\documentclass[preprint]{elsarticle}
\usepackage{amssymb}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{bm}


\title{
On
%strongly continuous one-parameter
$C_0$-semigroups
of holomorphic Carath\'eodory isometries in Hilbert space\tnoteref{t1}}
\tnotetext[t1]{
The author's research was supported by the project
T\'AMOP-4.1.1.C-12/1/KONV-2012-0005
Impulse Lasers for Use in Materials Science and Biophotonics
of the European Union and co-financed
by the European Social Fund.}

\author[bi]{L.L. Stach\'o} %\corref{cor1}\fnref{fn1}}
\ead{stacho@math.u-szeged.hu}

\address[bi]{Bolyai Institute, University of Szeged,
Aradi V\'ertan\'uk tere 1,\\ H-6720 Szeged, Hungary}


\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\long\def\DEL#1{}

\font\mathfrak=eufm10

\font\Fr=eufm10 scaled \magstep1
\font\Frr=eufm7
%\def\mathfrak#1{\hbox{\Fr #1}}

\def\FFF{\hbox{\mathfrak F}}

\def\RR{{\mathbb R}}
\def\TT{{\mathbb T}}
\def\PP{{\mathbb P}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\CC{{\mathbb C}}

\def\eps{\varepsilon}

\def\qed{\hfill\ifmmode\quad\hbox{$\Box$}\else\hbox{$\Box$}\par\fi}

\def\ATOP#1#2{\genfrac{}{}{0pt}{}{#1}{#2}}
\def\OVER#1#2{\genfrac{}{}{0.1pt}{}{#1}{#2}}

\def\SCS{{$C_0$S}\ \ }
\def\SCG{{$C_0$G}\ \ }

\def\xx{e}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
We establish closed formulas
for all strongly continuous one-parameter semigroups of
holomorphic Carath\'eodory isometries of the unit ball
of a Hilbert space
in terms of spectral resolutions of skew self-adjoint dilations
related to Vesentini's non-linear
infinitesimal generator.
\end{abstract}

\begin{keyword}
strongly continuous one-parameter group
\sep
holomorphic Carath\'eodory isometry
\sep
Hilbert space
\sep
dilation


\MSC 47D03 \sep 32H15 \sep 46G20
\end{keyword}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newdefinition{rmk}{Remark}
\newdefinition{example}{Example}
\newproof{pf}{Proof}
\newproof{pot}{Proof of Theorem \ref{thm2}}




%References
%\def\ABATE{ }    %Abate                        NINCS
%\def\BKU{ }      %Brau-Kaup-Upmeier            NINCS
\def\DED{1}       %Deddens
%\def\DINEEN{ }   %Dineen                       NINCS
\def\NAG{2}       %Engel-Nagel
\def\FRVE{3}      %Franzoni-Vesentini
%\def\GOEBEL{ }   %Goebel                       NINCS
%\def\HS { }      %Hayden-Suffridge             NINCS
%\def\IST{ }      %Isidro-Stacho                NINCS
\def\KAUP{4}      %Kaup1981
\def\KAUPR{5}     %Kaup Riemann mapping thm
\def\KHAS{6}      %Khaskewits-Reich-Shiket
%\def\NACHBIN{ }  %Nachbin                      NINCS
%\def\RSZ{ }      %Riesz-Sz-Nagy                NINCS
\def\STA{7}       %Stach\'o Ploiesti-cikk
\def\FSZ{8}       %Sz-Nagy - Foias
\def\VESi{9}      %Vesentini Adv. Math
\def\VESii{10}    %Vesentini note
%\def\VIG{        %Vigue                        NINCS


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle


\section{Introduction}
\ADS

Throughout this work ${\bf H}$ denotes an arbitrarily fixed complex Hilbert space
with
%the
scalar product $\langle x \vert\, y \rangle$ being antilinear in $y$ and the canonical norm $\Vert x\Vert :=\langle x\vert x\rangle^{1/2}$.
We also keep fixed the standard notations
${\bf B}:=\{ x\in{\bf H}: \Vert x\Vert<1\}$,
$a^*:=[x\mapsto \langle x\vert a\rangle]$
%and $I :={\rm Id}_{{\bf H}}$
for the open unit ball, and the adjoint representation of bounded linear functionals, respectively.
We regard the elements $h,h^*$ $(h\in{\bf H})$
as column resp. row matrices
and, given a linear map $A:{\bf S}\to {\bf H}$ on some linear submanifold of ${\bf H}$.
We use the canonical
matrix identifications
$x\oplus\xi \equiv \big[ \ATOP{x}{\xi}\big]$ resp.
% $(x\!\in\!{\bf H},\, \xi\!\in\!\CC)$,
$\big[ \ATOP{A \ b}{ \ c^* \, d}\big] \equiv \big[ x\oplus\xi\mapsto (Ax+b)\oplus(c^*x+d) \big]$
with $x,b,c\in{\bf H}$ and  $\xi,d\in\CC$.
This gives rise to the familiar linear representation of fractional linear maps on ${\bf H}$:
$$\FFF\Big( \Big[ \ATOP{A \ \ b }{ c^* \ d} \Big] \Big) := \big[ x\mapsto (c^*x+d)^{-1} (Ax+b) \big] .$$
Our object of chief interest will be the semigroup
${\rm Iso}(d_{{\bf B}})$
of all holomorphic isometries of ${\bf B}$ with respect to the Carath\'eodory metric $d_{{\bf B}}$.
Recall [\FRVE] that
all its elements
are fractional linear maps (restricted to ${\bf B}$),
namely they are
%exactly the
compositions of {\it M\"obius transformations}\footnote{Fractional linear transformations
mapping ${\bf B}$ injectively onto itself.}
with linear isometries of ${\bf H}$.

In 1987, in his pioneering work [\VESi] Vesentini
studied subsemigroups of ${\rm Iso}(d_{{\bf B}})$ arising from
strongly continuous one-parameter matrix-semigroups.\footnote{Given a topological space ${\bf X}$, a family $[T^t \!: t\!\in\!\RR_+]$ $\big[$resp. $[T^t \!: t\!\in\!\RR] \big]$
of self-maps $T^t:{\bf X}\to{\bf X}$
is a {\it strongly continuous one-parameter semigroup} [resp. {\it group}] if $T^{s+t}=T^s \circ T^t$ $\big(s,t\in\RR_+$ [resp. $\RR] \big)$ and all the orbits
$t\mapsto T^t(x)$ $(x\in{\bf X})$ are continuous.}
He established  that
the correspondence
$\FFF^\# : \big[ {\cal U}^t: t\!\in\!\RR_+ \big] \mapsto \big[ \FFF({\cal U}^t) \vert{\bf B}: \ t\!\in\!\RR_+ \big]$
maps
the family $\hbox{\mathfrak{S}}$ of all
strongly continuous one-parameter semigroups of $\CC$-linear isometries of the
indefinite norm $\Vert x\Vert^2-\vert \xi\vert^2$
on ${\bf H}\oplus\CC$
into the family $\hbox{\mathfrak{B}}$ of all
strongly continuous one-parameter semigroups $[\Psi^t: t\!\in\!\RR_+] \subset {\rm Iso}(d_{{\bf B}})$.
According to [\VESi, Th.VII],
given $[{\cal U}^t: t\in\RR_+] \in \hbox{\mathfrak{S}}$ with the infinitesimal generator 
${\cal A}= \OVER{d}{ dt}\big\vert_{t=0+} {\cal U}^t$,
for the corresponding non-linear objects $\Psi^t:=\FFF({\cal U}^t)\big\vert{\bf B}$
we have
$\big\{ p\in{\bf B}: t\mapsto \Psi^t(p)$ is differentiable$\big\} =
\big\{ x\in {\bf B}:\
x\!\oplus\! 1 \in{\rm dom}({\cal A})\big\}$ which is dense in the ball ${\bf B}$.
%
It is well-known [\VESi,\KHAS] that here we can identify the in ${\bf H}\!\oplus\!\CC$ densely defined linear operator ${\cal A}$ with
an  ${\bf H}\!\oplus\!\CC$-split matrix if and only if
the orbit $t\mapsto \Psi^t(0)$ is differentiable. This happens if and only if the generator
${\cal A}$
has the form
\ADT
\newcount\GenS \GenS=\SEC
\newcount\GenT \GenT=\THM
$${\cal A}= \Big[
\begin{matrix} iA + \nu  &b \\ b^*  &\nu \end{matrix}\Big], \qquad
\nu\!\in\!\CC, \ b\!\in\!{\bf H}, \ A\!\in\!{\rm Her}_{{\rm s}}({\bf H})
\eqno(\NUM)$$
%with some constant $\nu\in\CC$, a vector $b\in{\bf H}$
with ${\rm dom}({\cal A}) = {\rm dom}(A) \oplus\CC$
where ${\rm Her}_{{\rm s}}({\bf H})$ stands for the family of all unbounded ${\bf H}$-hermitian operators
(maximal symmetric, in ${\bf H}$ densely defined closed linear operators).
%(i.e. if
%$iR$ is the generator of a strongly continuous semigroup of $\CC$-linear isometries of ${\bf H})$.
Even the cases with non-differentiable $0$-orbit can be treated by
passing to a semigroup $[\Phi^t: t\in\RR_+]$ of the form $\Theta^{-1} \circ \Psi^t \circ \Theta$ with any M\"obius transformation $\Theta$
such that $\Theta(0) \in{\rm dom}(\Gamma)$.
Indeed, since the M\"obius group is transitive on ${\bf B}$, hence any strongly continuous one-parameter semigroup $[\Psi^t : t\in\RR_+]\in\hbox{\mathfrak{B}}$
is equivalent up to a M\"obius transformation
({\it M\"obius equivalent} for short in the sequel)
to a semigroup $[\Phi^t : t\in\RR_+]\in\hbox{\mathfrak{B}}$ whose infinitesimal generator
has the form
\ADT
\newcount\VfieldS \VfieldS=\SEC
\newcount\VfieldT \VfieldT=\THM
$$\Gamma(x) = \OVER{d}{ dt}\Big\vert_{t=0+} \!\!\!\Phi^t \ \, = b -\big\langle x \big\vert b \big\rangle x + iA x , \quad  x\in{\rm dom}(R) \cap {\bf B}
\eqno(\NUM)$$
with some maximal symmetric operator $A$ defined densely on ${\bf H}$ and some vector $b\in {\bf H}$.
Also conversely, if $iA$ is the infinitesimal generator for some strongly continuous one-parameter subsemigroup
of ${\cal L}({\bf H})$ then, for any $b\in{\bf H}$, the vector field (\the\VfieldS.\the\VfieldT) is the infinitesimal
generator of a strongly continuous one-parameter subsemigroup of
${\rm Iso}(d_{{\bf B}})$.
In [\VESii,\KHAS] these considerations were extended to semigroups of fractional linear transformations arising from
a strongly continuous one parameter semigroup of automorphisms of a Krein space,
concluding with a description of the fractional-linear image of a strongly continuous one parameter group
by means of the solutions of Ricatti type equations $\dot{x}=\Gamma(x)$ with analogous vector fields to (\the\VfieldS.\the\VfieldT) in Pontriagin spaces.
It should be noted that as far no argument appeared in the literature concerning the seemingly plausible surjectivity of the map $\FFF^\#$.
The question is rather harmless in our setting: in the case of the unit ball of a Hilbert space
an argument with joint fixed points
(Proposition 3.5) %{\the\ScS.\the\ScT})
furnishes positive answer.
However, e.g. in the case of the unit ball of ${\cal L}({\bf H})$ the surjectivity of the respective $\FFF^\#$ seems to be open and highly not trivial.




\ADS
\section{Results}

Henceforth, for short, {\it \SCS} [resp. {\it \SCG}] will abbreviate the terms
{\it strongly continuous one-parameter semigroup} [{\it -group}].
We shall write
$ {\rm gen}[U^t: t\in\RR_+]$ or  ${\rm gen}[\widetilde{U}^t: t\in\RR]$
for the infinitesimal generator of the \SCS $[U^t: t\in\RR_+]$ or \SCG $[\widetilde{U}^t: t\in\RR]$, respectively.
Given a closed subspace ${\bf K}$ in the Hilbert spaces ${\bf H}$ or $\CC\xx\oplus {\bf H}$
we shall write ${\rm P}_{{\bf K}}$ the orthogonal projection onto ${\bf K}$ without danger of confusion.

\medskip

In this paper we develop a triangularization method leading to explicit algebraic formulas for a \SCS generated by a vector field (\the\VfieldS.\the\VfieldT)
in terms of fixed points of $\Gamma$ and quadratures of a \SCS formed by complex linear isometries of a 1-codimensional subspace of ${\bf H}$.
As a consequence we conclude that any \SCS of holomorphic Carath\'eodory isometries of ${\bf B}$ admits
a {\it dilation} to a \SCG of surjective holomorphic Carath\'eodory isometries of the unit ball of some covering Hilbert space.
Our fixed point approach seems to be new even in finite dimensions 
(with uniformly continuous one-parameter groups).

\medskip
\rm
Recall [\FRVE] that any 
Carath\'eodory isometry
$\Psi\in{\rm Iso}(d_{\bf B})$ admits 
a continuous extension $\overline{\Psi}$ to the closed unit ball $\overline{{\bf B}}$.
Given a \SCS $\bm{\Psi}=[\Psi^t \!: t\!\in\!\RR_+] \!\subset\!{\rm Iso}(d_{{\bf B}})$, 
the extensions 
$\overline{\bm{\Psi}}\!:=\![\overline{\Psi^t} \!: t\!\in\!\RR_+]$ 
form also a \SCS (see [\STA] in more general setting).
According to [\VESi, Section 7] $\overline{\bm{\Psi}}$ admits
common fixed points whose family
${\rm Fix}(\overline{\bm{\Psi}})$
consists of one or two boundary points or
it is
the intersection of $\overline{{\bf B}}$ with some closed complex-affine submanifold
containing points from ${\bf B}$ in which case
%$[\Psi^t \!\!: t\!\in\!\RR_+]$
$\bm{\Psi}$
is simply M\"obius equivalent
to a \SCS\hskip-1mm of linear isometries of ${\bf H}$ restricted to ${\bf B}$.

\smallskip
Our main goal is the following classification of the remaining cases with explicit formulas up to M\"obius equivalence.


\bigskip\noindent
\ADT
\newcount\MainS \MainS=\SEC
\newcount\MainT \MainT=\THM
\bf Theorem \NUM. \it Suppose the vector field {\rm (\the\VfieldS.\the\VfieldT)} is the infinitesimal generator of a
\SCS $\bm{\Phi}:=[\Phi^t : t\in\RR_+] \subset {\rm Iso}(d_{{\bf B}})$ having a common boundary fixed point $\xx\in {\rm Fix}(\overline{\bm\Phi})\cap \partial{\bf B}$.
Then we have
\ADT
\newcount\QuadS \QuadS=\SEC
\newcount\QuadT \QuadT=\THM
\begin{equation*}
\begin{split}
&\hskip-8mm
{\rm P}_{\!\CC\xx} \, \Phi^t \!(\xi\xx \!+\! x_0) \!=\!
\Big[ 1 - (1 - \xi) e^{-2\lambda t} \big/ \varphi_{\lambda,\mu}(t,x_0,\xi) \Big] \xx \\
&\hskip-8mm
{\rm P}_{\!{\bf H}_0} \Phi^t \! (\xi\xx \!+\! x_0) \!=\!
\Big[ (1 \!-\! \xi) e^{-2\lambda t} \big( {{\hbox{$\int_0^t$}}}
e^{\lambda s} V_0^s \, ds \big) b_0 \!+\!  e^{-\lambda t} V_0^t x_0\Big] \Big/
\varphi_{\lambda,\mu}(t,x_0,\xi) \\
\end{split}
\tag\NUM
\end{equation*}
for all points $x_0+\xi \xx\in {\bf B}$ with $x_0\perp \xx$ where ${\bf H}_0:={\bf H}\ominus\CC\xx$, $\lambda:={\rm Re}\langle \xx\vert b\rangle$,
$\mu:= {\rm Im}\langle \xx\vert b\rangle$, $b_0:={\rm P}_{{\bf H}_0} b$,
$[V_0^t : t\in\RR_+]$ is the \SCS of linear ${\bf H}_0$-isometries
generated by the 
%in ${\bf H}_0$ densely defined closed skew-symmetric 
skew-${\bf H}_0$-hermitian operator $i P_{{\bf H}_0} (A - \mu) \big\vert {\bf H}_0$
and
\begin{equation*}
\begin{split}
\varphi_{\lambda,\mu}(t,x_0,\xi) &:= 1 +
(1 - \xi) \Big\langle \big( {{\hbox{$\int_0^t$}}} e^{-2\lambda s}
{{\hbox{$\int_0^s$}}} e^{\lambda r} V_0^r \, dr \, ds \big) b_0
\Big\vert  \ b_0 \Big\rangle - \\
& \hskip7mm - (1 - \xi) (\lambda+i\mu) {{\hbox{$\int_0^t$}}} e^{-2\lambda s} ds
+ \Big\langle
 \big( {{\hbox{$\int_0^t$}}} e^{-\lambda s} V_0^s \, ds \big) x_0 \Big\vert \ b_0 \Big\rangle .
\end{split}
\end{equation*}

\ADT
\newcount\RemS \RemS=\SEC
\newcount\RemT \RemT=\THM
\noindent
\bf Remark \NUM. \rm
The following converse can be discovered from the proofs later on (see Corollary 3.12).  %{\the\TupleS.\the\TupleT}.
Given any couple of
vectors $\xx,b_0\in {\bf H}$ such that $\Vert \xx\Vert=1$
and $b_0\perp\xx$
along with any %strongly continuous one-parameter group
\SCS $[V_0^t: t\in\RR_+]$ of
linear isometries of ${\bf H}_0={\bf H}\ominus(\CC\xx)$ and two real constants $\lambda,\mu$,
the maps $(\the\QuadS.\the\QuadT)$ form a
\SCS in ${\rm Iso}(d_{{\bf B}})$.

\ADT
\newcount\IntS \IntS=\SEC
\newcount\IntT \IntT=\THM
\bigskip
\noindent
\bf Remark \NUM. \rm
In case of $\lambda\ne 0$, one can express the integrated operators in $(\the\QuadS.\the\QuadT)$
in terms of the resolvent $R( \pm \lambda, iS_0)$ of the ${\bf H}_0$-hermitian operator
$S_0:=i^{-1} {\rm gen} [V_0^t: t\in\RR_+]$. 
Namely we have 
$\int_0^t e^{-\lambda\tau} V_0^\tau \, d\tau = 
\big( 1 - e^{-\lambda t} V_0^t \big) R(\lambda, iS_0)$,
$\int_0^t \! e^{-2\lambda\tau} \!\! \int_0^\tau \!\!\! e^{\lambda\sigma} V_0^\sigma  d\sigma \, d\tau
\!=\! \OVER{1}{ 2\lambda} (1 \!-\! e^{-2\lambda t}) 
R(-\lambda,iS_0) \!-\! \big( 1 \!-\! e^{-\lambda t} V_0^t \big) 
R(\lambda,iS_0) R(-\lambda,iS_0)$.


\ADT
\newcount\TwofixS \TwofixS=\SEC
\newcount\TwofixT \TwofixT=\THM
\newcount\NNNNN
\bigskip
\noindent
\bf Theorem \NUM. \it
Let $\bm{\Psi}:=[\Psi^t: t\in\RR_+] \subset{\rm Iso}(d_{{\bf B}})$ be a
{\SCS} with $\xx \in {\rm Fix}(\overline{\bm{\Psi}}) \subset \partial{\bf B}$.
Then, with the notations of Theorem {\rm \the\MainS.\the\MainT}, we have the alternatives
\begin{itemize}
\item[{\rm (i)}] ${\rm Fix}(\overline{\bm{\Psi}})$ consists of two points
and
$\bm{\Psi}$ is M\"obius equivalent to some
\SCS $[\Phi^t: t\in\RR_+] \subset{\rm Iso}(d_{{\bf B}})$ of the form
\ADT
%\newcount\TwoS \TwoS=\SEC
%\newcount\TwoT \TwoT=\THM
$$\Phi^t(\xi \xx + x_0) = \OVER{ \xi +\tanh(\lambda t) }{ 1 + \xi \tanh(\lambda t) } \, \xx +
\OVER{e^{-\lambda t} }{ \cosh(\lambda t) + \xi \sinh(\lambda t)} V_0^t x_0 ;
\eqno(\NUM)$$
\item[{\rm (ii)}] $\{ \xx\} = {\rm Fix}(\overline{\bm{\Psi}})$, there is a 
$\bm{\Psi}$-invariant disc of the form $\emptyset\ne(\xx+\CC v)\cap{\bf B}$
and
$\bm{\Psi}$ is M\"obius equivalent to
a \SCS $[\Phi^t : t\in\RR_+]$ of the form
\ADT
%\newcount\OneS \OneS=\SEC
%\newcount\OneT \OneT=\THM
$$\Phi^t(\xi \xx + x_0) =
\OVER{1+i\mu t }{ 1-i\mu t} \, \OVER{\xi - i\mu t/(1+i\mu t) }{ 1 +i\mu t \xi /(1-i\mu t)} \, \xx + 
\OVER{1 }{ 1 - i\mu t (1-\xi) } \, V_0^t x_0 ;
\eqno(\NUM)$$
\item[{\rm (iii)}] there is no 
$\bm{\Psi}$-invariant disc of the form $\emptyset\ne(\xx+\CC v)\cap{\bf B}$
and
$\bm{\Psi}$ is M\"obius equivalent to
a \SCS $[\Phi^t : t\in\RR_+]$ of the form {\rm (\the\QuadS.\the\QuadT)} with $\lambda=0$.
\end{itemize}
\rm
\ADT
\newcount\TwoS \TwoS=\SEC
\newcount\TwoT \TwoT=\THM
\ADT
\newcount\OneS \OneS=\SEC
\newcount\OneT \OneT=\THM

\ADT
\noindent
\bf Remark \NUM. \rm
In the setting of Theorem {\the\MainS.\the\MainT} a non-emty dics
$(\xx+\CC v)\cap{\bf B}$ is $[\Phi^t: t\in\RR_+]$-invariant
if and only if 
$\xx \not\perp v \in{\rm dom}(A)$ and 
$(iA+\langle \xx\vert\, b\rangle)v\in\CC\xx$
as established in Lemma 3.15. 
Hence, in finite dimensions only cases (i),(ii) may apper.
Example 3.16 with possible independent interest 
for physics or stochastic processes
shows that case (iii) is not void. 

\bigskip
\rm
Recall that, as an implicit simple special case\footnote{We begin Section 4
with an elementary proof in Banach space setting.}
of Deddens [\DED,Main Thm.],
every
\SCS $[U^t \!: t\!\in\!\RR_+]$ of isometries of ${\bf H}$ admits a 
{\it unitary group dilation}
in the following sense: there exists a Hilbert space $\widehat{{\bf H}}$ 
containing ${\bf H}$ as a subspace along with a
\SCG $[{\widehat{U}}^t \!: t\!\in\!\RR]$ of unitary operators of $\widehat{{\bf H}}$
such that $U^t \!=\! \widehat{U}^t \big\vert {\bf H}$ $(t\!\in\!\RR_+)$.
%
Applying a unitary dilation $[ \widehat{V}_0^t \!: r\!\in\!\RR]$ 
of the isometry semigroup $[ V_0^t \!: t\!\in\!\RR_+]$
in (\the\QuadS.\the\QuadT), 
we readily obtain the following result with non-linear dilations.


\bigskip\noindent
\ADT
\newcount\ExtS \ExtS=\SEC
\newcount\ExtT \ExtT=\THM
\bf Corollary \NUM. \it Given any \SCS $[\Psi^t: t\in\RR_+]$ of holomorphic Carat\'eodory isometries of ${\bf B}$,
there is a strongly continuous one parameter group $[\widehat{\Psi}^t: t\in\RR_+]$ of surjective holomorphic Carath\'eodory isometries of the unit ball
$\widehat{{\bf B}}$ of some Hilbert space $\widehat{{\bf H}}$ containing ${\bf H}$ as a subspace such that
$\Psi^t = \widehat{\Psi}^t \big\vert {\bf B}$ $(t\in\RR_+)$.

\bigskip
\rm
By means of the functional calculus of the skew self-adjoint generator $i\widehat{S}_0$
of the dilation group $[\widehat{V}_0^t : t\in\RR]$ of the \SCS $[V_0^t: t\in\RR_+]$
in the setting of Theorem {\the\MainS.\the\MainT}, we get the following.

\bigskip
\noindent
\ADT
\newcount\FuncS \FuncS=\SEC
\newcount\FuncT \FuncT=\THM
\bf Corollary \NUM. \it In $(\the\QuadS.\the\QuadT)$ we can write

\smallskip
\noindent
$\varphi_{\!\lambda,\mu}\!(t, \! x_0, \!\xi) \!=\!
\Big\langle \! x_0 \Big\vert  f_1(t,\lambda, \widehat{S}_0) b_0 \!\Big\rangle
\!+\! (1 \!-\! \xi) \Big[ \Big\langle \! f_2(t,\lambda, \widehat{S}_0) b_0
\Big\vert   b_0 \!\Big\rangle
- (\lambda+i\mu) \int_0^t \! e^{-2\lambda s} ds \Big]  \!+\! 1$,

$P_{{\bf H}_0} \Phi^t(x) =
\varphi_{\lambda,\mu}(t,x_0,\xi)^{-1} \Big[ e^{-\lambda t} \exp(i t \widehat{S}_0) x_0 + (1 - \xi) e^{-2\lambda t}
f_1(t,\lambda,\widehat{S}_0) b_0 \Big]$

\smallskip
\noindent
with the bounded analytic functions 
$f_j(t,\lambda, \cdot) : \RR\to\CC$ $(j=1,2;\ \lambda,t\in\RR)$
\begin{equation*}
\begin{split}
&f_1(t,\lambda,\sigma) := %\int_0^t e^{-\lambda\tau} e^{-i \sigma \tau} \, d\tau=
\OVER{1 - e^{-t(\lambda+i\sigma)} }{ \lambda + i\sigma} = \sum_{n=0}^\infty 
\OVER{(-1)^n (\lambda+i\sigma)^n }{ (n+1)!} t^{n+1} ,\qquad 
f_2(t,\lambda,\sigma) :=\\ 
&:= \OVER{ e^{-2\lambda t} }{ 2\lambda(\lambda\!+\! i\sigma)} \!+\! 
\OVER{1 }{ 2\lambda(\lambda\!-\! i\sigma)} \!-\! 
\OVER{e^{-t(\lambda \!-\! i\sigma)} }{\lambda^2\!+\!\sigma^2} =
\sum_{n=2}^\infty \Big[ 
\OVER{ (-2\lambda)^n}{ 2\lambda(\lambda+i\sigma)} - 
\OVER{ (-\lambda+i\sigma)^n }{ \lambda^2+\sigma^2} \Big] \OVER{t^n }{ n!} .
\end{split}
\end{equation*}

\rm

\ADS
\section{Triangularization with boundary fixed points}

\noindent
\ADT
\newcount\ScS \ScS=\SEC
\newcount\ScT \ScT=\THM
\bf Lemma \NUM. \it 
Assume $\bm{\Psi}=[\Psi^t: t\in\RR_+] \subset{\rm Iso}(d_{{\bf B}})$
is a \SCS where $\Psi^t =\FFF {\cal U}_t\big\vert{\bf B}$
with
${\cal U}_t \in {\cal L}({\bf H}\oplus\CC)$ $(t\in\RR_+)$.
Then there is a family $[\mu_t : t\in \RR_+] \subset\CC \setminus\{ 0\}$ such that
$\big[ \mu_t {\cal U}_t: t\in\RR_+\big]$ is a \SCS in ${\cal L}({\bf H}\oplus\CC)$.


\bigskip
\noindent
\bf Proof. \rm
Let $\xx$ be a common fixed point of the %continuous extensions of
the transformations $\overline{\Psi^t}=
\FFF{\cal U}_t \vert \overline{{\bf B}}$ $(t\in\RR_+)$.
We are going to show that the choice
$\mu_t:= \big[ {\cal U}_t (\xx\oplus 1) \big]_\CC^{-1}$ 
entailing $\mu_t {\cal U}_t (\xx\oplus 1) = \xx\oplus 1$
suits our requirements.
Consider the matrices ${\cal V}^t := \mu_t {\cal U}_t$. 
Clearly $\FFF {\cal V}^t = \FFF {\cal U}_t$ $(t\in\RR_+)$.
Since the map %${\cal U}\mapsto \FFF{\cal U}\vert \overline{{\bf B}}$ 
${\cal U}\mapsto \FFF{\cal U}\vert {\bf B}$
is a homomorphism with respect to compositions,
and since its preimages are unique up to non-zero factors, we have
$\FFF{\cal V}^{t+s} \vert {\bf B} = 
\Psi^{t+s} = \Psi^t \circ \Psi^s = \FFF ({\cal V}^t {\cal V}^s )\vert{\bf B}$
and hence ${\cal V}^{t+s} = d_{t,s} {\cal V}^t {\cal V}^s$ $(t,s\in\RR)$ 
with suitable constants $d_{t,s}\ne 0$.
The fixed point property
\ADT
\newcount\FprS \FprS=\SEC
\newcount\FprT \FprT=\THM
$${\cal V}^t (\xx\oplus 1)= \xx\oplus 1 \quad (t\in\RR_+) \eqno(\NUM)$$
ensures that $d_{t,s} \equiv 1$
that is the family $[{\cal V}^t: t\in\RR_+]$ is a one-parameter matrix semigroup.
To see its strong continuity,
recall [\FRVE, Ch. VI] that the {\it M\"obius shifts}
\ADT
\newcount\MobS \MobS=\SEC
\newcount\MobT \MobT=\THM
$$\Theta_a := \FFF {\cal M}_a , \ \ 
{\cal M}_a:=\Big[ 
\begin{matrix} Q_a &\!\!\!\! a \\ a^* &\!\!\!\! 1 \end{matrix}\Big], \ \
%\hbox{where} \ \ 
Q_a:= {\rm P}_{\!\CC a} + \sqrt{1 \!-\! \Vert a\Vert^2} (1-{\rm P}_{\CC a})
\quad (a \!\in\! {\bf B})
\eqno(\NUM)$$
act transitively on ${\bf B}$. 
Thus, since every element of ${\rm Iso}(d_{{\bf B}})$ 
keeping the origin fixed is a restriction of
a linear isometry of ${\bf H}$, we can write 
$\Psi^t = \Theta_{a_t} \circ U_t$ where $a_t:=\Psi^t(0)$
and $U_t$ is a suitable linear isometry of ${\bf H}$.
Since $U_t=\FFF\big[ \ATOP{U_t \ 0 }{ 0 \ 1}\big]$, 
%and since the $\FFF$ preimages are unique up to non-zero factors,
with suitable constants $\delta_t\ne 0$ we can write
$${\cal V}_t :=\delta_t {\cal M}_{a_t} \left[ \begin{matrix} U_t  &0 \\ 0 &1 \end{matrix}\right]  =
\delta_t \left[ \begin{matrix} Q_{a_t} U_t &a_t \\ [U_t^* a_t]^* &1 \end{matrix}\right] 
\qquad (t\!\in\!\RR_+) .$$
The value of
$\delta_t$ is determined unambiguously by (\the\FprS.\the\FprT): $\delta_t= \big[ 1 + \langle U_t \xx \vert a_t \rangle \big]^{-1}$.
%
Thus to complete the proof, it suffices to see the continuity of the functions
$t\mapsto a_t$, $t\mapsto [U_t x, Q_{a_t}x]$ $(x\in {\bf H})$.
It is an immediate consequence of [\NAG,App. A6] that the product $t\mapsto A_t B_t$ is strongly continuous
for any couple of uniformly bounded strongly continuous operator valued functions
$t\mapsto A_t\in{\cal L}({\bf X}_1,{\bf X}_2)$, $t\mapsto B_t\in{\cal L}({\bf X}_2,{\bf X}_3$
in case of normed spaces ${\bf X}_k$.
By assumption, the orbit $t\mapsto a_t=\Psi^t(0)$ is a norm-continuous map $\RR_+\to{\bf B}$
entailing the norm continuity of the function $t\mapsto Q_{a_t}$.
We deduce the strong continuity of the ${\bf H}$-isometry valued
function $t\mapsto U_t$ as follows. Consider any vector $x\in{\bf H}$.
We may assume $x\in{\bf B}$ without loss of generality.
Then, by the aid of the M\"obius shifts $(\the\MobS.\the\MobT)$
we can write
$$U_t x = \big[\Theta_{a_t}^{-1} \circ \Psi_t \big](x) = 
\Theta_{-a_t} \big( \Psi(x)\big)
\qquad (t\!\in\!\RR_+) $$
whence the continuity of
$t\mapsto U_t x = (1-\langle x\vert a_t\rangle)^{-1} [Q_{a_t} x - a_t]$
is immediate.


\bigskip
\ADT
\newcount\AssS \AssS=\SEC
\newcount\AssT \AssT=\THM
\noindent
\bf \NUM \ Standard  notations, assumptions.
\rm
Henceforth, for the proofs for Section 2, we assume without loss of generality the following facts.

\smallskip

\begin{itemize}

\item[(i)] \ $\bm{\Psi}:=[\Psi^t \!\!: t\!\in\!\RR_+]$ 
is an arbitrarily given \SCS of holomorphic Carath\'eodory isometries of ${\bf B}$
having no common fixed point within ${\bf B}$.

\medskip
\item[(ii)]
$\Phi^t:= \Theta\circ \Psi^t \circ \Theta^{-1}$ $(t\in\RR_+)$
with a suitable M\"obius transformation $\Theta$;

\medskip
\item[(iii)] \ the orbit $t\mapsto \Phi^t (0)$ is differentiable and 
%$[\Phi^t: t\in\RR_+] = \FFF^\# [{\cal U}^t: t\in\RR_+]
$\Phi^t = \FFF^\# {\cal U}^t \big\vert{\bf B}$
with some \SCS $[U^t: \, t\!\in\!\RR_+]$ of linear ${\bf H}$-isometries,
$${\cal A}:= {\rm gen}[{\cal U}^t : t\!\in\!\RR_+] = 
\Big[ \begin{matrix} iA &b \\ b^*  &0 \end{matrix} \Big] ,
\qquad b\!\in\!{\bf H}, \ iA ={\rm gen} [U^t : t\in\RR_+];$$

\item[(iv)] \ $\xx \in\partial{\bf B}$ is a joint boundary fixed point 
%fixed point of the continuous extensions 
of the maps $\overline{\Phi^t}$,
we write
$${\bf H}_0 \!:={\bf H}\ominus \CC \xx, \ \ 
P:={\rm P}_{\!\CC \xx}, \ \ P_0:={\rm P}_{{\bf H}_0}=1 \!-\! P, \ \
T \!:x \!\mapsto\! x \!+\! \xx, \ \ {\cal T}:= 
\Big[ \begin{matrix} {\rm id}_{{\bf H}} &\xx \\ 0 &1 \end{matrix} \Big] . $$
\end{itemize}


\bigskip
\ADT
\newcount\PropS \PropS=\SEC
\newcount\PropT \PropT=\THM
\noindent
\bf Proposition \NUM. \it
We have $\xx\!\in\!{\rm dom}(A)$ %moreover 
with ${\cal A}(\xx\oplus\!1) =\nu (\xx\oplus\!1)$ and 
$b=(\nu \!- \! iA)\xx$ for some $\nu\!\in\!\CC$.
The possibly unbounded operator

 $A_0 :=P_0 A\big\vert {\bf H}_0\cap{\rm dom}(A)$ is ${\bf H}_0$-hermitian and,
in terms of $(\CC\xx\oplus {\bf H}_0 \oplus \CC)$-matrices, we have
\ADT
\newcount\AaS \AaS=\SEC
\newcount\AaT \AaT=\THM
$${\cal T}^{-1} {\cal A} {\cal T} = 
\left[ \begin{matrix}  -\overline{\nu} &0 &0 \\ -b_0 &iA_0 &0 \\  
\nu &b_0^* & \nu \end{matrix} \right]
\quad \hbox{\rm where} \ \ b_0 := P_0 b, \ \ \nu=\langle \xx \vert\, b\rangle .
\eqno(\NUM)$$
\bf Proof. \rm
By assumption \the\AssS.\the\AssT(ii),
$\xx \oplus 1$
is a joint eigenvector of the
linear operators ${\cal U}^t$.
Hence
${\cal U}^t (\xx \!\oplus\! 1) = \zeta_t (\xx \!\oplus\! 1)$ $(t \!\in\! \RR_+)$
with a continuous solution $[t\mapsto \zeta_t]$
 of the Cauchy equation $\zeta_{s+t}=\zeta_s \zeta_t$. %$(s,t \!\in\!\RR_+)$.
Thus
for some $\nu\in\CC$, \
$\zeta_t=e^{\nu t}$
and we have
$$\xx\!\oplus\! 1 \in {\rm dom}({\cal A}) = 
\{ \hbox{\mathfrak{z}}: t\mapsto{\cal U}^t \hbox{\mathfrak{z}} \ 
\hbox{is differentiable}\}, \ \
{\cal A}(\xx\!\oplus\! 1) = \nu (\xx\!\oplus\! 1). $$
As a consequence,
$\xx \in {\rm dom}(A) = {\rm P}_{\!\bf H} {\rm dom}({\cal A})$
and the operator
$$\widetilde{A}_0:= A -PA-AP+PAP = (1-P)A(1-P)= P_0 A P_0$$
is a bounded perturbation ranging in ${\bf H}_0$ of
$A\in{\rm Her}_s({\bf H})$ with a self-adjoint operator of finite rank.
Hence its restriction $A_0$ to ${\bf H}_0$ is a well-defined unbounded ${\bf H}_0$-hermitian operator.
%with $iA_0$
%generating a \SCS $[U_0^t: t\in\RR_+]$ of linear ${\bf H}_0$-isometries.
Since $A$ is a $(\CC\xx\oplus{\bf H}_0)$-matrix %type 
operator, 
we can write
$${\cal A} \!=\!\! \left[ \begin{matrix} iA &b\\ b^* &0 \end{matrix} \right] \!=\!\!
\left[ \begin{matrix} i\alpha &ia_0^* &\beta \\ ia_0 &iA_0 &b_0 \\ 
\overline{\beta} & b_0^* &0\end{matrix} \right] \! , \ \
b_0 \!:=\! P_0 b, \ \beta \!:=\! \langle b \vert \xx \rangle, \ 
a_0 \!:=\! P_0A\xx, \ \alpha v\!:=\! \langle A\xx \vert \xx \rangle$$
in terms of $(\CC\xx\oplus{\bf H}_0\oplus\CC)$-matrices.
The eigenvector equation ${\cal A}(\xx\!\oplus\! 1) = \nu (\xx\!\oplus\! 1)$
means $iA\xx + b =\nu\xx$ with $\langle \xx \vert\, b\rangle=\nu$  
entailing $i\alpha + \beta =\nu$, $ia_0+b_0=0$, $\overline{\beta}=\nu$.
Since
$${\cal T}=
\left[ \begin{matrix} 1 &0 &1 \\ 0 &1 &0 \\ 0 &0 & 1 \end{matrix} \right], \quad
{\cal T}^{-1}=
\left[ \begin{matrix} 1 &0 &-1 \\ 0 &1 &0 \\ 0 &0 & 1 \end{matrix} \right] ,$$
in $(\CC\xx\!\oplus\!{\bf H}_0\!\oplus\!\CC$)-matrix form, 
hence (\the\AaS.\the\AaT) is immediate.


\bigskip
\noindent
\ADT
\bf Notation \NUM. \rm 
Henceforth $[U_0^t \!\! : \! t\!\in\!\RR_+]$ denotes the 
\hskip-0.5mm \SCS \hskip-1.2mm of ${\bf H}_0$-isometries generated by $iA_0$.


\bigskip
\noindent
\ADT
\newcount\TriS \TriS=\SEC
\newcount\TriT \TriT=\THM
\bf Lemma \NUM. \it Let ${\bf E}_1,{\bf E}_2$ be Banach spaces,
${\cal G}:=\big[ \ATOP{G_1 \ \ 0 \ }{ H \ \ \ G_2}\big]$
with
$H\in{\cal L}({\bf E}_1,{\bf E}_2)$ 
and
$G_{k} \!=\! {\rm gen}[W_{k}^t \! : t \!\in\!\RR_+]$ for some  \SCS  \hskip-1mm
$[W_{k}^t \!: t\!\in\!\RR_+] \!\subset\! {\cal L}({\bf E}_k)$. %$(k=1,2)$.
Then the family %$[{\cal W}^t : t\in\RR_+]$ of the operator matrices
$${\cal S}^t := \left[ \begin{matrix}W_{1}^t &0 \\ 
\int_0^t W_{2}^{t-s} H W_{1}^s ds & W_{2}^t\end{matrix}\right] \qquad (t\in\RR_+)$$
is a \SCS in ${\cal L}({\bf E}_1\oplus{\bf E}_2)$
such that \
${\rm gen}[{\cal S}^t : t\in\RR_+] = {\cal G}$.

\bigskip
\noindent
\bf Proof. \rm
The operator ${\cal G}$ is a bounded perturbation 
by the operator 
${\cal H}:= \big[ ATOP{ 0 \ \, 0 }{ H \ 0 } \big]$
of the generator
$G_{1}\!\oplus\! G_{2} (\equiv \big[ \ATOP{G_1 \ \ 0 \ }{ \ 0 \ \ G_2} \big])$ 
of the \SCS $[{\cal W}^t : t\!\in\!\RR_+]$
with ${\cal W}^t \!:= \! W_1^t \!\oplus\! W_2^t$ and 
${\rm dom}({\cal G}) \!=\! {\rm dom}(G_{1})\!\oplus\! {\rm dom}(G_{2})$.
According to [\NAG,Thm.III.1.10], 
%with well-defined and in $t$ differentiable terms, 
for every fixed
$\hbox{\mathfrak{z}} \!=\! x\!\oplus\! y \!\in\!{\rm dom}({\cal G})$ 
we have
$${\cal S}^t \hbox{\mathfrak{z}} =\sum_{n=0}^\infty S_n(t) \quad 
\hbox{where} \quad S_0(t):={\cal W}^t \hbox{\mathfrak{z}} , \ \
S_{n+1}(t)= \int_0^t \!\! {\cal W}^{t-s} {\cal H} S_n^{(k)}(s) \, ds .$$
Since ${\cal H}$ is an off-diagonal $2\!\times\! 2$ triangular operator matrix,
$S_n(t) =0$ for $n>1$.


\ADT
\bigskip
\noindent
\bf \NUM. Proof of Theorem {\the\MainS.\the\MainT} \rm

%\medskip
\noindent
Since ${\cal T}$ a bounded invertible ${\bf H}\!\oplus\!\CC$-operator 
and ${\cal A}={\rm gen}[{\cal U}^t: t\in\RR_+]$, we have
$${\cal T}^{-1}{\cal A}{\cal T} = 
{\rm gen} \big[ {\cal V}^t : t\in\RR_+\big] \quad \hbox{for} 
\quad {\cal V}^t := {\cal T}^{-1}{\cal U}^t{\cal T} .$$
Since
$\Phi^t = \FFF({\cal U}^t)\big\vert{\bf B}$ $(t\in\RR_+)$,
in terms of the translation
$T x:= x+\xx$ %$(x\in{\bf H})$
we can interpret the \SCS
$[ {\cal V}^t : t\in\RR^+]$
as the linear representation by means of $\FFF$ 
of the semigroup $[T^{-1}\circ\Phi^t\circ T: t\in\RR_+]$
formed by holomorphic isometries of the shifted ball ${\bf B}-\xx$
whose continuous extensions leave the origin fixed.
Due to the projective identities
$\FFF\big( {\cal T}^{-1}{\cal V}{\cal T} \big) = 
T^{-1} \circ \FFF({\cal V}) \circ T$ \
$\big({\cal V}\!\in\!{\cal L}({\bf H}\oplus\CC)\big)$,
for the points $x\in T^{-1}{\bf B} = {\bf B} -\xx$ we have
$$\FFF{\cal V}^t (x) = 
\big[\FFF({\cal T}^{-1} {\cal U}^t {\cal T}) \big](x) =
[T^{-1}\circ\Phi\circ T](x)
=\Phi(x+\xx)-\xx .$$
Therefore
$$\Phi^t (x) = \FFF {\cal V}^t (x-\xx) + \xx
\quad (x\in{\bf B}).$$
By the aid of Lemma \the\TriS.\the\TriT\ and (\the\AaS.\the\AaT) we calculate
a quadrature form for ${\cal V}^t$ as follows.
Regarding the top left $2\!\times\!2$-corner of the matrix ${\cal T}^{-1}{\cal A}{\cal T}$
we get
\ADT
\newcount\VtS \VtS=\SEC
\newcount\VtT \VtT=\THM
$$\left[ \begin{matrix} -\overline{\nu} &0 \\ -b_0 &iA_0 \end{matrix} \right] = {\rm gen} \big[ V^t: t\in\RR_+\big] , \quad
V^t = \left[ \begin{matrix} e^{-\overline{\nu} t} &0 \\ 
\int_0^t U_0^{t-s} e^{-\overline{\nu}s} (-b_0) ds &U_0^t \end{matrix} \right] .
\eqno(\NUM)$$
Another application of Lemma \the\TriS.\the\TriT\ to 
${\cal T}^{-1}{\cal A}{\cal T}$ yields
\ADT
\newcount\VVtS \VVtS=\SEC
\newcount\VVtT \VVtT=\THM
$${\cal V}^t = \left[ \begin{matrix} V^t &0 \\ 
\int_0^t e^{\nu(t-s)} b^* V^s ds & e^{\nu t} \end{matrix} \right] 
\qquad (t\in\RR_+). \eqno(\NUM)$$
As a consequence of (\the\VVtS.\the\VVtT), since
${\cal V}^t(x \!\oplus\! 1) \!=\! 
\big[ V^t x\big] \oplus e^{\nu t} 
\big[ \int_0^t \langle e^{-\nu s} V^\tau x \vert\, b\rangle d\tau + 1\big]$,
we get
\ADT
\newcount\PhiS \PhiS=\SEC
\newcount\PhiT \PhiT=\THM
$$\Phi^t(x) = 
\OVER{e^{-\nu t} V^t(x-\xx) }{
\big\langle {{\hbox{$\int_0^t$}}} e^{-\nu s} V^s (x-\xx)\, ds \big\vert\, b \big\rangle +1} 
+ \xx \qquad (x\in{\bf B}, \ t\in\RR_+).
\eqno(\NUM)$$
We substitute (\the\VtS.\the\VtT) into (\the\PhiS.\the\PhiT) 
in terms of the new parametrization
$$\lambda = {\rm Re}\,\nu , \ \ \mu := {\rm Im}\,\nu , \ \ 
V_0^t := e^{-i\mu t} U_0^t .$$
Given any vector $z= z_0 + \zeta \xx, z_0\in{\bf H}_0$, 
and recalling
the commutativity of convolutions,
%that $b=b_0+\overline\nu \xx$, we have
\begin{equation*}
\begin{split}
&e^{-\nu t} V^t z =
\zeta e^{-2\lambda t} \Big[ \xx - \! \int_0^t \!\! e^{\lambda s} V_0^s b_0 \, ds \Big] \!+
e^{-\lambda t} V_0^t z_0 ,\qquad
\int_0^t \!\!\big\langle e^{-\nu s} V^s z \big\vert b \big\rangle \, ds = \\
&=\zeta (\lambda \!+\! i\mu) \OVER{ 1 \!-\! e^{-2\lambda t} }{ 2\lambda } - \zeta\!
\int_{\! 0}^t \!\!\!\! e^{-2\lambda s} \!\! 
\int_{\! 0}^s  \!\!\!\! e^{\lambda r} 
\big\langle V_0^r b_0 \big\vert b_0 \big\rangle\, dr
+ \int_{\! 0}^t \!\!
e^{-\lambda s} \big\langle V_0^s z_0 \big\vert b_0 \big\rangle \, ds \, ds .
\end{split}
\end{equation*}
The statement of Theorem \the\MainS.\the\MainT\ is immediate from (\the\PhiS.\the\PhiT) 
with $z \!:\!=x \!-\! \xx \!=\! x_0 \!+\! (\xi \!-\! 1) \xx$.


\bigskip
\noindent
\ADT
\newcount\TupleS \TupleS=\SEC
\newcount\TupleT \TupleT=\THM
\bf Remark \NUM. \rm
It is discovered from the above proof that
any tuple
\vskip-3mm
$$\hbox{\mathfrak a}:=\big( {\bf H},\xx,[ V_0^t: t\!\in\!\RR_+], b_0, \lambda, \mu \big)$$
\vskip-1mm
\noindent
with
a Hilbert space ${\bf H}$, a unit vector $\xx\!\in\!{\bf H}$, 
a \SCS $[V_0^t: t\!\in\!\RR_+]$
of  ${\bf H}_0(:={\bf H}\ominus\CC\xx)$-isometries,
a vector $b_0 \!\in\! {\bf H}_0$ and two real constants
gives rise to a \SCS
 $[ \Phi^t_{\hbox{\mathfrak a}}: t\!\in\!\RR_+]$ 
of holomorphic Carath\'eodory isometries of
the open unit ball \ ${\bf B}$ of \ ${\bf H} \equiv \CC\xx\!\oplus\!{\bf H}_0$
whose generator
$\Gamma(x) =\OVER{d}{ dt}
\big\vert_{t=0+} \Phi^t_{\hbox{\mathfrak a}}(x) =
\OVER{d}{ dt}\big\vert_{t=0+} \FFF({\cal T}{\cal V}^t {\cal T}^{-1})x $
has the form $(\the\VfieldS.\the\VfieldT)$
with
\ADT
$$b=\left[ \ATOP{ \lambda-i\mu }{ b_0} \right], \quad A=
\left[ \ATOP{2\mu \  -\! ib_0^* }{ ib_0 \ \ \ A_0 }\right], \quad i A_0 =
{\rm gen}[V_0^t: t\in\RR_+] .
\eqno(\NUM)$$
In particular we can extend $[\Phi^t_{\hbox{\mathfrak a}}: t\in\RR_+]$ 
to a \SCG $[\Phi^t_{\hbox{\mathfrak a}}: t\in\RR] \subset {\rm Iso}(d_{{\bf B}})$
if and only if $[V_0^t: t\in\RR_+]$ consists of \ ${\bf H}_0$-unitary operators
(cf. [\VESi,Thm.II]).
Furthermore, given any tuple $\hbox{\mathfrak b}:=({\bf H},A,\xx,\lambda)$ 
with a densely defined maximal symmetric linear ${\bf H}$-operator $A$,
there is a unique {\SCS} $[\Psi_{{\mathfrak b}}^t : t\in\RR_+] \subset {\rm Iso}(d_{{\bf B}})$ whose infinitesimal generator is of the form
(\the\VfieldS.\the\VfieldT) with $b:=(\nu-iA)e$ 
where $\nu:=\lambda+i\mu$ and $\mu=\langle A\xx \vert\, \xx \rangle$.


\ADT
\bigskip
\noindent
\bf Lemma \NUM. \it The \SCS $[ \Phi^t: t\in \RR_+] \subset {\rm Iso}(d_{{\bf B}})$ 
with generator {\rm (\the\VfieldS.\the\VfieldT)}
and joint boundary fixed point $\xx\in\partial{\bf B}$ admits no
invariant disc of the form ${\bf B}\cap (\xx+\CC\xx) \ne\emptyset$ 
if and only if
the operator \ $iA+\langle \xx \vert b\rangle$ \ 
is not injective or \ $\xx\in{\rm range}(iA-\langle \xx \vert b\rangle)$.


\bigskip
\noindent
\bf Proof. \rm
Consider any vector $v\in{\bf H}$ such that $\xx+v\in {\bf B}$.
The disc $\Delta_{\xx}^v := {\bf B} \cap (\xx+\CC v)$ is
$[ \Phi^t: t\in \RR_+]$-invariant if and only if 
the vector field (\the\VfieldS.\the\VfieldT) is tangent to it
that is if
$b - \langle \xx+\tau v \vert b \rangle (\xx + \tau v) + iA (\xx+\tau v) \in \CC v$ 
whenever
$\xx+\tau v \in{\bf B}$. 
This happens if and only if
$-\langle v \vert b \rangle \xx + iA v =\zeta v$ for some $\zeta\in\CC$
because we have $\xx\in{\rm dom}(\Gamma) = %{\bf B}\cap (\xx+{\rm dom}(A)) =
{\rm dom}(A)$ and
$\Gamma(\xx) = b-\langle \xx \vert b \rangle +iA\xx = 0$ 
(due to the fact that the point $\xx$ is $[\overline{\Phi^t} : t\in\RR_+]$-invariant).
According to Proposition {\the\PropS.\the\PropT}, here we have
$b=(\nu - iA)\xx$ where $\nu=\langle \xx \vert b\rangle$.
Therefore 
$\zeta v = - \langle v \vert (\nu - iA)\xx \rangle \xx + iA\xx = 
\langle (-\overline{\nu} - iA)v \vert \xx \rangle \xx + iA\xx$.
Notice that, in general, 
${rm P}_{\CC\xx} x = \langle x\vert\xx\rangle \xx = 
x - {\rm P}_{{\bf H}_0} x$ $(x\in{\bf H})$.
Thus
the disc $\Delta_\xx^v$ is $[\Phi^t : t\in\RR_+]$-invariant if and only if
$-\overline{\nu} {\rm P}_{\CC\xx} v + {\rm P}_{{\bf H}_0} (iAv)- \zeta v =0$
i.e. ${\rm P}_{{\bf H}_0} (iAv-\zeta v) =0$ and 
${\rm P}_{\CC\xx} (-\overline{\nu}-\zeta)\xx =0$
for some $\zeta\in\CC$.
By assumption $\Delta_\xx^v\ne\emptyset$ which is possible 
if and only if ${\rm P}_{\CC\xx} v \ne 0$ implying $\zeta=-\overline{\nu}$.
Hence we conclude that the $[\Phi^t: t\in\RR_+]$-invariance of 
$\Delta_\xx^v$ is equivalent to the relation
${\rm P}_{{\bf H}_0} (iAv+\overline{\nu}) v =0$ 
i.e. to $(iA+\overline{\nu})v \in\CC\xx$
which completes the proof.

\ADT
\bigskip
\noindent 
\bf Example \NUM. \rm
The {\SCS} of the type $[\Psi_{\hbox{\mathfrak b}}^t : t\in\RR_+]$ 
in Remark {\the\TupleS.\the\TupleT} with
${\bf H}:=L^2(\RR)$, $A f := [x\mapsto xf(x)$
$\big({\rm dom}(A):=
\big\{ f: \int_{-\infty}^\infty \vert xf(x)\vert^2 \, dx <\infty\big\} \big)$,
%$\xx:=1_{(0,1)} :x\mapsto[1$ if $0\!<\! x \!<\! 1$ else $0]$ 
$\xx \!:=\! (2\pi)^{-1/2} \!\exp\big(\!\!-\!(x \!-\!1)^2/2\big)$
and $\lambda \!:=\! 0$ admits no invariant 1-dimensional disc.
Proof: We have $\langle A\xx\vert\,\xx\rangle =
(2\pi)^{-1}\int_{-\infty}^\infty x \exp(-(x-1)^2)\, dx =
(2\pi)^{1/2}\ne 0$. Thus, according to the construction of the 
{\SCS} \hskip-2mm
$\bm{\Psi}_{\hbox{\mathfrak{b}}}$, 
$\nu \!=\! \langle \xx \vert \, b\rangle \!=\! 
i\mu \!=\! i\langle A\xx \vert \,\xx\rangle/2 \!\in\! i\RR \!\setminus\!\{0\}$.
The relation $(iA+\overline{\nu})v=\zeta\xx$ would imply
$v=-i\zeta\exp(-(x-1)^2/2)/(x-\mu) \in L^2(\RR)$ which is possible only if $v=0$. 



\bigskip
\noindent
\ADT
\bf \NUM. Proof of Theorem {\the\TwofixS.\the\TwofixT} 

\noindent
\rm
Recall [\FRVE] that the 1-dimensional complex affine discs the form
$\Delta_{p,q}:=\big(p+\CC(q-p)\big)\cap{\bf B}$
$(q\ne p,q\in\partial{\bf B})$
are the ranges of
complex geodesics for the Carath\'eodory  $d_{{\bf B}}$, and
$d_{{\bf B}}$-isometries map complexmetric
%$d_{{\bf B}}$-geodesics onto complex $d_{{\bf B}}$-geodesics.
their family into itself.
In particular, in the case when $p\ne q\in\partial{\bf B}$ 
are joint fixed points of the continuous extensions $\overline{\Psi^t}$
the disc $\Delta_{p,q}$ is automatically $[\Psi^t : t\in\RR_+]$-invariant.
Suppose $\Psi^t(\Delta_{p,q}) =\Delta_{p,q}$ $(t\in\RR_+)$.
Then the restricted maps $\psi_{p,q}^t := \Psi^t\big\vert D_{p,q}$ form a
\SCS of  holomorphic automorphisms of a 1-dimensional Hilbert ball,
thus their continuous extensions $\overline{\psi_{p,q}^t}$ to 
$\overline{D_{p,q}}$ admit at least one fixed point
which is necessarily a joint fixed point for the maps $\overline{\Psi^t}$.
A 1-dimensional application of Theorem {\the\MainS.\the\MainT} 
shows that all the orbits
$t\mapsto \psi_{p,q}^t(x) = \Psi^t(x)$ $(x\in \Delta_{p,q})$ 
are automatically real analytic.
Hence, given any M\"obius transformation $\Theta$, 
the \SCS $[\Phi^t: t\in\RR_+]$ with $\Phi^t:=\Theta\circ\Psi^t\circ\Theta^{-1}$
leaves the $d_{{\bf B}}$-geodesic $D_{\Theta(p),\Theta(q)}$ invariant 
with differentiable 0-orbit $t\mapsto \Phi^t(0)$.
Also conversely: if $[\Phi^t: t\in\RR_+]$ is a {\SCS} 
leaving the disc $\Delta_{\xx,-\xx} (= \{ \zeta \xx: \vert\zeta\vert<1 \})$ invariant
and $\Phi^t(\xx)=\xx,\, \Psi^t=\Theta^{-1}\circ\Phi^t\circ\Theta$ $(t\in\RR_+)$ as in {\the\AssS.\the\AssT},
then the image $\Theta(\overline{\Delta_{\xx,-\xx}})$ 
is a $[\overline{\Psi^t}: t\in\RR_+]$-invariant 
1-dimensional affine section of $\overline{{\bf B}}$
containing a joint fixed point (the point $\Theta^{-1}(\xx))$ 
of $[\overline{\Psi^t}: t\in\RR_+]$.

Proof of (i),(ii). 
It remains only to verify the possibility of the simplified representations (\the\TwoS.\the\TwoT),(\the\OneS.\the\OneT)
by means of an appropriate choice 
for the coordinatizing M\"obius transformation $\Theta$ in \the\AssS.\the\AssT.
%
By setting $x_0:=0$ in (\the\QuadS.\the\QuadT), 
it is straightforward to check that a \SCS
$[\Phi^t: t\in\RR_+]$ of the form {\the\QuadS.\the\QuadT} leaves
the disc $\Delta_{\xx,-\xx}$ invariant
if and only if $b_0=0$
and $\Phi^t(\xi\xx) = \omega_{\lambda,\mu}(t,\xi) \xx$ \ $(\vert\xi\vert<1)$
with the function
$$\omega_{\lambda,\mu}(t,\xi) := 
1 - \OVER{2\lambda (1-\xi) e^{-2\lambda t} }{ 
2\lambda - (1-\xi) (\lambda+i\mu)(1-e^{-\lambda t})} .$$
It is also easy to see that the constant $1$ is a 
joint fixed point of all functions $\omega_{\lambda,\mu}(t, \cdot)$
and, for fixed $\lambda,\mu\in\RR$, the family 
$\omega_{\lambda,\mu}(t, \cdot)$ $t\in\RR_+)$ admits another fixed point
namely the constant $\xi_{\lambda,\mu}:=
\OVER{i\mu-\lambda }{ i\mu+\lambda}$ with modulus $1$
if and only if we have $\mu =0$.
%
Due to folklore 2-transitivity properties of the M\"obius group 
(for direct proof see [\STA]),
given any two couples $(e_1,e_2),(f_1,f_2) \in[\partial{\bf B}]^2$
of distinct boundary points
there exists a M\"obius transformation 
$\Theta^{(e_1,f_1,e_2,f_2)}$ with the effect $e_k\mapsto f_k$ $(k=1,2)$.
Thus in case if $[\overline{\Psi^t} : t\in\RR_+]$ has only 
a unique fixed point $p\in \partial{\bf B}$
but the disc $\Delta_{p,q}$ is $[\Psi^t: t\in\RR_+]$-invariant,
with any choice $\Theta:=\Theta^{(p,\xx,q,\kappa\xx)}$ 
where $\vert\kappa\vert=1$ we get
a formula for $\Phi^t$ by substituting 
$b_0=0$ and $\mu=0$ in (\the\QuadS.\the\QuadT) which is (\the\OneS.\the\OneT).

If $[\overline{\Psi^t} : t\in\RR_+]$ admits two 
distinct fixed points $p,q\in \partial{\bf B}$
then, as we have shown, the disc $\Delta_{p,q}$ is 
automatically $[\Psi^t: t\in\RR_+]$-invariant, and
with the choice $\Theta:=\Theta^{(p,\xx,q,-\xx)}$ we get
a formula for $\Phi^t$ by substituting $b_0=0$ and 
$\mu=0$ in (\the\QuadS.\the\QuadT) establishing (\the\TwoS.\the\TwoT).

\smallskip
Proof of (iii).
Suppose indirectly that $0\ne \lambda={\rm Re} \langle e\vert b\rangle$. 
Then the skew symmetry of $iA$ entails 
${\rm range}(iA-\langle e\vert b\rangle)={\bf H}$.
By the previous lemma, we have a non-trivial 
$\Gamma$-invariant disc and we are in the settings of (i) or (iii).
By assumption, (i) is not the case. However,
in the case of (iii) we have $\langle e\vert b\rangle=i\mu \in i\RR$ automatically.
%which is contradictory.


\bigskip
\noindent
\ADT
\bf \NUM. Proof for Remark \the\IntS.\the\IntT. \rm
The operator $S_0$ is closed with dense domain in ${\bf H}_0$.
Since $S_0$ is also symmetric, both $\pm iS_0$ are dissipative 
(namely ${\rm Re}\langle \pm iS x_0 \vert x_0\rangle =0$ for $x_0\in{\rm dom}(S)$)
with the properties that both ${\rm range}(\pm iS_0 + \delta)$ 
are dense in ${\bf H}$ for any $\delta>0$
and that the operators $(iS + \delta)^{-1}: 
{\rm range}(S) \to {\bf H}_0$ $(0\ne \delta\in\RR)$ 
are all bounded and densely defined.\footnote{
Indeed $y_0 \perp {\rm range}(\pm iS_0 + \delta)$ means 
$0=\langle \pm iS_0 x_0 - \delta x_0 \vert y_0\rangle$
that is  
$0=\langle x_0 \vert \mp iS_0 y_0 - \delta y_0\rangle$ for $(x_0\in{\rm dom}(S))$
entailing 
$\mp iS_0 y_0 + \delta y_0 =0$ with 
$\delta \Vert y_0\Vert^2 = \pm i \langle S_0 y_0 \vert y_0\rangle \in i\RR$
which is possible only if $y=0$.
Thus by the Lumer-Phillips theorem [\NAG, II. Thm. 3.15], also
the ope rator $- iS_0$ generates a
strongly continuous contraction (actually isometry) semigroup and 
all the values $0\ne \delta \in\RR$ belong to the resolvent set of $iS$.}
%
Given $\delta\!\in\!\RR\setminus\{ 0\}$,
by [\NAG, II. Lemma 1.3] for any $x_0\!\in\!{\rm range}(iS_0 \!-\! \delta)$
and $t \!>\! 0$ we have
$\int_0^t e^{-\delta\tau} V_0^\tau x_0 \, d\tau \!=\! 
\int_0^t e^{-\delta\tau} V_0^\tau (iS_0 \!-\!\delta) 
[(iS_0 \!-\!\delta)^{-1} x_0] \, d\tau  \!=\!
\big( e^{-\delta t} V_0^t \!-\! 1\big) (iS_0 \!-\!\delta)^{-1} x_0 $.
The boundedness of both the operators $V^t$ and the resolvents 
$R(\delta,iS)={\rm closure}\big( (\delta - iS_0)^{-1} \big)$ establishes
{\the\IntS.\the\IntT} %that
for %all
$t\in\RR_+$ and $0\ne \lambda\in\RR$
with integrals of strongly continuous bounded operator valued functions.


\ADS
 \section{Dilation}

\ADT
\newcount\DilS \DilS=\SEC
\newcount\DilT \DilT=\THM
\noindent
\bf Lemma \NUM. \it
Let $[ U^t: t\in\RR_+]$ be a \SCS of linear isometries of a Banach space ${\bf E}$.
Suppose ${\bf E}$ is a subspace of another Banach space ${\bf F}$ and there is
a surjective isometry $V \in{\cal L}({\bf F})$ such that $U^1 = V\big\vert {\bf E}$.
Then there is a subspace 
${\bf E} \subset\widehat{{\bf E}} \subset{\bf F}$ along with a \SCG
$[\widehat{U}^t : t\in\RR]$ of 
surjective linear isometries of $\widehat{{\bf E}}$
such that $U^t = \widehat{U}^t \big\vert {\bf E}$ $(t\in\RR_+)$
with 
${\rm dom}\big({\rm gen}[\widehat{U}^t : t\in\RR]\big) \supset 
{\rm dom}\big({\rm gen}[U^t : t\in\RR]\big)$

\bigskip\noindent
\bf Proof. \rm
Let \ $\widehat{{\bf E}} \!:=\! 
{\rm closure}\big( {\bf E}_\infty \big)$ in ${\bf F}$ where
${\bf E}_\infty := \bigcup_{n=0}^\infty$ with ${\bf E}_n:=V^{-n} {\bf E}$.
%${\bf E}_\infty \!:=\! \big\{ x\!\in\!{\bf F}\! : \exists\, n\!\in\!\ZZ_+  \ V^n x \!\in\! {\bf E} \big\}$.
By assumption $V{\bf E} = U^1 {\bf E} \subset {\bf E}$ whence, by induction we conclude that
the subspaces
${\bf E}_n$ $(n\in\ZZ_+)$ form an increasing sequence.
Therefore all the operators
$U_n^t \! := \! V^{-2n} U^{t+n} V^n \big\vert {\bf E}_n$ 
$(t \!\ge\!\! -n, \, n\!\in\!\!\ZZ_+ )$
are well-defined isometries ${\bf E}_n \!\to\! {\bf E}_{\lceil n-t \rceil}$.
We have $U_n^t = U_{n+1}^t\big\vert {\bf E}_n$ 
for all indices $n\!\in\!\ZZ_+$.
Indeed if $\widehat{x}\!\in\!{\bf E}_n$ and $t\!\ge\!-n$ then
\begin{equation*}
\begin{split}
U_{n+1}^t \widehat{x} &=
V^{-2n-2} U^{t+n+1} V^{n+1} \widehat{x} =
V^{-2n-2} U^{t+n+1} U^1 V^n \widehat{x} =\\
&= V^{-2n-2} U^{t+n+2} V^n \widehat{x} =
V^{-2n-2} V^2 U^{t+n} V^n \widehat{x} =
V^{-2n} U^{t+n} V^n \widehat{x} = U_n^t \widehat{x}
\end{split}
\end{equation*}
since 
%$U^1$ is a restriction of $V$ 
$V$ extends $U^1$
and we have
$ V^{n+1} \widehat{x} \in {\bf E}$ implying 
$ V^{n+1} \widehat{x} = U^1 V^n \widehat{x}$.
Hence
$$U_\infty^t \widehat{x} := 
\lim_{n\to\infty} U_n^t \widehat{x} = \big[ U_n^t \widehat{x}: \ n \in\ZZ_+ , \ n\ge t\big]
\qquad (\widehat{x}\in {\bf E}_\infty)$$
is
a well-defined linear isometry of the linear manifold ${\bf E}_\infty$
for any %parameter
$t\in\RR$.
Since
${\rm range}(U_n^t) \supset  V^{-2n} U^{\lceil t\rceil +n} {\bf E}  =
V^{\lceil t\rceil -n} {\bf E}$ for $t\ge -n$, we have
${\rm range}(U_\infty^t) = {\bf E}_\infty$ $(t\in\RR)$.
Thus the operators
$\widehat{U}^t := {\rm closure}(U_\infty^t)$  $(t\in\RR)$
are well-defined surjective linear $\widehat{{\bf E}}$-isometries, 
each of which extending the respective $U^t$.
We check they form a \SCG as follows.
Since $[\widehat{U}^t : t\in\RR]$ is an equilipschitzian family, 
%(actually consisting of isometries),
it suffices to see that its restriction $[\widehat{U}^t : t\in\RR]$
to the dense submanifold ${\bf E}_\infty$ of 
$\widehat{{\bf E}}$ is a \SCG\hskip-1mm .
Given $s,t\in\RR$ and $\widehat{x}\in\widehat{E}_\ell$, we have
$\widehat{U}^t \widehat{x} = V^{-2n} U^{t+n} V^n \widehat{x}
\in \widehat{{\bf E}}_{2n}$ whenever $n\ge \-t\ell$
and
$\widehat{U}^s ( \widehat{U}^t \widehat{x} ) = 
V^{-2m} U^{t+m} V^m  \widehat{U}^t \widehat{x}$
whenever $m\ge \max\{-s,2n\}$. 
It follows $\widehat{U}^s \widehat{U}^t \widehat{x} =  \widehat{U}^{s+t} \widehat{x}$
because hence, with $k\ge2( \vert s \vert + \vert t \vert + \ell)$ we have
\begin{equation*}
\begin{split}
\widehat{U}^s ( \widehat{U}^t \widehat{x} ) &= V^{-4k} U^{s+2k} V^{2k}  V^{-2k} U^{t+k} V^k \widehat{x} =\\
&= V^{-4k} U^{s+t+3k} V^k \widehat{x} =   V^{-4k} U^{s+t+2k}  V^{2k} \widehat{x}  V^{k} =
\widehat{U}^{s+t} \widehat{x} .
\end{split}
\end{equation*}
To see strong continuity, consider
any vector $\widehat{x}\in\widehat{E}_\ell$. 
Then for any integer $n\ge\ell$ the orbit
$(-n,\infty)\ni t \mapsto \widehat{U}^t \widehat{x} =  
V^{-2n} U^{t+n} (V^n \widehat{x})$
is continuous since $V^{-2n}$ is an isometry and 
$(V^n \widehat{x}) \in{\bf E}$.
Hence we can see also the required generator domain inclusion property:
with $\widehat{x}:=x\in{\rm dom}\big( {\rm gen}[U^t: t\in\RR_+]\big)$ we have
$V^n x = U^n x \in{\rm dom}\big( {\rm gen}[U^t: t\in\RR_+]\big)$ 
entailing even the differentiability of
the orbits $(-n,\infty)\ni t\mapsto \widehat{U}^t x$.


\bigskip
In particular, since every linear isometry of a 
Hilbert space admits a unitary dilation [\FSZ],
in our setting of interest we conclude the following.

\bigskip\noindent
\ADT
\newcount\HilbS \HilbS=\SEC
\newcount\HilbT \HilbT=\THM
\bf Corollary \NUM. \it If \ $[U^t: t\in\RR_+]$ is a 
\SCS of linear ${\bf H}$-isometries,
there exists a Hilbert space $\widehat{{\bf H}}$ containing 
${\bf H}$ as a subspace along with a \SCG
$[\widehat{U}^t : t\in\RR]$ of \ $\widehat{{\bf H}}$-unitary operators
such that \ $U^t = \widehat{U}^t \big\vert {\bf H}$ $(t\in\RR_+)$
%and
whose %infinitesimal
generator is
%a $($possibly unbounded$)$ densely defined skew self-adjoint
an extension of \
${\rm gen}[U^t : t\in\RR_+]$.


\bigskip\noindent
\ADT
\bf \NUM. Proof of Corollaries {\the\ExtS.\the\ExtT}-{\the\FuncT}
\rm

\noindent
Given any Hilbert space $\widehat{{\bf H}}$ containing ${\bf H}$ as a subspace,
every M\"obius transformation %$\Theta_c$
of ${\bf H}$ extends to a
M\"obius transformation
of $\widehat{{\bf H}}$. Hence it suffices to see only that any \SCS of the form
%treated
of Theorem {\the\MainS.\the\MainT} admits a group a dilation
of the same algebraic form in a larger Hilbert space.
Let $[\Phi^t: t\!\in\!\RR_+]$ be given as in Theorem {\the\MainS.\the\MainT}.
According to
Corollary {\the\TupleS.\the\TupleT}, for some tuple
$\hbox{\mathfrak a}:=
\big( {\bf H},\xx,[ V_0^t: t\!\in\!\RR_+], b_0, \lambda, \mu \big)$
we have
\ADT
$\Phi^t = \Phi^t_{\hbox{\mathfrak a}} = 
\FFF({\cal U}^t)$ $(t\in\RR_+)$ with
$${\rm gen}[{\cal U}^t : t\in\RR_+] = 
\left[ \begin{matrix} iR & b \\ b^* & 0 \end{matrix} \right] =
\left[ \begin{matrix} i(S_0+\mu) &-b_0 & b_0 \\ 
b_0^* & 2i\mu &\lambda-i\mu \cr b_0^* &\lambda+i\mu & 0 \end{matrix}\right] .
\eqno(\NUM)$$
in terms of
$\big[{\bf H}_0\oplus(\CC\xx)\oplus\CC\big]$-matrices.
Let $[\widehat{V}_0^t : t\in \RR]$ be the dilation \SCG of 
$[V_0^t: t\in\RR_+]$ consisting of unitary operators
of a covering Hilbert space $\widehat{{\bf H}}_0$ of ${\bf H}_0$ 
with the skew self-adjoint extension $i\widehat{S}_0 = 
{\rm gen}[\widehat{V}_0^t : t\!\in\! \RR]$ of $iS_0$
guaranteed by Corollary {\the\HilbS.\the\HilbT}.
Also by Corollary {\the\TupleS.\the\TupleT}, the tuple
$\widehat{\hbox{\mathfrak a}}:=
\big( \widehat{{\bf H}},\xx,[ \widehat{V}_0^t: t\!\in\!\RR_+], b_0, \lambda, \mu \big)$
where $\widehat{{\bf H}}:=\widehat{{\bf H}}_0\oplus(\CC\xx)$
gives rise to a \SCG $\Phi_{\widehat{\hbox{\mathfrak a}}}^t : t\in\RR]$
such that
$\Phi_{\widehat{{\mathfrak a}}}^t = \FFF(\widehat{{\cal U}}^t)$ $(t\in\RR)$
whose infinitesimal generator can be written in the form 
of the right hand side of (\NUM)
when the entry $S_0$ is replaced with $\widehat{S}_0$.
Hence, by Theorem {\the\MainS.\the\MainT}, the transformations 
$\Phi^t_{{\mathfrak a}}$ can be written in the form (2.3)
with  $\widehat{S}_0$ in place of $S_0$ and $\widehat{V}_0^t$ in place of $V_0^t$.
Since $\widehat{V}_0^t \big\vert {\bf H}_0 = V_0^t$ $(t\in\RR_+)$, 
it readily follows
$\Phi_{\widehat{\hbox{\mathfrak a}}}^t \big\vert {\bf H} = 
\Phi_{\hbox{\mathfrak a}}^t$ $(t\in\RR_+)$
which completes the proof of Corollary {\the\ExtS.\the\ExtT}.

To prove Corollary {\the\FuncS.\the\FuncT}, consider any 
\SCS $[\Phi_{{\mathfrak a}}^t : t\in\RR_+]$ with its dilation group
$[\widehat{\Phi}_{\hbox{\mathfrak a}}^t : t\in\RR_+]$ as above.
By construction, the dilation \SCG $[\widehat{V}_0^t : t\in\RR]$ 
consists of $\widehat{{\bf H}}_0$-unitary operators.
Thus, in view of Stone's classical theorem, 
we can apply the functional calculus [\FSZ] with 
its skew self-adjoint generator $i\widehat{S}_0$
when evaluating the transformations 
$\Phi_{\hbox{\mathfrak a}}^t$ by means of (\the\PhiS.\the\PhiT).
Actually, for any $t\in\RR$ we have
$$\int_0^t e^{-\lambda\tau} \widehat{V}_0^\tau d\tau = 
g_{1,t}(\widehat{S}_0) , \quad
\int_0^t e^{-2\lambda\tau} \int_0^\tau 
e^{\lambda\sigma} \widehat{V}_0^\sigma \, d\sigma \, d\tau = g_{2,t}(\widehat{S}_0)$$
with the functions \
${\bf s} \!\mapsto\! \int_0^t e^{-\lambda\tau} e^{i\tau{\bf s}} d\tau $ 
resp.
${\bf s} \!\mapsto\! \int_0^t e^{-2\lambda\tau} \int_0^\tau 
e^{\lambda\sigma}  e^{i\sigma{\bf s}} d\sigma\,d\tau$ \
which are real analytic $\RR\to\CC$.
% as being bounded integrals of the same type.
Straightforward calculation establishes their %simplified
algebraic form and the Taylor series
appearing in Corollary {\the\FuncS.\the\FuncT}.

\bigskip
\bigskip

\noindent
\bf References \rm

\begin{itemize}

\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[ {[\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[ {[\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[ {[\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[ {[\STA]} ] L.L. Stach\'o,
{\it A fixed point approach to unitary $C_0$-groups in Hilbert space},
Rev. Roum. Math. Pures Appl., submitted 2015.

\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

\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.

\end{itemize}


\end{document}

\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-szeged.hu}



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@article{greenwade93,
    author  = "George D. Greenwade",
    title   = "The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})",
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@book{goossens93,
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    title     = "The LaTeX Companion",
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	%month		= "",
	%keywords	= "",
	%note		= "",
}

@proceedings 
    The proceedings of a conference.

        Required fields: title, year.
        Optional fields: editor, volume/number, series, address, month, organization, publisher, note.

@proceedings{Xproceedings,
	title		= "",
	%editor		= "",
	%volume		= "",
	%number		= "",
	%series		= "",
	%address	= "",
	%organization	= "",
	%publisher	= "",
	year		= "",
	%month		= "",
	%note		= "",
}

@techreport 
    Technical report from educational, commercial or standardization institution.

        Required fields: author, title, institution, year.
        Optional fields: type, number, address, month, note.

@techreport{Xtreport,
    author    = "",
    title     = "",
    institution = "",
    %type     = "", 
    %number   = "",
    %address  = "",
    year      = "XXXX",
    %month    = "",
    %note     = "",
}

@unpublished 
    An unpublished article, book, thesis, etc.

        Required fields: author, title, note.
        Optional fields: month, year.

@unpublished{Xunpublished,
	author	= "",
	title	= "",
	%year	= "",
	%month	= "",
	note	= "",
}



@article{AbedonHymanThomas2003,
  author = "Abedon, S. T. and Hyman, P. and Thomas, C.",
  year = "2003",
  title = "Experimental examination of bacteriophage latent-period evolution as a response to bacterial availability",
  journal = "Applied and Environmental Microbiology",
  volume = "69",
  pages = "7499--7506"
},

@incollection{Abedon1994,
  author = "Abedon, S. T.",
  title = "Lysis and the interaction between free phages and infected cells",
  pages = "397--405",
  booktitle = "Molecular biology of bacteriophage T4",
  editor = "Karam, Jim D. Karam and Drake, John W. and Kreuzer, Kenneth N. and Mosig, Gisela
            and Hall, Dwight and Eiserling, Frederick A. and Black, Lindsay W. and Kutter, Elizabeth
            and Carlson, Karin and Miller, Eric S. and Spicer, Eleanor",
  publisher = "ASM Press, Washington DC",
  year = "1994"
},

If you have to cite a website you can use @misc, for example:

@misc{website:fermentas-lambda,
      author = "Fermentas Inc.",
      title = "Phage Lambda: description \& restriction map",
      month = "November",
      year = "2008",
      url = "http://www.fermentas.com/techinfo/nucleicacids/maplambda.htm"
},

The note field comes in handy if you need to add unstructured information, for example that the corresponding issue of the journal has yet to appear:

@article{blackholes,
      author="Rabbert Klein",
      title="Black Holes and Their Relation to Hiding Eggs",
      journal="Theoretical Easter Physics",
      publisher="Eggs Ltd.",
      year="2010",
      note="(to appear)"
}

Getting current LaTeX document to use your .bib file

At the end of your LaTeX file (that is, after the content, but before \end{document}), you need to place the following commands:

\bibliographystyle{plain}
\bibliography{sample1,sample2,...,samplen} 
% Note the lack of whitespace between the commas and the next bib file.

Bibliography styles are files recognized by BibTeX that tell it how to format the information stored in the .bib file when processed for output. And so the first command listed above is declaring which style file to use. The style file in this instance is plain.bst (which comes as standard with BibTeX). You do not need to add the .bst extension when using this command, as it is assumed. Despite its name, the plain style does a pretty good job (look at the output of this tutorial to see what I mean).

The second command is the one that actually specifies the .bib file you wish to use. The ones I created for this tutorial were called sample1.bib, sample2.bib, . . ., samplen.bib, but once again, you don't include the file extension. At the moment, the .bib file is in the same directory as the LaTeX document too. However, if your .bib file was elsewhere (which makes sense if you intend to maintain a centralized database of references for all your research), you need to specify the path as well, e.g \bibliography{/some/where/sample} or \bibliography{../sample1} (if the .bib file is in the parent directory of the .tex document that calls it).

Now that LaTeX and BibTeX know where to look for the appropriate files, actually citing the references is fairly trivial. The \cite{ref_key} is the command you need, making sure that the ref_key corresponds exactly to one of the entries in the .bib file. If you wish to cite more than one reference at the same time, do the following: \cite{ref_key1, ref_key2, ..., ref_keyN}.






