\in{\bf Z})
\leqno(\NUM)$$
where $A:{\bf Z}\to{\bf Z}$ is a densely defined %closed
symmetric linear operator
and $\lambda$ is some real number.

\bigskip
\ADT
\bf Remark \NUM. \rm
We know already that any vector field of the form
$$\overline{x}+{\bf Z} \ni \overline{x} + z \mapsto
-[\Lambda z] (\overline{x}+z)+Bz$$
is tangent to the unit sphere if and only if
$B$ and $\Lambda$ are of the form (3.10) with a symmetric operator $A$
and any real number $\lambda$ if $\Vert \overline{x}\Vert =1$ and with $\lambda=0$
in case of $\Vert\overline{x}\Vert=1$.

Our next goal is to establish that the operator $A$



In order to pass to symmetric operators. Thus for any vector
$\big[ {x\atop\xi}\big] \in{\cal D}$ we have
$${\cal A} \Big[ {x\atop\xi} \Big]= {d\over dt}\Big\vert_{t=0} \widehat{\Psi}^t \Big[ {x\atop\xi} \Big] =
\left[{ (iA+\lambda I) (x-\xi\overline{x}) \atop \langle (iA-\lambda I) (x-\xi\overline{x}) \vert \overline{x} \rangle} \right]
$$
and for any point $x \in [\overline{x}+{\bf Z}]\cap\overline{\bf B}$ we have
$${d\over dt}\Big\vert_{t=0} \Psi^t (x) =
-\big\langle (iA-\lambda I)(x-\overline{x}) \big\vert \overline{x} \big\rangle x +
(iA+\lambda I) (x-\overline{x})$$
where $A:{\bf Z}\to{\bf Z}$ is a densely defined closed symmetric linear operator
and $\lambda$ is some real number.
}%ENDDEL


\DEL{

\bigskip\bf Proposition. \it
There exists a constant $\rho\in\RR$ such that
$$\Lambda v = \big\langle (B \!+\! \rho I)v \,\big\vert\, \overline{x} \big\rangle, \quad
2\,{\rm Re}\, \big\langle Bv \big\vert v \big\rangle = \rho
\big\langle v \big\vert v \big\rangle \qquad (v \in {\bf Z}).$$
\bf Proof. \rm Consider any vector $0\ne v\in{\bf Z}$ and let
$$z:= \zeta v \quad \hbox{where} \quad
\zeta := -{ \langle \overline{x} \vert v \rangle \over \langle v
\vert v \rangle } .$$ Observe that \ $\overline{x} \!+\! z \perp z
\in{\bf Z}$. Hence we may apply the lemma with the conclusion
$\zeta\, \Lambda v = \big\langle \zeta Bv \,\big\vert\, \overline{x}
\big\rangle + 2\,{\rm Re} \big\langle \zeta Bv \,\big\vert\, \zeta
v\big\rangle$ that is $\Lambda v = \big\langle Bv \,\big\vert\,
\overline{x} \big\rangle + 2 \,\overline{\zeta}\, {\rm Re}
\big\langle Bv \,\big\vert\, v\big\rangle$ implying
$$\Lambda v = \big\langle Bv \,\big\vert\, \overline{x} \big\rangle +
2 \,{ {\rm Re} \big\langle Bv \,\big\vert\, v\big\rangle \over
\big\langle v \,\big\vert\, v\big\rangle } \, \big\langle v
\,\big\vert\, \overline{x} \big\rangle \ .$$ Hence we see that
$$v\mapsto 2 \,{ {\rm Re} \big\langle Bv \,\big\vert\, v\big\rangle
\over  \big\langle v \,\big\vert\, v\big\rangle } = {\Lambda v -
\big\langle Bv \,\big\vert\, \overline{x} \big\rangle
\over\big\langle v \,\big\vert\, \overline{x} \big\rangle }$$ is a
real valued G\^ateaux holomorphic function on the algebraically open
domain $\{ v\!\in\!{\bf Z} : \ v\not\perp \overline{x} \}$ which
is possible only if being constant. Hence the statement is
immediate. %\qed

}%ENDDEL





\bigskip\bigskip
\ADS
\bf \the\SEC. The Jordan case
%$0 \in {\rm dom}\big({d\over dt}\vert_{t=0} \Psi^t \big)$
\rm

\bigskip
We continue the previous investigations
with unchanged notations but under the additional hypothesis that
\ADT
\newcount\NivpiS \NivpiS=\SEC
\newcount\NivpiT \NivpiT=\THM
$$0 \in {\bf D} ={\rm dom}\Big( {d\over dt}\Big\vert_{t=0} \Psi^t \Big) \ .\leqno(\NUM)$$
As we know,
${\bf D} = \big\{ x\in{\bf B}:\ \big[ {x\atop 1}\big] \in {\cal D}\big\}
= {\bf B}\cap(\overline{x} + {\bf Z})$ where
${\cal D}={\rm dom}({\cal A})$ with ${\cal A}={d\over dt}\big\vert_{t=0} \widehat{\Psi}^t_{\overline{x}}$
\break
and
${\bf Z}= \big\{ x\in{\bf H}:\ \big[ {x\atop 0}\big] \in {\cal D} \big\}$
is a dense complex linear submanifold in ${\bf H}$.
Thus, as a consequence of (\the\NivpiS.\the\NivpiT),
for the distinguished common fixed point of the extended automorphisms
$\overline{\Psi}^t$ we have $\overline{x} \in {\bf Z} ={\rm dom}(B) ={\rm dom}(\Lambda)$.
Therefore also
$$\Big[ {0 \atop 1}\Big] \!\in\! {\cal D} \!=\! \Big[ {{\bf Z}\atop\CC} \Big], \quad
{\cal A} \Big[ {x\atop \xi}\Big] \!=\!
\Big[ {B(x-\xi \overline{x}) \atop \Lambda(x-\xi \overline{x})} \Big] \!=\!
\Big[ { B \ \ -\!B\overline{x} \atop \Lambda \ \ -\!\Lambda\overline{x} }\Big] \Big[ {x\atop \xi}\Big]\quad
(x \!\in\! {\bf Z}, \, \xi\!\in\!\CC).$$
\ADT
\bf Remark \NUM. \rm
Recall [\KAUP] that the complete holomorphic vector fields on ${\bf B}$ are the infinitesimal generators
of the uniformly continuous one-parameter subgroups of ${\rm Aut}({\bf B})$ and,
with suitable
$c\in{\bf H}$ and a bounded self-adjoint operator $C\in{\cal L}({\bf H})$
they can be written in the form $x\mapsto a - \{ x c^* x\} + iC x$
by means of the
Jordan triple product.
\ADT
$$\big\{ x y^* z \big\} := {1\over 2} \big\langle x \big\vert y\big\rangle z +
{1\over 2} \big\langle z \big\vert y\big\rangle x
\leqno(\NUM)$$

In terms of the factorization $\Psi^t=\Theta_{a_t}\circ U_t\vert {\bf B}$ we introduce the following vector resp. not necessarily
bounded symmetric linear operator:
$$b:={d\over dt}\Big\vert_{t=0} \Psi^t(0) = \lim_{t\to 0} {1\over t} a_t , \quad
R:=-i{d\over dt}\Big\vert_{t=0} U_t : x\mapsto \lim_{t\to 0} {1\over it} (U_t-I)x . $$
\DEL{
Throughout this section we assume
that
a common fixed point $\overline{x}$ of $\big[ \overline{\Psi}^t:\, t\!\in\!\RR\big]$
lyes in the boundary of the unit ball and the origin belongs to the domain
${\bf D}$ of the non-linear generator
that is we have
$$0\in {\bf D}, \quad \Vert \overline{x} \Vert =1 .
\leqno(Ab)$$

\bf Remark, \rm
Since ${\bf D}=\overline{x}+{\bf Z} \cap \overline{{\bf B}}$
and since ${\bf Z}$ is a complex linear submanifold of ${\bf H}$,
we have $0\in{\bf D}$ if and only if
$\overline{x} \in{\bf Z}$
which is equivalent to
$\big[{0\atop 1}\big]= \big[{\overline{x}\atop 1}\big] - \big[{\overline{x}\atop 0}\big] \in{\cal D}$.
In this case the terms $B\overline{x},\Lambda\overline{x}$
are also well-defined
and in their terms we can write the infinitesimal generator ${\cal A}$ in a matrix form
as
$$\eqalign{
&{\cal A} \Big[ {0\atop 1}\Big]
= {\cal A} \Big( \Big[ {\overline{x}\atop 1}\Big] - \Big[ {\overline{x}\atop 0}\Big] \Big) =
- \Big[ {B\overline{x}\atop \Lambda \overline{x}} \Big] ,\cr
&{\cal A} \Big[ {x\atop \xi}\Big] =
\Big[ {B(x-\xi \overline{x}) \atop \Lambda(x-\xi \overline{x})} \Big] =
\Big[ { B \ -\!B\overline{x} \atop \Lambda \ -\!\Lambda\overline{x} }\Big] \Big[ {x\atop \xi}\Big] .\cr}$$
Notice that in case of
$\big[{0\atop 1}\big] \not\in{\cal D}$ we have only
${\cal A}\big[{x\atop \xi}\big] =
\big[ {B \atop \Lambda} \big]\Big( [I \ -1]\big[{x\atop \xi}\big] \Big)$
while the formal product $\big[ {B \atop \Lambda} \big] [I \ -1]$ cannot be defined.
}%ENDDEL

\bigskip
\ADT
\newcount\NivpivS \NivpivS=\SEC
\newcount\NivpivT \NivpivT=\THM
\bf Proposition \NUM. \it
Under hypothesis $(\the\NivpiS.\the\NivpiT)$, we have ${\bf D}={\bf B}\cap{\bf Z}$ along with ${\rm dom}(R) \supset {\bf Z}$
and the vector field
$\Omega:={d\over dt}\big\vert_{t=0}\Psi^t$ admits the Jordan form
\ADT
\newcount\NivpvS \NivpvS=\SEC
\newcount\NivpvT \NivpvT=\THM
$$\Omega(x) = b-\{ xb^*x \} + iR x \qquad (x\in{\bf D}) .
\leqno(\NUM)$$
\bf Proof. \rm
The relation ${\bf D}={\bf B}\cap{\bf Z}$ is clear since $\overline{x}\in{\bf Z}$.
By the definition of the generator ${\cal A}$,
$${\cal A} \Big[ {0\atop 1}\Big] =
\lim_{t\to 0} {1\over t} \big( \widehat{\Psi}^t - \widehat{\Psi}^0\big) \Big[ {0\atop 1}\Big] =
\lim_{t\to 0} {1\over t} \Big\{ {1\over 1+\langle U_t \overline{x} \vert a_t \rangle}
\Big[ {a_t \atop 1} \Big] - \Big[ {0 \atop 1}\Big] \Big\} .$$
Since $\lim\limits_{t\to 0} a_t =0$
and $\Vert U_t \overline{x} \Vert\le 1$ $(t\in\RR)$,
taking $(\the\NiipviiiS.\the\NiipviiiT)$ into account, we see
that the limit
$$b:=\lim_{t\to 0} {1\over t} a_t = {d\over dt} \Big\vert_{t=0} \Psi^t(0)$$
is well-defined and
$$B \overline{x} = -\Big[ {\cal A} \Big[ {0\atop 1}\Big] \Big]_{{\bf H}} =-b ,
\quad
\Lambda \overline{x} = -\Big[ {\cal A} \Big[ {0\atop 1}\Big] \Big]_{\CC} =
\big\langle \overline{x} \,\big\vert\, b \big\rangle .
%\langle U_t \overline{x} \vert a_t \}rangle = t \Lambda \overline{x} + o(t) , \quad
%a_t= t b + o(t) .
$$
As a consequence we also have
$$\eqalign{
\beta_t &=\sqrt{1-\Vert a_t\Vert^2} = \sqrt{1-\Vert tb+o(t)\Vert^2} =
1 - {1\over 2} \Vert b\Vert^2 t^2 + o(t^2) ,\cr
Q_t &= P_t + \beta_t (I-P_t)= I + (1-\beta_t)P_t = I + o(t) \quad (\hbox{in operator norm}).\cr}$$
Since also $U_t z = z + o(1)$ in norm,
hence we deduce that
%$1-\beta_t = o(t)$ and therefore,
for any vector $z\in{\bf Z}$,  and $\zeta\in\CC$,
\ADT
\newcount\NivpviS \NivpviS=\SEC
\newcount\NivpviT \NivpviT=\THM
$$\eqalign{
\widehat{\Psi}^t \Big[ {z\atop \zeta}\Big] &=
{1- \langle\overline{x}\vert a_t\rangle \over 1- \Vert a_t\Vert^2}
\left[ {Q_t U_t \ \ a_t \atop a_t^* U_t z \ \ 1 } \right] \Big[ {z\atop \zeta} \Big] =
%\big( 1 - t \langle \overline{x} \vert b\rangle +o(t) \big)
%\left[ { (I+o(t)) U_t z \atop t \langle U_t z \vert b\rangle +o(t)} \right] =\cr
%&=
\left[ { U_t z - t \langle \overline{x} \vert b\rangle z  %+ \zeta a_t
\atop t \langle U_t z \vert b\rangle + \zeta
}\right]
 + o(t) \quad \hbox{in norm}.
\cr} \leqno(\NUM)$$


\DEL{
$$\eqalign{
{1\over t} \big( \widehat{\Psi}^t - I \big) \Big[ {z\atop 0} \Big] &=
{1\over t} \left[ {(P_t + \beta_t \overline{P}_t) U_t z -
\langle U^t\overline{x} \vert a_t\rangle z
\atop
t \Lambda z + o(t) } \right] =\cr
&= \left[ {
t^{-1} (U_t-I) z + t^{-1} (1-\beta_t) \overline{P}_t U_t z -
\langle \overline{x} \vert b \rangle z
\atop
\Lambda z} \right] + o(1) .\cr}$$
}

By definition,
$\big[ {Bz  \atop \Lambda z} \big]= \lim\limits_{t\to 0}
{1\over t} \big( \widehat{\Psi}^t - I \big) \big[ {z\atop 0} \big]$.
Hence with well-defined limits we conclude that
\ADT
\newcount\NivpviiS \NivpviiS=\SEC
\newcount\NivpviiT \NivpviiT=\THM
$$Bz =  \lim_{t\to 0} {1\over it} (U_t \!-\! I) z -
\big\langle \overline{x} \, \big\vert  b \big\rangle z
\qquad (z\in {\bf Z}) .
\leqno(\NUM)$$
The strong limit of $t^{-1}(U_t-I)\vert{\bf Z}$ is necessarily symmetric due to the fact that each $U_t$ is unitary.
Thus comparing (\NUM) with Proposition {\the\NiiipviiS.\the\NiiipviiT} stating that the operator $B$
has the form $iA+\lambda I$ with some $\lambda\in\RR$ and a symmetric operator $A$ with ${\rm dom}(A)={\bf Z}$,
we get
\DEL{
In particular, from Proposition {\the\NiiipviiS.\the\NiiipviiT}
we obtain that
for any vector $z\in{\bf Z}$ we have
$i \langle Az \vert z \rangle + \lambda = \lim\limits_{t\to 0} t^{-1} \langle (U_t-I)z \vert z\rangle
- \langle \overline{x} \vert b \rangle$
whence the the symmetry of $A$ along with the properties
$0= {d\over dt}_{t=0} \Vert U_t z\Vert^2 = 2\,{\rm Re} \lim\limits_{t\to 0} t^{-1}\langle (U_t-I)z \vert z\rangle$
and $\lambda\in\RR$ implies
}%ENDDEL
\ADT
\newcount\NivpviiiS \NivpviiiS=\SEC
\newcount\NivpviiiT \NivpviiiT=\THM
$$\lambda= -{\rm Re} \big\langle \overline{x} \,\big\vert\, b \big\rangle \, , \quad
A = \lim_{t\to 0} \hskip-4mm{\phantom{\big\vert}}^{{\rm strong}} \ \big(U^t -I \big)
- {\rm Im} \big\langle \overline{x} \,\big\vert\, b \big\rangle I \Big\vert{\bf Z}
\quad\ \hbox{in \the\NiiipviiS.\the\NiiipviiT} .
\leqno(\NUM)$$
We calculate $\Omega$ by substituting
%the above values of $A$ and $\lambda$
(\the\NivpviiS.\the\NivpviiT-\the\NivpviiiT)
into its form $\Omega(x)= \big[\Lambda(\overline{x}-x)\big]x + B(x-\overline{x})$
applying also the relations $B\overline{x}=-b$, $\Lambda\overline{x}=\langle x\vert b\rangle$,
$B=iA+\lambda I$, $\Lambda x=\langle (iA-\lambda I)x\vert\overline{x}\rangle = \langle (B-2\lambda I)x\vert\overline{x}\rangle$.
Namely, given any vector $x\in{\bf D}$,
taking into account the antisymmetry of the operator $iA = B \!-\!\lambda I$,
we can write
$$\eqalign{
\Omega(x) &= \big[\Lambda \overline{x}\big]x - \big[\Lambda x\big]x+ Bx - B\overline{x} =\cr
&= \langle \overline{x} \vert b \rangle x - \langle (B-2\lambda I)x \vert \overline{x}\rangle x + Bx + b = \cr
&= b + \big[ \langle \overline{x} \vert b \rangle I + B \big] x - \langle (B-\lambda I)x \vert \overline{x}\rangle x
+ \lambda \langle x \vert \overline{x}\rangle x = \cr
&= b + iR x + \langle x \vert (B - \lambda I)\overline{x} \rangle x + \lambda \langle x \vert \overline{x}\rangle x =\cr
&= b + i R x + \langle x \vert B\overline{x} \rangle x =
b + i R x - \langle x\vert b\rangle x .\cr}$$
%The proof is complete since, by Proposition \the\NiiipviiS.\the\NiiipviiT we have $\lambda=0$ in case of $\Vert \overline{x}\Vert\ne 1$.

\DEL{
\bigskip
\bf Lemma. \it We have
$(1+\langle U_t \overline{x}\vert a_t\rangle)^{-1}
\Big[  {
%P_t+\beta_t \overline{P}_t
Q_t \ a_t \atop \ \ \ a_t^* \ \ \ 1} \Big] =
\Big[ {I \ 0 \atop 0 \ 1 }\Big] +
t \Big[ \!\!\!\!
{ -\langle \overline{x} \vert b\rangle I\ b \atop \quad b^* \ -\langle \overline{x} \vert b\rangle} \Big]
+ o^{{\rm norm}}(t).$

\bigskip
\bf Proof. \rm
This is a direct consequence of the facts that
$a_t = tb+ o^{{\rm norm}}(t)$, $\beta_t=1+o(t)$,
$(1+\langle U_t \overline{x}\vert a_t\rangle)^{-1} =
(1-\langle \overline{x} \vert a_t\rangle)(1-\langle a_t \vert a_t\rangle)^{-1} =
(1-t\langle \overline{x} \vert b\rangle +o(t)) \beta^{-2} = 1-t\langle \overline{x} \vert b\rangle +o(t)$
and $P_t +\beta_t \overline{P}_t = (1-\beta_t)P_t +\beta_t (P_t+\overline{P}_t) = I +o^{{\rm norm}}(t)$
as $t\to 0$.
%\qed
}%ENDDEL

\bigskip
\ADT
\bf Corollary \bf\NUM. \it %We have
${\bf Z} = {\rm dom}(R)$
%${\bf Z} = {\rm dom} \big( \lim\limits_{t\to 0} t^{-1} (U_t-I) \big)$,
that is $x\in{\bf Z}$ if and only if the limit $\lim\limits_{t\to 0} t^{-1}(U^tx-x)$ exists.

\bigskip
\bf Proof. \rm
Recall that ${\bf Z} = \big\{ x\in{\bf H} :\ {d\over dt}\big\vert_{t=0} \widehat{\Psi}^t \big[ {x\atop 0}\big]$
{\rm is well-defined}$\big\}$.
From Proposition {\the\NivpivS.\the\NivpivT} we know that
$iRx =\lim\limits_{t\to 0} t^{-1}(U_t-I)x$ is well-defined for every vector $x\in{\bf Z}$.
Conversely, suppose $u=\lim\limits_{t\to 0} t^{-1}(U_tx-x)$ is well-defined. Then
$U_tx= x+tu+o^{{\rm norm}}(t)$ and (\the\NivpviS.\the\NivpviT) establishes that
$\widehat{\Psi}^t \big[ {x\atop 0} \big] =
\big[ {x\atop 0} \big] + t \big[ {u - \langle \overline{x} \vert a_t\rangle x \atop
\langle x \vert b \rangle} \big] + o^{{\rm norm}}(t)$.
\DEL{
$$\eqalign{
\widehat{\Psi}^t \Big[ {x\atop 0} \Big] %&=
%(1+\langle U_t \overline{x} \vert a_t\rangle )
%\Big[ %{P_t +\beta_t \overline{P}_t
%{Q_t \ \ a_t \atop a_t^* \ \ \ 1 } \Big] \,
%\Big[ {U_t \ 0 \atop 0 \ \ 1} \Big] \, \Big[ {x\atop 0} \Big] =\cr
%&= \Big( \Big[ { I \ 0 \atop 0 \ 1} \Big] -
%t \Big[ { \langle \overline{x} \vert a_t\rangle \ -b \atop -b^* \ \langle \overline{x} \vert a_t\rangle} \Big]
%+ o^{{\rm norm}}(1) \Big) \,
%\Big[ {x+t u+o^{{\rm norm}}(t) \atop 0} \Big] =\cr
&= \Big[ {x\atop 0} \Big] + t \Big[ {u - \langle \overline{x} \vert a_t\rangle x \atop
\langle x \vert b \rangle} \Big] + o^{{\rm norm}}(t) . \cr}
$$
}%DEL

\bigskip
\ADT
\bf Lemma \NUM. \it We have \ \ $U_{-t} = U_t^{-1} = U_t^*$, \ $a_{-t} = -U_t^* a_t$ \ \ $(t\in\RR)$.

\bigskip
\bf Proof. \rm
Given any %parameter
$t\in\RR$, we have $\Psi^{-t} = \Psi_t^{-1}$ that is
$\Theta_{a_{-t}} U_{-t} = \big[ \Theta_{a_t} U_t \big]^{-1} =
U_t^{-1} \Theta_{a_t}^{-1} = U_t^{-1} \Theta_{-a_t} = \big[ U_t^{-1} \Theta_{-a_t} U_t\big] U_t^{-1} =
\Theta_{U_t^{-1} (-a_t)} U_t^{-1}$.
By the
%uniqueness of the
unambiguous decomposability of holomorphic automorphisms of circular domains into
M\"obius and unitary parts [\BKU], hence we deduce that
$\Theta_{a_{-t}} = \Theta_{-U_t^{-1} a_t}$ and $U_{-t} = U_t^{-1}$.
%\qed

\bigskip
\ADT
\newcount\NivpxiS \NivpxiS=\SEC
\newcount\NivpxiT \NivpxiT=\THM
\bf Lemma \NUM. \it The operator $R$ is self-adjoint with ${\rm dom}(R)={\bf Z}$.

\bigskip
\bf Proof. \rm
In view of (\the\NivpviS.\the\NivpviT),
and since $U_t x = x+it Rx +o^{{\rm norm}}(t)$ for any
$x\in {\bf Z}={\rm dom}(R)={\rm dom}({d\over dt}\big\vert_{t=0}U_t)$,
we conclude that
\ADT
\newcount\JorgenS \JorgenS=\SEC
\newcount\JorgenT \JorgenT=\THM
$$\eqalign{
{d\over dt}\Big\vert_{t=0} \widehat{\Psi}^t \Big[ {x\atop \xi}\Big]
%&= {d\over dt}\Big\vert_{t=0} (1+\langle U_t \overline{x}\vert a_t\rangle)^{-1}
%\Big[  { %P_t+\beta_t \overline{P}_t
%Q_t \ \ a_t \atop a_t^* \ \ 1} \Big]
%\Big[ {U_t \ 0 \atop 0 \ \ 1} \Big] \Big[ { x\atop \xi}\Big] =\cr
%&= {d\over dt}\Big\vert_{t=0} (1+\langle U_t \overline{x}\vert a_t\rangle)^{-1}
%\Big[  { %P_t+\beta_t \overline{P}_t
%Q_t \ \ a_t \atop a_t^* \ \ \ 1} \Big] \Big[ { x\atop \xi}\Big] +
%{d\over dt}\Big\vert_{t=0} \Big[ {U_t \ 0 \atop 0 \ \ 1} \Big] \Big[ { x\atop \xi}\Big] =\cr
&= \Big[\hskip-3mm { iR \!-\! \langle \overline{x} \vert b\rangle I \ \ \ \ b \atop \qquad\ \ b^*  \quad \ \ -\!\langle \overline{x} \vert b\rangle} \Big]
\Big[ { x\atop \xi}\Big] \cr}
\quad \hbox{for any} \ \ \Big[ {x\atop \xi}\Big] \in{\cal D} = \Big[ {{\bf Z}\atop \CC}\Big].
\leqno(\NUM)$$
The linear operator in ${\cal L}({\cal H})$ with matrix
$\big[ \!\!\!\! {-\!
\langle \overline{x} \vert b\rangle I \ \ b \atop \quad b^* \quad -\langle \overline{x} \vert b\rangle} \big] $
is bounded. Since ${\cal A} ={d\over dt}\big\vert_{t=0} \widehat{\Psi}^t$ with domain ${\cal D}$
is the generator of a strongly continuous semigroup in ${\cal L}({\cal H})$,
by the theorem of bounded perturbations [\NAG], also
the operator with matrix
$\big[ {iR \ 0 \atop \ 0 \ 0}\big]$ with domain ${\cal D}$ is the
generator of a
strongly continuous one-parameter subgroup of ${\cal L}({\cal H})$
entailing that $iR$ is the
generator of a strongly continuous group $\big[ V_t: t\in\RR\big]$ in ${\cal L}({\bf H})$.
Since $U_{-t}=U_t^{-1}=U_t^*$, the arguments on sun adjoint semigroups in [\NAG, p. 69]
show that $\lim\limits_{t\to 0} t^{-1}(U_t^* -I) =-iR$ is the generator of the sun adjoint group
$\big[ V_t^*: t\in\RR\big]= \big[ V_{-t}: t\in\RR\big]$
and we have $-iR=(iR)^*$ which completes the proof.
%\qed




\bigskip
\ADT
\bf Theorem \NUM. \it Any vector field of the form $(\the\NivpvS.\the\NivpvT)$
where $R$ is a not necessarily bounded self-adjoint operator with dense domain ${\bf Z} \subset {\bf H}$,
is the infinitesimal generator defined on ${\bf D}:={\bf Z} \cap{\bf B}$
of a pointwise continuous one-parameter group
$[\Phi^t:\ t\in\RR]$ of holomorphic automorphisms of ${\bf B}$.\footnote{$^3$}{\rm
That is, for all $x\in {\bf D}:={\bf Z}\cap{\bf B}$,
the functions $t\mapsto \Phi^t(x)$ range in ${\bf Z}$, they are differentiable and satisfy
the identity ${d\over dt} \Phi^t(x)={\cal V}\big(\Phi^t(x)\big)$ $(t\in\RR)$.}

\bigskip
\bf Proof. \rm
It suffices to see that
there is a strongly (i.e. pointwise) continuous one-parameter group
$\big[ {\cal V}^t:\ t\in\RR\big]$ of
bounded linear operators of the space ${\cal H}$
%${\bf H}\oplus \CC$
such that
$${d\over dt}\Big\vert_{t=0} {\cal V}^t \Big[ {x \atop \xi} \Big] =
\Big[ {iR \ b \atop b^* \ \ 0}\Big] \Big[ {x \atop \xi} \Big] \quad (x\in{\bf Z},\ \xi\in\CC),
\qquad {\cal V}^t {\cal K} \subset {\cal K} := \Big\{ \Big[ {x\atop \xi}\Big] : \ \Vert x\Vert^2 \ge \vert \xi\vert^2 \Big\}. $$
Namely, in this case the maps
$$\Phi^t (x) := \Big[ {\cal V}^t \Big[ {x\atop 1} \Big] \Big]_{\CC}^{-1} \Big[ {\cal V}^t \Big[ {x\atop 1} \Big] \Big]_{{\bf H}}
\qquad (t\in\RR,\ x\in{\bf D})$$
suit the requirements of the theorem since
$x\in{\bf D} \Rightarrow \big[ {x\atop 1}\big] \in{\cal K}
\Rightarrow {\cal V}^t \big[ {x\atop 1}\big] \Rightarrow \Phi^t(x)\in{\bf D}$ and
$x\!\in\!{\bf D} \!\Rightarrow\! {d\over dt}\big\vert_{t=0} \! \Phi^t(x) \!=\!
- \!\big[ {\cal V}^0 \big[ {x\atop 1} \big] \big]_{\!\CC}^{\!-2}
{d\over dt}\big\vert_{t=0} \big[ {\cal V}^t \big[ {x\atop 1} \big] \big]_{\!\CC}
\big[ {\cal V}^0 \big[ {x\atop 1} \big] \big]_{\!{\bf H}} +
\big\{ {\cal V}^0 \big[ {x\atop 1} \big] \big\}_{\!\CC}^{\!-1}
{d\over dt}\big\vert_{t=0} \big[ {\cal V}^t \big[ {x\atop 1} \big] \big]_{\!{\bf H}}$
$=
- \big[ \big[ {iR \ b \atop b^* \ 0}\big] \big[ {x\atop 1}\big] \big]_{\CC} x \!+\!
\big[ \big[ {iR \ b \atop b^* \ 0}\big] \big[ {x\atop 1} \big] \big]_{\!{\bf H}} =
-\langle x\vert b\rangle x + iRx +b = \Omega(x)$.
%
Notice that, by Corollary \the\NiipxS.\the\NiipxT, a strongly continuous one parameter group of linear operator
leaves the cone ${\cal K}$ invariant if all its members map the boundary
$\partial {\cal K} \!\!=\! \big\{ \!\big[ {x\atop \xi} \big] \!\! : \Vert x\Vert \!\!=\!\!\vert \xi\vert\big\} \!\!=\!
\big\{ \!\big[ {x\atop e^{i\tau}\Vert x\Vert} \big] \!\! :  x\!\!\in\!\!{\bf H},\, \tau\!\!\in\!\!\RR \big\}$
\break
into itself.
\DEL{
\footnote{$^*$}{In general, if a Hilbert space ${\bf W}$ is the orthogonal
%direct
sum
of the subspaces ${\bf W_k}$ $(k\!=\!1,2)$ and $C$ is the infinitesimal generator of a
strongly continuous one-parameter group $[T^t: t\!\in\!\RR ]$ in ${\cal L}({\bf W})$
then we have $T^t {\bf C} \!=\!{\bf C}$ $(t\!\in\!\RR)$ for the cone
${\bf C} \!:=\!\big\{ {\bf w}_1\!\oplus\!{\bf w}_2: {\bf w}_k\!\in\!{\bf W}_k, \,
\Vert{\bf w}_1\Vert \!>\! \Vert{\bf w}_2\Vert \big\}$
if and only if $C$ is tangent to the boundary of ${\bf C}$
that is if
\hfill\break
$(*)$ ${\rm Re}\big\langle\! C({\bf w}_1\!\oplus\!{\bf w}_2),\!{\bf w}_1 \!\big\rangle \!=\!
{\rm Re}\big\langle\! C({\bf w}_1\!\oplus{\bf w}_2),\!{\bf w}_2 \!\big\rangle$ whenever
${\bf w}_1\!\oplus\!{\bf w}_2 \!\in\!{\rm dom}(C)$ with
$\Vert{\bf w}_1\Vert \!=\! \Vert{\bf w}_2\Vert$.
\hfill\break
Proof. It is immediate that $T^t{\bf C\!}\!\subset\! {\bf C}$ $(t\!\in\!\RR)$
$\Rightarrow T^t\partial{\bf C}\!\subset\! \partial{\bf C}$ $(t\!\in\!\RR)$
$\Rightarrow {d\over dt}\vert_{t=0} T^t({\bf w}_1\oplus{\bf w}_2) \in
{\rm Tan}_{{\bf w}_1\oplus{\bf w}_2}({\bf C})$ for ${\bf w}_1\oplus{\bf w}_2\in{\rm dom}(C)$
$\Rightarrow (*)$.
Assume $(*)$ and
let $P$ denote the canonical projection of ${\bf W}$ onto ${\bf W}_1$
and define
$T^{t,s}:=\exp(tC+sP)$ $(s,t\in\RR)$.
By the theorem of bounded perturbations [\NAG, p.158]
the operators $T^{t,s}$ are all well-defined.
Moreover, by [\NAG, Corollary 1.7 p. 161]
(applied with $B:=sP$ and $A:=C$ there)
we have $\lim_{s\to 0} T^{t,s}{\bf w} = T^t{\bf w}$ $({\bf w}\in{\rm dom}(C), t\in\RR)$.
Therefore, to establish that $T^t{\bf C}\subset {\bf C}$ $(t\in\RR)$,
it suffices to see only that \ $(**)$
$T^{t,s}{\bf w} \!\in\!{\bf C}$ whenever
${\bf  w}\!\in\!{\rm dom}(C)\!\cap\! {\bf C}$ and $t,s\!>\! 0$.
To proceed to contradiction, let
$s,t>0$ and ${\bf w}:={\bf w}_1\oplus {\bf w}_2 \in{\rm dom}(C)$ with
$\Vert{\bf w}_1\Vert > \Vert{\bf w}_2\Vert$ but
$\Vert [T^{t,s}{\bf w}]_1\Vert \le \Vert [T^{t,s}{\bf w}]_2\Vert$.
The function
$\delta(\tau):=\Vert [T^{\tau,s}{\bf w}]_1\Vert^2 - \Vert [T^{\tau,s}{\bf w}]_2\Vert^2$
is differentiable in $\tau$ on the whole $\RR$ and
$\delta(0)> 0 \ge \delta(t)$. Thus there exist a point $t_*\in(0,t]$
such that $\delta(\tau)>0=\delta(t_*)$ $(0\le\tau<t_*)$.
Since $\delta(t_*)=0$,
${\bf w}_*:=T^{t_*,s}{\bf w} \in\partial{\bf C}$
and hence
${\rm Re}\langle C{\bf w}_*,[{\bf w}_*]_1\rangle=
{\rm Re}\langle C{\bf w}_*,[{\bf w}_*]_2\rangle$.
We get the contradiction $0\ge \delta^\prime(t_*) =
2{\rm Re}\langle (C+sP){\bf w}_*,[{\bf w}_*]_1\rangle -
2{\rm Re}\langle (C+sP){\bf w}_*,[{\bf w}_*]_2\rangle =
2s \Vert[{\bf w}_*]_1\Vert^2 >0$.
}
}%ENBDDEL
Therefore it suffices to check that
there is a (necessarily unique) strongly continuous one-parameter group
in ${\cal L}({\cal H})$ with domain ${\bf Z}\oplus\CC=\big[ {{\bf Z}\atop\CC}\big]$
%${\bf Z}\oplus\CC$
such that
$${d\over dt} {\cal V}^t \Big[ {x \atop \xi} \Big] =
\Big[ {iR \ b \atop b^* \ \ 0}\Big] {\cal V}^t \Big[ {x \atop \xi} \Big] ,
\quad \left\Vert \Big[ {\cal V}^t \Big[ { x\atop \Vert x\Vert } \Big] \Big]_{{\bf H}} \right\Vert=
\left\vert \Big[ {\cal V}^t \Big[ { x\atop \Vert x\Vert } \Big] \Big]_{\CC} \right\vert
\qquad (x\in{\bf Z},\ t\in\RR) . $$
By Stone's theorem, the ${\cal H}$-unitary operators
${\cal W}^t \big[{x\atop\xi}\big] := \big[{\exp(itR)x \atop \xi}\big]$
form a strongly continuous one-parameter group whose infinitesimal generator
is defined on ${\rm dom}(R) \oplus\CC ={\bf Z}\oplus\CC$ with the diagonal matrix
$\big[ {iR \ 0 \atop 0 \ \ 0}\big]$.
Since the matrix $\big[ {0 \ \ b \atop b^* \ 0} \big]$ represents a bounded linear operator
in ${\cal H}$,
by the theorem of bounded perturbations [\NAG],
there is a strongly continuous one-parameter group
$\big[ {\cal V}^t: \, t\in\RR\big]$
whose generator is defined on
${\bf Z}\oplus\CC$ with the matrix $\big[ {iR \ b \atop b^* \ 0}\big]$.
In particular
${d\over dt} {\cal V}^t \big[ {x\atop \xi}\big] =
\big[ {iR \ b \atop b^* \ 0}\big] {\cal V}^t \big[ {x\atop \xi}\big]$ $(t\in\RR,\ x\in{\bf Z})$.
To complete the proof, we show that necessarily
$${d\over dt} \left[ \Big\Vert\Big[ {\cal V}^t \Big[ { x\atop \Vert x\Vert } \Big] \Big]_{{\bf H}} \Big\Vert^2 -
\Big\vert \Big[ {\cal V}^t \Big[ { x\atop \Vert x\Vert } \Big] \Big]_{\CC} \Big\vert^2 \right] =0
\qquad (t\in\RR,\ x\in{\bf Z}). $$
Consider any vector
$x\in{\bf Z}$ and write $\big[ {x_t \atop \xi_t} \big] :=  {\cal V}^t \big[ {x\atop \Vert x\Vert }\big]$
for all parameters $t\in\RR$.
Then
$$\eqalign{
{d\over dt} \Big[ \Vert x_t \Vert^2 - \vert \xi_t\vert^2 \Big] &=
2\,{\rm Re} \Big[ \big\langle dx_t/dt \big\vert x_t \big\rangle - \big( d\xi_t/dt) \overline{\xi_t} \Big] =\cr
&= 2\,{\rm Re} \left\{ \Big\langle \Big[ \Big[ {iR \ b \atop b^* \ \ 0}\Big]\Big[ {x_t\atop\xi_t}\Big] \Big]_{\!{\bf H}}
\Big\vert x_t \Big\rangle -
\Big[ \Big[ {iR \ b \atop b^* \ \ 0}\Big]\Big[ {x_t\atop\xi_t}\Big] \Big]_{\!\CC}
\big( d\xi_t/dt) \overline{\xi_t} \right\} =\cr
&= 2\,{\rm Re} \left[ \big\langle iRx_t + \xi_t b \big\vert x_t \big\rangle -
\big\langle x_t \big\vert b\big\rangle \overline{\xi_t} \right] =\cr
&= -2\,{\rm Im} \big\langle Rx_t \big\vert x_t \big\rangle +
2\,{\rm Im}\big( \big\langle \xi b\big\vert x_t\big\rangle - \big\langle x_t \big\vert \xi b\big\rangle \big)
=0 . %\hbox{\qed}
\cr}$$




\bigskip
\bigskip

\ADS
\bf \the\SEC. Proof of the main results

\bigskip
\bf Proof of Theorem \the\MainS.\the\MainT. \rm
Consider any pointwise continuous one-parameter subgroup $\big[ \Psi^t:\ t\in\RR\big]$ of ${\rm Aut}\big({\bf B}\big)$
and let us fix any common fixed point $\overline{x}\in\overline{{\bf B}}$
of the continuous extensions
of the maps $\Psi^t$ $(t\in\RR)$ (guaranteed by Lemma 2.4).
From Proposition \the\NiiipviiS.\the\NiiipviiT, we know already that (\the\NipiiS.\the\NipiiT-\the\NipiiiT) hold 
for some dense linear complex submanifold ${\bf Z}$ of the underlying Hilbert space
with a symmetric linear operator $A$ with ${\rm dom}(A)={\bf Z}$.
We have to see that $A$ is even self-adjoint in any case and, conversely
every self-adjoint operator with domain ${\bf Z}$ may appear in (\the\NipiiiS.\the\NipiiiT)
with any constant $\lambda\in\RR$ if $\overline{x}\in\partial{\bf B}$ or $\lambda=0$ if $\overline{x}\in{\bf B}$.
In order to establish a link to the Jordan case,
fix any point
$c \!\in\! {\bf D}\!=\!{\rm dom}\big({d\Psi^t\over dt}\big\vert_{t=0}\big)$
and let
%introduce the automorphisms
$$\Phi^t := \Theta_{-c} \Psi^t \Theta_c \quad (t\in\RR), \quad \overline{y}:=\Theta_c(\overline{x})$$
by means of the M\"obius transformations (2.3).
%
Clearly $\big[ \overline{\Phi}^t:\ t\in\RR\big]$ is a
strongly continuous one-parameter subgroup of
${\rm Aut}\big({\bf B}\big)$
such that
\ADT
\newcount\NvpiS \NvpiS=\SEC
\newcount\NvpiT \NvpiT=\THM
$$0=\Theta_{-c} (c) \in \Theta_{-c} \left({\rm dom}\Big( {\partial \Psi^t \over dt}\Big\vert_{t=0}\Big) \right) =
{\rm dom}\Big( {\partial \Phi^t \over dt}\Big\vert_{t=0}\Big) , \quad
\overline{y} \in
%=\Theta_c(\overline{x} \in \Theta_c\Big( \bigcap_{t\in\RR} {\rm Fix}\big( \overline{\Psi}^t\big) \Big) =
\bigcap_{t\in\RR} {\rm Fix}\big( \overline{\Phi}^t\big) .
\leqno(\NUM)$$
Thus we can apply the results of Section 4 in particular Lemma {\the\NivpxiS.\the\NivpxiT} to $[\Phi^t:\ t\in\RR]$ to conclude
that there is a dense complex linear submanifold ${\bf Y} \subset {\bf H}$ along with a self-adjoint operator
$R$ with ${\rm dom}(R)={\bf Y}$ and a vector $b\in{\bf H}$ such that
$$\widehat{\Phi}^t_{\overline{y}} = e^{-\langle \overline{x} \vert b\rangle t}
\exp \Big( t \Big[ { iR \ \ b \atop b^* \ \ 0}\Big] \Big) \qquad (t\in\RR). $$
Hence Corollary 2.11 establishes the existence of a constant $\nu\in\CC$ with
$$\widehat{\Psi}^t_{\overline{x}} = e^{\nu t} \beta^{-2} \widetilde{\Theta_c} \widehat{\Phi}^t_{\overline{y}} \widetilde{\Theta_{-c}}
\qquad (t\in\RR)$$
due to the identity
$\widetilde{\Theta_c} \widetilde{\Theta_{-c}} = \beta^2 {\cal I} = (1-\Vert c\Vert^2) \big[ {I \ 0\atop 0 \ 1}\big]$
for the canonical representations
$$\widetilde{\Theta_{\pm c}} = \Big[ {Q \ \ \pm c \atop \pm c^* \ \ 1 }\Big]
\quad \hbox{where} \ \ Q:=\beta I + (1-\beta) P , \ \beta:=\sqrt{1-\Vert c\Vert^2}, \ P:= {\rm P}_c .$$
By passing to infinitesimal generators, with $\mu:=\nu-\langle \overline{x} \vert b\rangle$, we get
\ADT
\newcount\NvpiiS \NvpiiS=\SEC
\newcount\NvpiiT \NvpiiT=\THM
$${d\widehat{\Psi}^t_{\overline{x}}\over dt}\Big\vert_{t=0} \Big[{ x\atop \xi}\Big] = \mu \Big[ {x\atop \xi}\Big] +
\beta^{-2} \widetilde{\Theta_c} \Big[ {iR \ \ b\atop b^* \ 0}\Big] \widetilde{\Theta_{-c}} \Big[ {x\atop \xi}\Big] \quad
\hbox{for} \ \ \Big[{ x\atop \xi}\Big]\in{\rm dom}\Big( {d\widehat{\Psi}^t_{\overline{x}}\over dt}\Big\vert_{t=0}\Big).
\leqno(\NUM)$$
From Lemma {\the\NiiipiiiS.\the\NiiipiiiT} and (\the\NvpiS.\the\NvpiT) we see that
$$\Big[ {{\bf Z} \atop 0}\Big] + \CC\Big[ {\overline{x} \atop \xi}\Big] =
{\rm dom}\Big( {d\widehat{\Psi}^t_{\overline{x}}\over dt}\Big\vert_{t=0}\Big) =
\widetilde{\Theta_c} {\rm dom}\Big( {d\widehat{\Phi}^t_{\overline{y}}\over dt}\Big\vert_{t=0}\Big) =
\widetilde{\Theta_c} \Big[ {{\bf Y} \atop \CC}\Big] .$$
Thus, given any vector $z\in{\bf H}$, we have $z\in{\bf Z}$ if and only if
$\big[ {z \atop 0}\big] = \widetilde{\Theta_c} \big[ {y\atop \eta}\big]$ for some $y\in{\bf Y}$ and $\eta\in\CC$
that is if $z=Qy-\langle y\vert c\rangle c =\big [Q-\Vert c\Vert^2  P] y  $ for some $y\in{\bf Y}$.
It follows
\ADT
$${\bf Z} = \big[ Q - \Vert c\Vert^2 P\big] {\bf Y} = Q^{-1} {\bf Y}
\leqno(\NUM)$$
because the operators $P,Q$ commute, we have $\beta^2=1-\Vert c\Vert^2>0$ and
$Q \big[ Q - \Vert c\Vert^2 P\big] = \big[ \beta I + (1-\beta) P] \big[ \beta I + (\beta^2 -\beta)P \big] =
\beta^2 I$.
We are now ready to establish the self-adjointness of the operator $A$ in (\the\NipiiiS.\the\NipiiiT).
By (\the\NvpiiS.\the\NvpiiT) we have
$$(iA + \lambda I) z = \left[ {d\widehat{\Psi}^t_{\overline{x}} \over dt} \Big\vert_{t=0} \Big[ {z\atop 0}\Big] \right]_{{\bf H}} =
\mu z + \beta \big[ iQRQ z  - Qbc^*z + cb^*Q z \big] \quad (z\in{\bf Z}) .$$
That is, with the bounded self-adjoint operator $S:=i\beta [  Qbc^* - cb^*Q z] = [iQbc^*]+[iQbc^*]^*$ we have
$A =  \beta Q R Q + S + i(\lambda-\mu) I$. We know the symmetry of $A$ already entailing the relations $\mu=\lambda$
with $A=\beta QRQ +S$. Here the operator $QRQ$ self-adjoint with ${\rm dom}(QRQ)=Q^{-1}{\rm dom}(R)=Q^{-1}{\bf Y} ={\bf Z} = {\rm dom}(A)$
since $R$ is a self-adjoint operator with ${\rm dom}(R)={\bf Y}$ while $Q$ is an invertible bounded self-adjoint operator,
Therefore, as being the bounded self-adjoint perturbation, the operator $A$ is necessarily self-adjoint.

To see the converse, we need only to check the reversibility of some of our previous arguments.
Assuming $A$ to be self-adjoint in (\the\NipiiS.\the\NipiiT-3), it is the theorem of bounded perturbations [\NAG]
ensures  that the operator
${\cal A}\big[{z+\xi \overline{x}\atop\xi}\big] :=
\big[ {(iA+\lambda I)z \atop \langle (iA-\lambda I) z\vert \overline{x} \rangle (z+\xi \overline{x}) } \big] $
$(z\in{\bf Z},\ \xi\in\CC)$
is the infinitesimal generator of a strongly continuous one-parameter subgroup $[ {\cal U}^t : t\in\RR]$ of ${\cal L}({\bf H})$ with graph
being tangent to the boundary of cone ${\cal K}$ in Corollary \the\NiipxS.\the\NiipxT(iii) and
we have ${\cal K} = {\cal U}^t {\cal K}$ $(t\in\RR)$. Hence
the holomorphic maps $\Psi^t(x) := \big[ {\cal U}^t \big[ {x\atop 1}\big] \big]_{\CC}^{-1} \big[ {\cal U}^t \big[ {x\atop 1}\big] \big]_{{\bf H}}$
are well-defined on the unit ball ${\bf B}$ leaving it invariant, and form a strongly continuous one-parameter subgroup in ${\rm Aut}({\bf B})$.

\bigskip
\bf Proof of Corollary \the\CorIS.\the\CorIT. \rm

By assumption $A \overline{x} =\rho \overline{x}$ in particular $\overline{x}\in{\rm dom}(A)$.
Thus we are in the Jordan case $0\in{\rm dom}(\Omega)$ and hence
$$\eqalign{
\Omega(x) &=
\big\langle x-\overline{x} \big\vert (iA+\lambda I)\overline{x} \big\rangle x + \big( iA + \lambda I\big) (x-\overline{x} =\cr
&= b - \langle x \vert b \rangle x + iR \qquad \hbox{with} \ \
b= \Omega(0) = (-\lambda-i\rho)\overline{x}, \ \
iR = \Omega^\prime(0)= iA+ i\rho I \ .\cr} $$
Recall that we have $\Psi^t(x)= \big[ \widehat{\Psi}^t \big( {x \atop 1} \big)\big]_{\CC}^{-1} \big[ \widehat{\Psi}^t \big( {x \atop 1} \big)\big]_{{\bf H}}$
$x\in{\bf H})$ for the strongly continuous one-parameter group $[ \widehat{\Psi}^t : t\in\RR]$ which,
according to (\the\JorgenS.\the\JorgenT), has the infinitesimal generator
$${d\over dt}\Big\vert_{t=0} \widehat{\Psi}^t =
\left[ \matrix{iR - \langle \overline{x} \vert b\rangle & b \cr b^* &(-\langle \overline{x} \vert b\rangle \cr } \right] =
\left[ \matrix{iA +\lambda I & (-\lambda-i\rho)\overline{x} \cr (-\lambda+i\rho)\overline{x}^* &\lambda-i\rho \cr} \right] .$$
Since the operator $A$ is self-adjoint with eigenvector $\overline{x}$,
it is reduced by the subspaces $\CC\overline{x}$ and ${\bf H}_0:= \overline{x}^\perp (=\{ x\in{\bf H}: x\perp \overline{x}\})$.
Thus in terms of ${\bf H}_0 \oplus \CC\overline{x} \oplus \CC$-matrices,
$${d\over dt}\Big\vert_{t=0} \widehat{\Psi}^t =
\left[ \matrix{ (iA+\lambda I)\big\vert{\bf H}_0 &0 &0 \cr 0 & \lambda+i\rho & -\lambda-i\rho \cr 0 & -\lambda+i\rho &\lambda-i\rho \cr}\right] $$
entailing that \ \ (a) in case $0\ne\lambda\in\RR$ we have
$$\eqalign{
\widehat{\Psi}^t &=
\left[ \matrix{ e^{i\lambda t} \exp(itA)\big\vert{\bf H}_0 &0 \cr 0 &
\exp t \left[ {\lambda+i\rho \ \  -\lambda-i\rho \atop -\lambda+i\rho  \ \ \lambda-i\rho} \right] \cr}\right] =\cr
&= \left[ \matrix{ e^{i\lambda t} \exp(itA)\big\vert{\bf H}_0 &0 &0 \cr 
0 &(\lambda+i\rho) e^{2\lambda t} + (\lambda+i\rho) &(-\lambda-i\rho) e^{2\lambda t} + (\lambda-i\rho) \cr 0 &(-\lambda+i\rho) e^{2\lambda t} + (\lambda+i\rho) &(\lambda-i\rho) e^{2\lambda t} + (\lambda-i\rho) \cr}\right] , \cr}$$
(b) in case of $\lambda=0$ we have
$$\eqalign{
\widehat{\Psi}^t &=
\left[ \matrix{ e^{i\lambda t} \exp(itA)\big\vert{\bf H}_0 &0 \cr 0 &
\exp it\rho \left[ {1 \ \ -1 \atop 1  \ \ -1} \right] \cr}\right] =
\left[ \matrix{ e^{i\lambda t} \exp(itA)\big\vert{\bf H}_0 &0 &0 \cr 
0 & 1+it\rho & -it\rho \cr 
0 & it\rho & 1-it\rho \cr}\right] . \cr}$$
Hence the statement is straightforward from the relation
$$\Psi^t(x_0+\xi\overline{x}) =
\big[\widehat{\Psi}^t[x_0,\xi,1]^{{\rm T}}\big]_3^{-1} 
\Big( \big[\widehat{\Psi}^t[x_0,\xi,1]^{{\rm T}}\big]_2 \overline{x} +  
\big[\widehat{\Psi}^t(x_0,\xi,1]^{{\rm T}}\big]_1 \Big) .$$

\bigskip
\ADT
\bf Lemma \NUM. \it The group ${\rm Aut}(\overline{{\bf B}})$ is 2-transitive on $\partial{\bf B}$
i.e. given any tuple $(e_1,f_1,e_2,f_2)$ of unit vectors with $e_1\ne e_2,\, f_1\ne f_2$, there is a transformation
$\Theta\in{\rm Aut}({\bf B})$such that $\overline{\Theta}(e_k)=f_k$ $(k=1,2)$.

\bigskip
\bf Proof. \rm Let $e_1,f_1,e_2,f_2 \!\in\!\partial{\bf B}$ with $e_1\!\ne\! e_2,\, f_1\!\ne\! f_2$.
\DEL{

Consider the (necessarily unit) vectors
$$g_1:= \Phi(e_1), g_2:= \Phi(e_1) \quad \hbox{where} \quad 
\Phi: x\mapsto {Q_{\alpha e_2} x + \alpha e_2 \over \langle x \vert \alpha e_2 \rangle + 1}
\quad\hbox{with} \ \ \alpha:=-\langle e_1\vert e_2 \rangle .$$
We have $Q_{\alpha e_2} = Q_{e_2} = {\rm P}_{e_2} + \sqrt{1-\vert\alpha\vert^2}(I-{\rm P}_{e_2})$
with
$Q_{\alpha e_2} e_2 = e_2$ and $\langle Q_{\alpha e_2} e_1 \vert e_2 \rangle = 
\langle e_1 \vert  Q_{\alpha e_2} e_2 \rangle = -\alpha$. 
It follows 
$$\big\langle g_1 \big\vert \, g_2 \big\rangle = 
\Big\langle  {Q_{\alpha e_2} e_1 +\alpha e_1 \over \overline{\alpha} \langle e_1 \vert e_2\rangle + 1} 
\Big\vert {1+\alpha\over 1+\overline{\alpha}} e_2 \Big\rangle =
{ \langle e_1 \vert e_2 \rangle + \alpha \over 1 + \overline{\alpha}  \langle e_1 \vert e_2 \rangle }  
\,  {1+\overline{\alpha}\over 1+{\alpha}} = 0 \quad 
\hbox{if} \ \ \big\vert \langle e_1 \vert e_2 \rangle \big\vert <1 .$$ 


}%ENDDEL
Let $c$ be the 
center of the circlar arc $C$ in the 
1-dimensional complex disc $D:={\bf B}\cap\{ (1-\zeta) e_1 +\zeta e_2 : \zeta\in\CC\}$
connecting the points $e_1,e_2$ and being orthogonal in real sense to the 
boundary circle of $D$. 
Since holomorphic fractional linear transformations preserve affine lines,
the M\"obius shift
$\Theta_c^{-1} =\Theta_{-c} (x) =(1 - \langle x \vert c\rangle)^{-1} (Q_c x - c)$
maps $D$ onto an affine disc passing through $0=\Theta_{-c}(c)$ and 
containg the points $g_k :=\overline{\Theta_{-c}}(e_k)$ $(k=1,2)$ as boundary points.
Since the continuous extension $\overline{\Theta_{-c}}$ preserves the unit sphere $\partial{\bf B}$, 
The image $\Theta_{-c}(D)$ is a disc of the form 
$\Theta_{-c}(D) = \{ \zeta g_k: \vert\zeta\vert<1\}$ $(k=1,2)$.
Since holomorphic maps are conformal, the image $\Theta^{-1}(C)$ is an arc passing through 
the origin and being orhogonal to the boundary circle which is possible only if it is 
a diameter of the form $\{ \tau g: -1<\tau <1 \}$ with $\Vert g\Vert=1$
that is we have $\overline{\Theta_{-c}}(e_k)=g_k=(-1)^k g$ $(k=1,2)$ for some unit vector $g$.
Similarly, $\overline{\Theta_{-d}}(f_k)=(-1)^k h$ $(k=1,2)$ 
for some unit vector $h$ and a M\"obius shift $\Theta_{-d}$.
It is a well-known elementary fact that any the unitary group is transitive on $\partial{\bf B}$,
actually e.g. we have $Ug=h$ with a twisted reflection $U:=\kappa[I-2{\rm P}_{h-\kappa g}]$ where
the constant $\kappa\in\CC$ is of modulus 1 such that $\zeta g+h \perp \zeta g -h$ i.e. 
$\kappa\langle g\vert h\rangle \in\RR$. 
Therefore the transformation $\Theta := \Theta_d\circ U \circ \Theta_{-c}$ 
suits the requirements of the lemma.


\bigskip
\bf Proof of Corollary \the\CorIIS.\the\CorIIT. \rm

Due to the 2-transitivity on the boundary, 
there exists $\Theta\in{\rm Aut}({\bf B})$ such that
$\Theta(\overline{x})=\overline{x}$, $\Theta(\overline{x}^\prime)=-\overline{x}$.
Thus, by passing to the group $[\Theta\circ\Psi^t\circ\Theta^{-1} : t\in\RR]$
instead of  $[\Psi^t : t\in\RR]$, we may assume without loss of generality that
$\{ \pm\overline{x} \} =
{\rm Fix}\big(\overline{\Phi}^{t_0}\big)$
with the effect 
$$-\big\langle (iA-\lambda I)(x-\overline{x}) \big\vert \overline{x} \big\rangle x + 
(iA + \lambda I) (x-\overline{x}) = 0 \quad \hbox{for} \ \ x=\pm\overline{x} .$$
Regarding the case $x=-\overline{x}$, since $\langle \overline{x} \vert \overline{x} \rangle =1$,
it follows \ 
$\big\langle A \overline{x} \big\vert \overline{x} \big\rangle \overline{x} + A\overline{x} =0$.
However, since the operator $A$ is self-adjoint, necessarily also
$A\overline{x}=\big\langle A \overline{x} \big\vert \overline{x} \big\rangle \overline{x}$
implying $A\overline{x}=0$. Hence the statement is immediate from Corollary \the\CorIS.\the\CorIT.



 \DEL{

\bigskip

\def\xx{\overline{x}}

\rm
Assume $\bigcap_{t\in\RR_+} {\rm Fix}(\overline{\Phi}^t) = \{ \xx_1,\xx_2\} \subset \partial{\bf B}$ with $\xx_1\ne\xx_2$.
Observe that in this case there exists $t_0$ with ${\rm Fix}(\Phi^{t_0})=\{ \xx_1,\xx_2\}$.
Indeed, otherwise %in the contrary case,
%let $T:= \big\{ \tau\in\RR_+: {\rm Fix}(\Phi^\tau) \setminus \{ \xx_1,\xx_2\} \big\}$.
the whole real segment $[\xx_1,\xx_2]$ would be contained in ${\rm Fix}(\Phi^t)$ for any index $t$.
It is well-known that there exists a M\"obius transformation $\Theta=F({\cal M})$ and a vector $\xx\in\partial{\bf B}$ such that
%${\rm Fix}\big(\Theta\circ\overline{\Phi}^{t_0} \circ\Theta^{-1}\big) =
$\big\{ \Theta(\xx_1),\Theta(\xx_2) \big\} =\{ \pm\xx \}$.
It follows
$\bigcap_{t\in\RR_+} {\rm Fix}\big(\Theta\circ\overline{\Phi}^t \circ\Theta^{-1}\big) =  \{ \pm\xx \}$.
Since the orbits $t\mapsto \overline{\Phi}^t (\pm \xx)$ are differentiable (actually constant), also the orbit of $0 =-\xx+\xx$ is differentiable.
In particular,
$(\pm\xx)\oplus 1$ are joint eigenvectors of the linear representation operators ${\cal M}{\cal U}^t {\cal M}^{-1}$ with $F({\cal U}^t)=\Phi^t$.
Thus, by passing to the M\"obius equivalent semigroup
$[\Theta\circ\Phi^t \circ\Theta^{-1}: t\in\RR_+]$ instead of  $[\Phi^t : t\in\RR_+]$,
%in this case
we may assume without loss of generality that
$$\bigcap_{t\in\RR_+} {\rm Fix}\big({\overline{\Phi}}^t\big) = \{ \pm\xx \}, \quad
{\cal R}[(-\xx)\oplus 1] = \nu_1 [(-\xx)\oplus 1], \ \ {\cal R}[\xx\oplus 1] = \nu_2 [\xx\oplus 1]$$
for some %constants
$\nu_1,\nu_2\in\CC$.
%Recall that the generator ${\cal R}$ of $[{\cal U}^t: t\in\RR]$ has the form ${\cal R}=\left[ {iR \ b \atop b^* \ 0} \right]$.
Hence
from (?????) %(\the\AaS.\the\AaT)
we see that
$b=-(\nu_1 - iR)\xx=(\nu_2 - iR)\xx$
entailing that $\xx$ is an eigenvector of $R$ with eigenvalue ${1\over 2i}(\nu_1+\nu_2) =\langle R\xx \vert \xx\rangle \in \RR$.
%With the notation $\lambda_k := {\rm Re}(\nu_k)$, $\mu_k := {\rm Im}(\nu_k)$,
Taking also (?????)
%(\the\AaS.\the\AaT$^\prime$)
in account we deduce
$b= 2^{-1}(\nu_2-\nu_1)\xx$,
$b_0= P_0 b=0, \ \ b= -\overline{\nu_1}\xx = \overline{\nu_2}\xx$,
$2{\rm Im}(\nu_k) = \langle R\xx \vert \xx \rangle =(2i)^{-1}(\nu_1+\nu_2) =0$.
Thus, for some $\lambda \in\RR$, necessarily
$\nu_1=\lambda$, $\nu_2=-\lambda$ entailing also
$R\xx=0$, $b=\lambda \xx$, ${\cal R}[(\pm \xx)\oplus 1] = \pm\lambda [(\pm \xx)\oplus 1]$.
With the choice $\nu:=\nu_1$, in terms of
%(\the\VtS.\the\VtT)
(?????)
this means that $b_0=0$, $S_0=P_0 R P_0 = R$  whence (\the\CorIIS.\the\CorIIT) is immediate.
%$$\eqalign{
%P\Phi^t(x) &= { (1-\lambda)(1-\xi) e^{-2\lambda t} + 1 \over 1 - (1-\xi)\lambda e^{-2\lambda t} } \xx \cr
%P_0\Phi^t(x) &= {e^{-\lambda t}\over 1 - (1-\xi)\lambda e^{-2\lambda t} } V_0^t x_0 \ .\cr}$$


\bigskip
\bigskip
\hrule
\bigskip





\vfill
\eject

}%ENDDEL

\bigskip
\bigskip

\bf References
\rm
\medskip
\item{[\ABATE]} M. Abate, \it Common fixed points of commuting holomorphic maps, \rm
Math. Ann. \bf 283 \rm (1989) 645-655.

\medskip
\item{[\BKU]} H. Braun -- W. Kaup -- H. Upmeier, \it On the automorphisms of
circular and Reinhardt domains in complex Banach spaces,
\rm Manuscripta Math., \bf 25 \rm (1978), 97-133.

\medskip
\item{[\DINEEN]} S. Dineen, \it Complete holomorphic vector fields on the second dual of a Banach space,
\rm Math. Scand. \bf 59 \rm (1986) 131-142.

\medskip
\item{[\NAG]} K.-J. Engel -- R. Nagel,
One-Parameter Semigroups for Linear Evolution Equations,
Graduate Texts in Mathematics Vol. 194, Springer-Verlag, New York, 2000.

\medskip
\item{[\FRVE]} T. Franzoni -- E. Vesentini, Holomorphic Maps and Invariant Distances,
North Holland Math. Studies 40, North Holland Publ. Co., Amsterdam-New York-Oxford, 1980.

\medskip
\item{[\GOEBEL]} K. Goebel, 
\it Fixed points and invariant domains of holomorphic mappings of the Hilbert ball,
\rm Nonlinear Analysis, Theory, Methods and Applications, \bf 6/12 \rm (1982) 1327-1334.

\medskip
\item{[\HS]} T.L. Hayden -- T.J. Suffridge, \it Biholomorphic maps in Hilbert space have a fixed point,
\rm Pacific J. Math., \bf 38 \rm (1971) 419-422.

\medskip
\item{[\IST]} J.-M. Isidro -- L.L. Stach\'o,
{\it Holomorphic Automorphism Groups in Banach Spaces},
North Holland Math. Studies 105,
North Holland Publ. Co., Oxford--New York--Amsterdam, 1985.

\medskip
\item{[\KHAS]} V. Katshkevich - S. Reich - D. Shoiket,
\it One-parameter semigroups of fractional-linear transformations, \rm
Operator Theory Advances and Applications, \bf 123 \rm (2001) 401--411.

\medskip
\item{[\KAUP]} W. Kaup, \it
\"Uber die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I.,
\rm Math. Ann., \bf 357 \rm (1981), 463--481.

\medskip
\item{[\KAUPR]} W. Kaup, \it
A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces,
\rm Math. Z., \bf 83 \rm (1983), 503--529.

\medskip
\item{[\NACHBIN]} L. Nachbin,
{\it Holomorphic functions, domains of holomorphy and local properties},
\rm North-Holland Pub. Co., Oxford--New York--Amsterdam, 1970.

\medskip
\item{[\RSZ]} F. Riesz -- B. Sz.-Nagy,
Vorlesungen \"uber Funktionalanalysis,
VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.

\medskip
\item{[\FSZ]}
B. Sz.-Nagy -- C. Foias, {\it Harmonic Analysis of Operators on Hilbert Space}, 
Akadémiai Kiadó/North-Holland Publ. Co., Budapest--Amsterdam--London, 1970.

\medskip
\item{[\VESi]} E. Vesentini,
\it Semigrioups of holomorphic isometries, \rm
Advances in Mathemetics, \bf 65 \rm (1987), 272--306.

\medskip
\item{[\VESii]} E. Vesentini,
\it Semigrioups of holomorphic isometries, \rm
in: P.M. Gauthier (ed.), Complex Potential Theory, pp. 475--548.
Kluwer Academic Pulbl., Netherlands, 1994.

\medskip
\item{[\VIG]} J.-P. Vigu\'e,
{\it Le group des automorphismes analytiques d$^{\,\prime}$un
domaine born\'e d$^{\,\prime}$un espace de Banach complexe.
Applications aux domaines born\'es symmetriques},
Ann. Sci. Ecole Norm. sup., \bf  9  \rm (1976), 203--282.

\bigskip
\bigskip
L.L. STACH\'O

Bolyai Institute

University of Szeged

Aradi V\'ertan\'uk tere 1

H-6720 SZEGED, HUNGARY

E-mail: {\tt stacho@math.u-szged.hu}



\bye


\bf References
\rm



\medskip
\item{[\DED]} J.A. Deddens, \it On extending semigroups of contractions, \rm
Trans. Amer. Math. Soc., \bf 145 \rm (1969), 233-239.

\medskip
\item{[\BKU]} H. Braun -- W. Kaup -- H. Upmeier, \it On the automorphisms of
circular and Reinhardt domains in complex Banach spaces,
\rm Manuscripta Math., \bf 25 \rm (1978), 97-133.

\medskip
\item{[\DINEEN]} S. Dineen, \it Complete holomorphic vector fields on the second dual of a Banach space,
\rm Math. Scand. \bf 59 \rm (1986) 131-142.

\medskip
\item{[\NAG]} K.-J. Engel -- R. Nagel,
One-Parameter Semigroups for Linear Evolution Equations,
Graduate Texts in Mathematics Vol. 194, Springer-Verlag, New York, 2000.

\medskip
\item{[\FRVE]} T. Franzoni -- E. Vesentini, Holomorphic Maps and Invariant Distances,
North Holland Math. Studies 40, North Holland Publ. Co., Amsterdam-New York-Oxford, 1980.

\medskip
\item{[\GOEBEL]} K. Goebel, \it Fixed points and invariant domains of holomorphic mappings of the Hilbert ball,
\rm Nonlinear Analysis, Theory, Methods and Applications, \bf 6/12 \rm (1982) 1327-1334.

\medskip
\item{[\HS]} T.L. Hayden -- T.J. Suffridge, \it Biholomorphic maps in Hilbert space have a fixed point,
\rm Pacific J. Math., \bf 38 \rm (1971) 419-422.

\medskip
\item{[\IST]} J.-M. Isidro -- L.L. Stach\'o,
{\it Holomorphic Automorphism Groups in Banach Spaces},
North Holland Math. Studies 105,
North Holland Publ. Co., Oxford--New York--Amsterdam, 1985.

\medskip
\item{[\KAUP]} W. Kaup, \it
\"Uber die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I.,
\rm Math. Ann., \bf 357 \rm (1981), 463--481.

\medskip
\item{[\NACHBIN]} L. Nachbin,
{\it Holomorphic functions, domains of holomorphy and local properties},
\rm North-Holland Pub. Co., Oxford--New York--Amsterdam, 1970.


\medskip
\item{[\RSZ]} F. Riesz -- B. Sz.-Nagy,
Vorlesungen \"uber Funktionalanalysis,
VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.
}%ENDDEL

\medskip
\item{[\FSZ]}
B. Sz.-Nagy -- C. Foias, {\it Harmonic Analysis of Operators on Hilbert Space}, Akadémiai Kiadó/North-Holland Publ. Co.,
Budapest--Amsterdam--London


\DEL{
\medskip
\item{[\VIG]} J.-P. Vigu\'e,
{\it Le group des automorphismes analytiques d$^{\,\prime}$un
domaine born\'e d$^{\,\prime}$un espace de Banach complexe.
Applications aux domaines born\'es symmetriques},
Ann. Sci. Ecole Norm. sup., \bf  9  \rm (1976), 203--282.
}%ENDDEL





%\bye
