Content-Type: multipart/mixed; boundary="-------------0102142116445"

This is a multi-part message in MIME format.

---------------0102142116445
Content-Type: text/plain; name="01-67.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-67.keywords"

Markovian semigroups, CCR algebras, quasi-free states,
---------------0102142116445
Content-Type: application/x-tex; name="dfccr.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="dfccr.tex"

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb}

\setlength{\hoffset}{-1in}
\setlength{\voffset}{-1in}
  \setlength{\oddsidemargin}{15mm}
 \setlength{\evensidemargin}{15mm}
  \setlength{\topmargin}{20mm}
\setlength{\headheight}{5mm}
\setlength{\headsep}{8mm}
\setlength{\textheight}{225mm}
\setlength{\textwidth}{170mm}
\setlength{\marginparsep}{0mm}
\setlength{\marginparwidth}{0mm}
\setlength{\marginparpush}{0mm}
\setlength{\footskip}{14mm}


\newcommand{\cW}{{\cal W}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cQ}{{\cal Q}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cA}{{\cal A}}

\def\R{{\mathbb R}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\C{{\mathbb C}}

\newcommand{\Con}{\operatorname{Const.}}
\newcommand{\Proj}{\,\rm{Proj}\,}
\newcommand{\ch}{{\mathfrak h}}

\newcommand{\1}{{\bf 1}}

\newtheorem{thm}{Theorem}[section]

\newtheorem{prop}{Proposition}[section]
\newtheorem{lemm}{Lemma}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{rema}{Remark}[section]

\makeatletter

\newenvironment{pf}{\bigskip\par\noindent{\it Proof.}}%
                    {$\square$\bigskip}


\@addtoreset{equation}{section}
\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
%\renewcommand{\thefootnote}{\alph{footnote}}
%\renewcommand{\thethm}{\thesection.\arabic{thm}.}
%%%%%%%%%%%%%////%%%%%%%%%%%%%?????????

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

\title{Dirichlet Forms and Symmetric Markovian Semigroups on CCR Algebras with respect to Quasi-free States}

\author{Changsoo Bahn, Chul Ki Ko and Yong Moon Park \\
Department of Mathematics,  Yonsei University, Seoul 120-749, Korea \\
E-mail: ympark@yonsei.ac.kr }

%\address{Department of Mathematics,  Yonsei University, Seoul 120-749, Korea
%\\E-mail: changsoo@math.yonsei.ac.kr}

%\author{Chul Ki Ko}

%\address{Department of Mathematics,  Yonsei University, Seoul 120-749, Korea
%\\E-mail: kochulki@math.yonsei.ac.kr}

%\author{Yong Moon Park}

%\address{Department of Mathematics,  Yonsei University, Seoul 120-749, Korea
%\\E-mail: ympark@yonsei.ac.kr}
%\date{}

\maketitle

\begin{abstract}
Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in \cite{Pa}, we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let $\cA(\ch_0)$ be the CCR algebra over a complex separable pre-Hilbert space $\ch_0$ and let $\omega$ be a quasi-free state on $\cA(\ch_0)$. For any normalized admissible function $f$ and complete orthonormal system (CONS) $\{g_n \} \subset \ch_0$, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of $(\cA(\ch_0), \omega)$. It turns out that the form is independent of admissible function $f$ and CONS $\{g_n\}$ chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The purpose of this paper is to construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to gauge invariant quasi-free states, and then investigate detailed properties such as ergodicity of the semigroups. Let $\cA(\ch_0)$ be the CCR algebra over a complex separable  pre-Hilbert space $\ch_0$ and let $\omega$ be a gauge invariant quasi-free state on $\cA(\ch_0)$ \cite{BR}. For any normalized admissible function $f$ and complete orthonormal system (CONS) $\{g_n\} \subset \ch_0$, we use the general construction method of Dirichlet forms developed in \cite{Pa} to construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of the pair $(\cA(\ch_0), \omega)$. We show that the Dirichlet form we constructed is independent of the admissible function $f$ and the CONS $\{ g_n\}$ chosen.  By establishing a (chaos) decomposition of the quasi-free Hilbert space (see Section 5) and investigating the spectrum of the generator (Dirichlet operator) of the semigroup, we prove that the semigroup is ergodic and tends to the equilibrium exponentially fast.

The study of noncommutative Dirichlet forms was pioneered by Albeverio and H\o egh-Krohn~\cite{AH-K}, Sauvageot~\cite{Sau1,Sau2,Sau3} and extensively developed by Davies and Lindsay~\cite{DL}, and Guido, Isola and Scarlatti~\cite{GIS}. All these authors considered Markovianity of forms and semigroups only with respect to a tracial state $\phi_0$.  Recently, Goldstein and Lindsay~\cite{GL1,GL2}, and Cipriani~\cite{Cip} have extended the abstract theory to a faithful normal state $\phi_0$ in the context of standard forms.

The need to construct Markovian semigroups on von Neumann algebras, which are symmetric with respect to a non-tracial state, is clear for various applications to open systems~\cite{Dav}, quantum statistical  mechanics~\cite{BR}, and  quantum probability theory~\cite{Acc,AFL,Part}.  Although on an abstract level we have quite well-developed theory as mentioned above, the progress in concrete application is very slow.  We would like to mention a few recent works in this direction. The completely positive Hamiltonian semigroup for quantum spin chains in the ground state representation has been considered in \cite{FNW, Mat, Na}.
 In \cite{MZ1}
 and \cite{MZ2}
  Majewski and Zegarlinski used the generalized conditional expectation to construct generators of spin-flip type dynamics for quantum spin systems. In \cite{Pa}, one of authors gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras and applied the method to construct translation invariant    Markovian semigroups for quantum spin systems. In \cite{CFL}, quantum Ornstein-Uhlenbeck semigroups were constructed by means of noncommutative Dirichlet forms. Extending the methods of \cite{Pa} and this paper we construct symmetric Markovian semigroups on CAR algebras with respect to quasi-free states \cite{BKP}.

Let us describe the content of this paper briefly. Let $\ch_0$ be a complex separable pre-Hilbert space and $\ch$ the completion of $\ch_0$. Denote by $\cA(\ch_0)$ the $C^*$-algebra over $\ch_0$ generated by the Weyl operators $W(g)$, $g\in \ch_0$. Let $A$ be a self-adjoint operator on $\ch$ satisfying
\begin{equation} \label{1.1}
0 < A \le \alpha \1 < \1
\end{equation}
for some $\alpha \in (0,1)$. The gauge invariant quasi-free state $\omega$ on $\cA(\ch_0)$ is given by
$$
\omega(W(f)) = \exp \{-(f,\frac1 4(\1+A)(\1-A)^{-1}f)\}, \quad
f \in \ch_0.
$$
We assume that $A^{-1}$ exists as a (unbounded) self-adjoint operator and that any vector in $\ch_0$ is an analytic vector for $A^{-1/2}$ (Assumption 3.1).

Let $(\cH_\omega, \pi_\omega, \Omega_\omega)$ be the GNS
representation of the pair $(\cA(\ch_0), \omega)$ and $ \cM=
\pi_\omega (\cA(\ch_0))''$. We suppress $\omega$ and $\pi_\omega$
from the notation, i.e., $\cH=\cH_\omega$, $W(f) = \pi_\omega
(W(f))$, etc. We also write $\xi_0 =\Omega_\omega$. Let $\sigma_t
: \cM \to \cM$ be the one parameter group of automorphisms defined
by
$$
\sigma_t(W(f)) = W(A^{it} f),\quad f \in \ch_0, \,\,t \in \R.
$$
Then $\omega$ satisfies $\sigma$-KMS conditions \cite{BR}. We use $\Delta$ and $J$ to denote the modular operator and the modular conjugation respectively. Then $\sigma_t$ becomes the modular group ; $\sigma_t(B) =\Delta^{it} B \Delta^{-it}$, $B\in \cM$. Let $\cM'$ be the commutant of $\cM$. The map $j: \cM \to \cM'$ is the antilinear $*$-isomorphism defined by $j(B) = JBJ$, $B\in \cM$. The natural positive cone $\cP$ associated with $(\cM, \xi_0)$ is the closure of the set $\{Bj(B)\xi_0:\, B\in \cM \}$. The form $(\cM, \cH, \cP, J)$ is the standard form associated with $(\cM, \xi_0)$.

For any $g\in \ch_0$, let $\Phi (g)$ be the infinitesimal generator of the unitary group $W(tg)$, $t\in \R$, and
\begin{eqnarray*}
a(g) &= &2^{-1/2}(\Phi(g)+i \Phi (ig)), \\
a ^*(g) &=& 2^{-1/2}(\Phi(g)-i \Phi(ig)).
\end{eqnarray*}
For any $f,g \in \ch_0$, $a(f)$ and $a^*(g)$ are densely defined, closed and $a(f)^* =a^*(f)$, and satisfy the canonical commutation relations (CCRs)~\cite{BR}. Notice that $a^\# (g) $ and $j(a^\#(g))$, $g\in \ch_0$, are affiliated with $\cM$ and $\cM'$ respectively, where $a^\#(g)$ stands for either $a(g)$ or $a^*(g)$.

For any $B\in \cM$ and $n\in \N$, we write
$$
B_n = \left(\frac n \pi\right)^\frac12 \int \sigma_t(B)e^{-nt^2}dt.
$$
Let $\cW$ be the algebra generated by $W(f)$, $f\in \ch_0$. We use $\cW_0$ and $\cM_0$ to denote the algebras generated by $W_n(f)$, $f\in \ch_0$, $n \in \N$ and by $B_n$, $B\in \cM$, $n \in \N$ respectively. We also use $\cH_{fin}$ to denote the subspace of $\cH$ spanned by the vectors of the form $(\prod_{j=1}^n \Phi (g_j)) \xi_0$, $g_j \in \ch_0$, $j=1,2,\cdots,n$. Obviously, $\cW \xi_0$, $\cW_0 \xi_0$, $\cM_0 \xi_0$ and $\cH_{fin}$ are dense in $\cH$.

For any normalized admissible function $f$ (Definition \ref{def2.1}) and CONS $\{g_n \} \subset \ch_0$ for $\ch$, we define a sesquilinear form $\cE : D(\cE) \times D(\cE) \to \C$ by
\begin{eqnarray}
\nonumber D(\cE) &=& \cW_0 \xi_0
\,(\,\text{or}\,\cW\xi_0,\,\cM_0\xi_0 ),
%\sum_{n=1}^\infty \cE^{(n)}(\xi,\xi) < \infty \}
\\
\label{1.2}
\cE(\eta,\xi) &=& \sum^\infty_{n=1} \cE^{(n)} (\eta,\xi),\quad\eta,\xi \in D(\cE),
\end{eqnarray}
where for each $n \in \N$, $\eta, \xi \in D(\cE^{(n)} )$= $\cW_0 \xi_0 $ ( or $\cW\xi_0,\,\cM_0 \xi_0  $)
\begin{eqnarray}
\label{1.3}
\cE^{(n)}(\eta, \xi)
&= &\int \langle\big(\sigma_{t-i/4} (a(g_n))-j\,
(\sigma_{t-i/4}(a^*(g_n)))\big)\eta,  \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(a(g_n))-j\,(\sigma_{t-i/4}(a^*(g_n)))\big)\xi\rangle f(t) \,dt
\\[12pt]
 &&\nonumber +\int \langle\big(\sigma_{t-i/4} (a^*(g_n))-j\,
(\sigma_{t-i/4}(a(g_n)))\big)\eta,  \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(a^*(g_n))-j\,(\sigma_{t-i/4}(a(g_n)))\big)\xi\rangle f(t) \,dt .
\end{eqnarray}
See Section 3 for the details.

 It turns out that the forms $(\cE,\cW\xi_0)$ and $(\cE,\cW_0 \xi_0)$ are closable and independent of the admissible function $f$
and the CONS $\{g_n \}$ chosen ( Proposition \ref{prop3.1}). For
each $ n \in \N$, the form $(\cE^{(n)}, \cM_0 \xi_0)$ is closable
and its closure $(\overline{\cE}^{(n)}, D(\overline{\cE}^{(n)}))$
is a Dirichlet form (Proposition \ref{prop3.2}). Let
$\overline{\cE}$ be the form defined by
$$
\overline{\cE} = \sum^\infty_{n=1} \overline{\cE}^{(n)}.
$$
Then $(\overline{\cE},D( \overline{\cE}))$ is a densely defined Dirichlet form (Theorem \ref{thm3.1}). As a Corollary of Theorem \ref{thm3.2}, the form $(\overline{\cE},D( \overline{\cE}))$ is also independent of $f$ and $\{g_n \}$ we have chosen.

Let $H$ be the infinitesimal generator of the Markovian semigroup
$\{T_t \}_{t\ge 0}$ associated to $(\overline{\cE},D(
\overline{\cE}))$. In order to analyze the spectrum $\sigma(H)$ of
$H$, we introduce a (chaos) decomposition of $\cH$. Let $B$ be the
operator given by
\begin{equation}
   B := A^{-1/2} - A^{1/2}.
\end{equation}
 For any $g \in \ch_0$, let $D_1(g)$ and $D_2(g)$ be the
operators on $\cH$ defined by
\begin{eqnarray*}
D_1(g) & :=& \sigma_{-i/4} (a(B^{-1/2} g)) -j ( \sigma_{-i/4}
(a^* (B^{-1/2} g))), \\
\nonumber
D_2(g) &:= &  \sigma_{-i/4} (a^*(B^{-1/2} g))-j( \sigma_{-i/4} (a(B^{-1/2} g))).
\end{eqnarray*}
Then $D_i (f) \xi_0 =0$, $i=1,2$, $ f\in \ch_0$ (Lemma
\ref{lem5.1}) and the following CCRs hold (Proposition
\ref{prop5.1}) : for any $f,g \in \ch_0$
\begin{eqnarray*}
{[D_1 (f), D_1 (g) ^* ]} & = & ( f,g ) \1,  \\
{[D_2 (f), D_2 (g) ^* ]} & = & ( g,f ) \1,  \\
{[D_i (f), D_i (g)]} &=& 0,  \quad i=1,2, \\
{[D_1 (f) ^\#, D_2 (g) ^\# ]} &=& 0,
\end{eqnarray*}
where $D_i (f)^\#$ is either $D_i(f)$ or $D_i (f)^*$, $i=1,2$.
Thus $D_i(f)$ and $D_i(g)^*$ can be considered as annihilation and
creation operators respectively and $\xi_0$  is the vacuum vector.
Thus $\cH$ has the following decomposition (Theorem \ref{thm5.1}):
$$
\cH =  \bigoplus^\infty_{m,n=0} \cH^{(m,n)},  % \sum_{n=1}^\infty\sum_{n=1}^\infty \cH^{(m,n)}
$$
where for each $m,n \in \N \cup \{0\}$, $\cH^{(m,n)}$ is the closure of the subspace spanned by vectors of the form
$$
(\prod_{j=1}^{m} D_1 (g_j )^*)(\prod_{l=1}^{n}
    D_2 (h_l )^* )\xi_0 ,\quad g_j ,\,h_l \in\ch_0.
$$

It follows that $\cH_{fin} \subset D(H)$ and for any CONS $\{g_n\}\subset \ch_0$
$$
  H=\sum_{n=1}^{\infty}\{D_1 (B^{1/2}g_n )^* D_1
  (B^{1/2}g_n )+ D_2 (B^{1/2}g_n )^* D_2
  (B^{1/2}g_n )\}
$$
as a bilinear form on $\cH_{fin} \times \cH_{fin}$ (Lemma \ref{lem5.4} (b)). Let $\cF$ be the symmetric Fock space over $\ch$ \cite{BR}. Then there is an antiunitary operator $V:\cF \to \cF$ and a unitary operator $U :\cH \to \cF \otimes V\cF$ (Proposition \ref{prop5.2}) such that
$$
 UHU^{-1} =d\Gamma (B)\otimes \text{\bf 1} +\text{\bf
 1}\otimes d\Gamma (B),
$$
where $d\Gamma (B)$ is the second quantization of $B$ \cite{BR}. Thus $H$ is essentially self-adjoint on $\cH_{fin}$ and independent of the admissible function $f$ and CONS $\{g_n\} \subset \ch_0$ chosen. Moreover the zero is a simple eigenvalue of $H$ with eigenvector $\xi_0$ and $(0, \alpha^{-1/2} -\alpha^{1/2}) \cap \sigma(H) = \emptyset$ (Theorem \ref{thm3.2}).

We should mention that the main results in this paper can be generalized. For an instance, if one replace $g_n$ by $B^\lambda g_n$, $ n\in \N$, $\lambda \in [-\frac12, \infty)$, in the definition of $\cE(\eta, \xi)$ in (\ref{1.2}) and (\ref{1.3}), the results in Section 3 still hold with an appropriate modification on the spectral gap. See Remark 3.1.
%( $(0, (\alpha^{-1/2} -\alpha^{1/2} )^{1+2\lambda } \cap \sigma (H) = \emptyset  $).

We organize this paper as follows; In Section 2 we first introduce
some terminologies in the theory of noncommutative Dirichlet forms
\cite{Cip} and then review the general construction method of
\cite{Pa}. We extend the method of \cite{Pa} slightly and produce
its proof. In section 3, we give an explicit expression of a
Dirichlet form for given normalized admissible function $f$ and
CONS $\{g_n \}\subset \ch_0$ for $\ch$, and then state main
results in this paper.  Section 4 is devoted to the proofs of
Proposition \ref{prop3.1}, Proposition \ref{prop3.2} and Theorem
\ref{thm3.1}.  In Section 5, we introduce a chaos decomposition of
$\cH$ and then prove the ergodicity of the semigroup and the
existence of a spectral gap. In Appendix, we give the proofs of
technical lemmas (Lemma \ref{lem4.3} and Lemma \ref{lem5.4}).

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

\section{Review on Construction of Dirichlet Forms on Standard Forms of von Neumann Algebras}

In this section, we first introduce necessary terminologies in the theory of noncommutative Dirichlet forms in the sense of Cipriani \cite{Cip}, and then describe the general construction method of Dirichlet forms on the standard form of von Neumann algebras developed in \cite{Pa}. We extend the result of \cite{Pa} slightly and give its proof.

Let $\cM$ be a $\sigma$-finite von Neumann algebra acting on a complex Hilbert space $\cH$.  A self-dual cone $\cP$ in $ \cH$ is a subset satisfying the property
$$
\{\xi\in\cH:\langle\xi,\eta\rangle\ge0, \quad\forall\eta\in\cP\}=\cP.
$$
$\cP$ is then a closed convex cone and $\cH$ is the
complexification of the real subspace $\cH^J:= \{\xi \in \cH :
\langle\xi, \eta\rangle \in \R, \,\,\, \forall \eta \in \cP\}$,
which elements are called $J$-{\it real} : $\cH=\cH^J+i\cH^J$.
Such a $\cP$ gives rise to a structure of ordered Hilbert space
on $\cH^J$ (denoted by $\le)$ and to an anti-unitary involution
$J$ on $\cH$, which preserves $\cH^J$ : $J(\xi+i\eta):=
\xi-i\eta$ for all $\xi, \eta \in \cH^J$. Any $J$-real element
$\xi \in \cH^J$ can be decomposed uniquely as a difference,
$\xi=\xi_+-\xi_-$, of two orthogonal, positive elements, called
the positive and the negative part of $\xi: \xi_+,\xi_- \in \cP,
\quad \langle\xi_+, \xi_-\rangle =0$.

A {\it standard form} $(\cM, \cH,\cP,J)$ of the von Neumann algebra $\cM$
acting faithfully on the Hilbert space $\cH$ consists of self-dual,
closed, convex cone $\cP$ in $\cH$ and the anti-unitary involution $J$
satisfying the properties:
\begin{enumerate}
\item[(a)] $J\cM J = \cM' $
\item[(b)] $JxJ = x^*, \quad \forall x \in \cM\cap \cM'$
\item[(c)] $J\xi =\xi, \quad \forall \xi \in \cP$
\item[(d)] $xJxJ(\cP)\subset \cP, \quad \forall x \in \cM, $
\end{enumerate}
where $\cM'$ is the commutant of $\cM$.

A bounded operator $A$ on $\cH$ is called {\it $J$-real} if $AJ=JA$
and {\it
positive preserving} if $A\cP \subset \cP$. The semigroup $\{T_t\}_{t\ge
0}$ is said to be {\it $J$-real} if $T_t$ is $J$-real for any $t\ge0$ and it
is called {\it positive preserving} if $T_t$ is  positive preserving for any
$t\ge 0$.

Let us fix a cyclic and separating vector $\xi_0$ in $\cP$. A bounded
operator $A:\cH \to \cH$ is called {\it sub-Markovian} (with respect to
$\xi_0$) if $0\le \xi\le \xi_0$ implies $0\le A\xi \le \xi_0$.  $A$
 is called {\it Markovian} if it is sub-Markovian and also
 $A\xi_0=\xi_0$. A semigroup $\{T_t\}_{t\ge 0}$ is said to be {\it
 sub-Markovian} (with respect to $\xi_0$) if $T_t$ is sub-Markovian for
 every $t\ge 0$. The semigroup $\{ T_t\}_{t\ge 0}$ is called {\it
 Markovian} if $T_t$ is Markovian for every $t\ge 0$.

Next, we consider a sesquilinear form on some linear manifold of
$\cH$ : $\cE(\cdot,\cdot): D(\cE)\times D(\cE) \to \C$. We also
consider the associated quadratic form: $\cE[\cdot]: D(\cE)\to
\C$,  $\cE[\xi]:= \cE(\xi,\xi)$.  A real valued quadratic form
$\cE[\cdot]$ is said to be {\it semi-bounded} if $\inf \{
\cE[\xi] : \xi \in D(\cE), \,\, ||\xi|| =1\} =-b > -\infty$. A
quadratic form $(\cE, D(\cE))$ is said to be {\it $J$-real} if $J
D(\cE) \subset D(\cE)$ and $\cE[J\xi] = \overline{\cE[\xi]}$ for
any $\xi \in D(\cE)$. For a given semi bounded quadratic form
$\cE$, one consider the inner product given by
$\langle\xi,\eta\rangle_\lambda:= \cE(\xi,\eta) + \lambda
\langle\xi,\eta\rangle$, for $\lambda>b$. The form $\cE$ is {\it
closed} if $D(\cE)$ is a Hilbert space for some of the above
norms. The form $\cE$ is called {\it closable} if it admits a
closed extension.

Associated to a semi bounded closed form $\cE$, there are a self-adjoint
operator $(H, D(H))$ and a strongly continuous, symmetric semigroup
$\{T_t\}_{t\ge0}$. Each of the above objects determines uniquely the
others according to well known relations (see Section 3.1 of \cite{BR} and
Section 1.3 of \cite{Fuk}).

From now on we will consider only $J$-real, real-valued, semi bounded, densely defined
quadratic forms. It is easy to check that these forms satisfy the
relation : $\cE[\xi+i\eta]=\cE[\xi]+\cE[\eta]$ for all $\xi +i\eta \in
D(\cE)^J+ iD(\cE)^J= D(\cE)$ where $D(\cE)^J := D(\cE) \cap H^J$.

Let us denoted by $\Proj(\xi,\cQ)$ the projection of the vector
$\xi \in \cH^J$ onto the closed, convex cone $\cQ \subset \cH^J$.
For $\xi,\eta \in \cH^J$, define
\begin{eqnarray*}
\xi \vee \eta &:=& \Proj (\xi, \eta+\cP) \\
\xi \wedge \eta &:=&\Proj (\xi, \eta-\cP).
\end{eqnarray*}

A $J$-real, real-valued, densely defined quadratic form
$(\cE, D(\cE))$ is called {\it Markovian} with respect to $\xi_0\in \cP$ if
$$
\xi \in D(\cE)^J \text{  implies  } \xi\wedge\xi_0 \in D(\cE) \text{ and
} \cE[\xi\wedge \xi_0] \le \cE[\xi].
$$
A closed Markovian form is called a {\it Dirichlet form}.
Let $\{T_t \}_{t\ge 0}$ be the semigroup associated to a semi bounded closed form $(\cE, D(\cE))$. Then one of the main results in \cite{Cip} is that $\{ T_t \}_{t\ge 0}$ is  sub-Markovian if and only if $(\cE, D(\cE))$ is a Dirichlet form (Theorem 4.11 of \cite{Cip}).

Next, we describe a general construction method of  Dirichlet forms on the natural standard
forms of von Neumann algebras associated with the Tomita-Takesaki
theory \cite{BR,Tak}.  The method has been developed in \cite{Pa}. However we generalize the result of \cite{Pa} slightly and produce the proof.

Let $\cM$ be a $\sigma$-finite von Neumann algebra acting on a Hilbert
space $\cH$ and let $\xi_0 \in \cH$ be a cyclic and separating vector
for $\cM$.
We use $\Delta$ and $J$ to denote respectively, the modular operator and
the modular conjugation associated with the pair $(\cM, \xi_0)$\cite{Ara,BR}. The
associated modular automorphism group  is denoted by $\sigma_t:
\sigma_t(A) = \Delta^{it} A \Delta^{-it}, A \in \cM$. Finally, $j:\cM \to \cM'$ is the anti linear $*$-isomorphism defined by $j(A)=JAJ$, $A\in \cM$.
The natural positive cone $\cP$ associated with $(\cM, \xi_0)$ is the
closure of the set
$$
\{Aj(A) \xi_0 : A\in \cM\}.
$$
By a general result the cone $\cP$ can be obtained by the closure of the
set
$$
\{ \Delta^{1/4}AA^* \xi_0 : A\in \cM\}.
$$
The natural cone $\cP$ is self-dual \cite{Ara,BR}. For the
detailed properties of $\cP$, we refer to Section 2.5.4 of
\cite{BR}. The form $(\cM, \cH, \cP, J)$ is the standard form
associated with the pair $(\cM, \xi_0)$.

In order to express Dirichlet forms, let us introduce the notion of
admissible functions:

\begin{defn} \label{def2.1}
An analytic function $f: D\to \C$ on a domain $D$ containing the
strip $Im\,\, z \in [-1/4, 1/4]$ is said to be admissible if the
following properties hold:
\begin{enumerate}
\item[(a)] $f(t) \ge 0$ for $ \forall t\in \R$,
\item[(b)] $f(t+i/4) + f(t-i/4) \ge 0$ for $\forall t\in \R$,
\item[(c)] there exist $M>0$ and $p>1$ such
that the bound
$$
|f(t+is)| \le M(1+|t|)^{-p}
$$
\end{enumerate}
holds uniformly in $s \in [-1/4,1/4]$.
\end{defn}
%\vspace*{0.1cm}
Let us give an example of an admissible function. Using the residue integration method it is easy to check that
$$
\int^\infty_{-\infty} (\cosh k)^{-1} e^{ikt}\,dk = 2\pi (e^{\frac {\pi}2 t}+e^{-\frac{\pi}{2} t })^{-1}.
$$
See also the expression in p. 94 of \cite{BR}.
Using the above, one may check that the function given by
 \begin{equation}\label{2.1}
f(t)=\frac2{\sqrt{2\pi}}\int (e^{k/4}+e^{-k/4})^{-1}e^{-\frac12
k^2}e^{-ikt}\,dk
\end{equation}
is an admissible function.

We are ready to give a construction of Dirichlet forms on the
standard form $(\cM, \cH, \cP, J)$ associated with the pair $(\cM,
\xi_0)$. Denote by $\cM_{an}$ the dense subset of $\cM$ consisting of
every $\sigma_t$-analytic element with a domain containing the
strip $I_{1/2} := \{z : | Im\,\,z| \le \frac 12 \} $ \cite{BR}.
  By Prop. 2.5.21 of \cite{BR},
any $A\in \cM_{an}$ is strongly analytic. In the following, the inner
product $\langle\cdot, \cdot \rangle$ on $\cH$ is conjugate linear in the first and
linear in the second variable. For given admissible
function $f$ and $x \in \cM_{an}$, define a sesquilinear form $\cE:
\cH\times \cH \to \C$ by
\begin{eqnarray}\label{2.2}
\cE(\eta, \xi)
&= &\int \langle\big(\sigma_{t-i/4} (x)-j\,
(\sigma_{t-i/4}(x^*))\big)\eta,  \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(x)-j\,(\sigma_{t-i/4}(x^*))\big)\xi\rangle f(t) \,dt
\\[12pt]
 &&\nonumber +\int \langle\big(\sigma_{t-i/4} (x^*)-j\,
(\sigma_{t-i/4}(x))\big)\eta,  \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(x^*)-j\,(\sigma_{t-i/4}(x))\big)\xi\rangle f(t) \,dt  \\[12pt]
&\equiv& \cE^{(1)}(\eta, \xi) +\cE^{(2)}(\eta, \xi). \nonumber
\end{eqnarray}
Then the associated quadratic form is given by
\begin{eqnarray} \label{2.3}
\cE[\xi]& = &\int || \,[\sigma_{t-i/4}(x) -j(\sigma_{t-i/4}(x^*))]\xi ||^2 f(t)\,dt \\
&& \nonumber +\int || \,[\sigma_{t-i/4}(x^*) -j(\sigma_{t-i/4}(x))]\xi ||^2 f(t)\,dt \\[12pt]
&\equiv& \cE^{(1)}[\xi] +\cE^{(2)}[\xi]. \nonumber
\end{eqnarray}
In \cite{Pa}, we have considered the case $x=x^* \in \cM_{an}$. The following is the result corresponding to Theorem 3.1 of \cite{Pa}.

\begin{thm} \label{thm2.1}
For a given admissible function $f$ and  $x \in
\cM_{an}$, let $(\cE,\cH)$ be defined as in (\ref{2.2}). Let $H$ be the
self-adjoint operator associated with $(\cE,\cH)$. Assume that
 there exists a constant $M >0$ such that the bound
$$
\sup_{s\in [-1/4,1/4]} ||\, \sigma_{t+is}(x) \,|| \le M
$$
holds  uniformly in $t\in \R$. Then the following
properties hold:
\begin{enumerate}
 \item[(a)] $H \xi_0 =0$,
 \item[(b)] $\cE$ is $J$-real,
 \item[(c)] $\cE(\xi_+,\xi_-) \le 0 \quad $   $\forall \xi \in \cH^J$.
\end{enumerate}
Furthermore the form $(\cE,\cH)$ is a Dirichlet form.
\end{thm}

We will produce the proof of Theorem \ref{thm2.1} at the end of this section.
The following is a  consequence of Theorem \ref{thm2.1}:

\begin{thm}  \label{thm2.2}
Let $\{T_t\}_{t\ge 0}$ be the semigroup generated by the form $(\cE,
\cH)$ in Theorem \ref{thm2.1}. Then $\{T_t\}_{t\ge 0}$ is a $J$-real, strongly
continuous, symmetric Markovian semigroup.
\end{thm}

\begin{pf}
It follows from Theorem \ref{thm2.1} (a) that $T_t(\xi_0) =\xi_0$ for any $t\ge
0$. Thus the theorem follows from Theorem 4.11 of \cite{Cip}.
\end{pf}

We now produce the proof of Theorem \ref{thm2.1}.

\vspace{0.2cm} \noindent
 {\bf Proof of Theorem \ref{thm2.1}.}
 (a) Notice that $JA\xi_0= \Delta^{1/2}A^*\xi_0$ for any $A\in \cM$.
 Thus we have that
 \begin{eqnarray*}
 \Big(\sigma_{t-i/4}(x)-j\,(\sigma_{t-i/4}(x^*))\Big)\,\xi_0 &=& \Delta^{1/4}
 \sigma_t(x)\xi_0 -J \Delta^{1/4} \sigma_t(x)^* \Delta^{-1/4} \xi_0
 \\
 &=& \Delta^{1/4}\sigma_t(x)\xi_0 -\Delta^{1/4} \sigma_t(x) \xi_0 \\
 &=& 0.
 \end{eqnarray*}
Replacing $x$ by $x^*$ in the above, we get
$$
 \Big(\sigma_{t-i/4}(x^*)-j\,(\sigma_{t-i/4}(x))\Big)\,\xi_0 =0.
$$
 Thus (a) follows from (\ref{2.2}) and the above facts.

(b) A direct estimate shows that
\begin{eqnarray*}
&&||\,\Big(\sigma_{t-i/4}(x) -j\,(\sigma_{t-i/4}(x^*))\Big) \, J\xi \, ||^2  \\
&& \quad = ||\,- J\, \big(\sigma_{t-i/4}(x^*) -j\,
(\sigma_{t-i/4}(x))\big)\, \xi\,||^2\\
&&\quad = ||\,\Big(\sigma_{t-i/4}(x^*)-j\,(\sigma_{t-i/4}(x))\Big)\,\xi \,||^2,
\end{eqnarray*}
which implies $\cE^{(1)}[J\xi] =\cE^{(2)}[\xi]$. The method used in the above also implies that $\cE^{(2)}[J\xi] =\cE^{(1)}[\xi]$.

(c) By the expression of $\cE(\eta, \xi)$ in (\ref{2.2}), $\cE(\xi_+, \xi_-)$ can be written as
\begin{eqnarray}
\cE(\xi_+, \xi_-) &=&\cE^{(1)} (\xi_+, \xi_-)+\cE^{(2)}
(\xi_+, \xi_-)\nonumber  \\[10pt]
&=& (\text{I}^{(1)} + \text{II}^{(1)}) +(\text{I}^{(2)} +\text{ II}^{(2)}) \label{2.4}
\end{eqnarray}
where
\begin{eqnarray} \nonumber
{\text I^{(1)}} &=&
 \int \Big(\langle\sigma_{t-i/4}(x) \xi_+,
\sigma_{t-i/4}(x) \xi_-\rangle\\
&& \quad \label{2.5}
+\langle\sigma_{t-i/4}(x^*) \xi_-, \sigma_{t-i/4}(x^*) \xi_+\rangle
\Big) f(t) \,dt,
 \\
\text{II}^{(1)} &=&
 -\int \Big(\langle \sigma_{t-i/4}(x) \xi_+,
j\,(\sigma_{t-i/4}(x^*))\xi_- \rangle \nonumber\\
 &&\quad  +\langle j\,(\sigma_{t-i/4}(x^*))\xi_+,
\sigma_{t-i/4}(x)\xi_-\rangle \Big) f(t)\,dt\nonumber
\end{eqnarray}
and $\text{I}^{(2)}$ and $\text{II}^{(2)}$ are obtained from  $\text{I}^{(1)}$   and $\text{II}^{(1)}$  respectively
  replacing $x$ by $x^*$ in the above.

As a consequence of  Theorem 4(7) of \cite{Ara}, $\cM \xi_+ \perp \cM \xi_-$,
which implies
$\text{I}^{(1)}=0$ and $\text{I}^{(2)}=0$. See also the proof of Proposition 5.3 (ii) of \cite{Cip}.
Next, we first consider $\text{II}^{(1)}$. It can be checked that
 $\sigma_{t-is}(x)^*=
\sigma_{t+is}(x^*)$, for any $x \in \cM_{an}$ and $ s\in \R$, and so
 \begin{equation}
 \begin{array}{l}
    \langle\sigma_{t-i/4}(x) \xi_+\, , \,
j\,(\sigma_{t-i/4}(x^*))\xi_-\rangle \\[10pt]
\quad \quad\quad =
 \langle\, \xi_+\, ,\,
 \sigma_{t+i/4}(x^*)j(\sigma_{t+i/4}(x)^*)\xi_-\rangle
 \end{array}
 \label{2.6}
 \end{equation}
 and
 \begin{equation}
 \begin{array}{l}
    \langle j\,(\sigma_{t-i/4}(x^*)) \xi_+\, ,\,
\sigma_{t-i/4}(x)\xi_-\rangle \\[10pt]
\quad \quad  \quad
   =
      \langle\xi_+\, ,\, \sigma_{t-i/4}(x)j(\sigma_{t-i/4}(x^*)^*)
      \xi_-\rangle.
      \end{array}
       \label{2.7}
  \end{equation}
It follows from (\ref{2.5})-(\ref{2.7}) that
 \begin{eqnarray*}
  \text{II}^{(1)} &=& -\int \langle
      \,
      \xi_+\, ,\,\sigma_{t+i/4}(x^*)j(\sigma_{t+i/4}(x)^*)\xi_-\rangle\,f(t)\,dt
      \\
      &&-\int \langle
       \xi_+\, , \,\sigma_{t-i/4}(x)j(\sigma_{t-i/4}(x^*)^*)\xi_-\, \rangle
        f(t)\,dt.
      \end{eqnarray*}
Notice that the map $A \mapsto j(A^*)$ from $\cM$ to $\cM'$ is linear.
It can be shown that for any $x\in \cM_{an}$ and $\xi\in \cH$, the map
$$
z \mapsto j(\sigma_z(x)^*)\xi
$$
is analytic on a domain containing the strip $I_{1/2}$. In fact,  the analyticity follows from the facts that $\langle\eta\, ,\,j(\sigma_z(x)^*)\xi\rangle$ $= $
$\langle\sigma_z(x)^*\,J \xi\, ,\,J\eta\,\rangle$ $=$ $\langle J\xi, \sigma_z(x) J\eta\,\rangle$ for any $\eta, \xi \in \cH$, and
 that weak analyticity implies strong
analyticity ( see Theorem VI.4 of \cite{RS}).

Using the Cauchy integral theorem, the assumption in the
          theorem, the property (c) in the Definition
        \ref{def2.1} and
          $\sigma_t(x)^*=\sigma_t (x^*)$, we obtain
          that
    \begin{eqnarray*}
 \text{II}^{(1)} =&& -\int
      \langle \xi_+,\sigma_{t}(x)^* j(\sigma_t(x)^*)\xi_-\rangle
      f(t-i/4)\,dt \\
&&  -\int
      \langle \xi_+,\sigma_{t}(x) j(\sigma_t(x))\xi_-\rangle
      f(t+i/4)\,dt .
          \end{eqnarray*}
Replacing $x$ and $x^*$ in the above, we obtain the expression of $\text{II}^{(2)}$.
Thus we get
\begin{eqnarray*}
\text{II}&=& -\int
      \langle \xi_+,[\sigma_{t}(x) j(\sigma_t(x))+ \sigma_t(x)^* j(\sigma_t(x)^*)]\xi_-\rangle  \\
         &&\quad \cdot ( f(t-i/4)+f(t+i/4))\,dt .
\end{eqnarray*}
  Since $\sigma_t(x)j(\sigma_t(x)) \xi_- \in \cP$, $\langle\xi_+,
  [\sigma_t(x)j(\sigma_t(x))+ \sigma_t(x)^* j(\sigma_t(x)^*)] \xi_-\rangle
  \ge 0 $ for $t\in \R$. By the property (b) in the Definition \ref{def2.1}, we
  conclude that II $\le 0$. This proved the part (c) of the theorem.

Clearly $\cE[\cdot] \ge 0$. Note that $\cE(\xi_0, \xi)=0$, $\forall \xi \in \cH$.  By Theorem \ref{thm2.1} (b)-(c), Proposition 4.5 (b) of \cite{Cip} and Proposition 4.10(ii) of \cite{Cip}, $(\cE,\cH)$ is a Dirichlet form.  $\quad \square$

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

\section{Dirichlet Forms on CCR Algebras with respect to Quasi-free States : Preliminaries and Main Results}

Let $\ch_0$ be a complex separable pre-Hilbert space. Denote by $\ch$ the completion of $\ch_0$. The inner product $(\cdot,\cdot)$ on $\ch$ is conjugate linear in the first and linear in the second variable. Let $\cA(\ch_0) $ be the $C^*$-algebra generated by the Weyl operators $W(f)$, $f \in \ch_0$, satisfying
\begin{eqnarray}
W(-f)&=&W(f)^*,   \nonumber \\
W(f) W(g)&=&e^{-\frac{i}2\text{Im} (f,g)}W(f+g) ,    \quad \forall f,g\in \ch_0.
\label{3.1}
\end{eqnarray}
For the abstract properties of $\cA(\ch_0)$, see Theorem 5.2.8 of  \cite{BR}.

Next, we describe quasi-free states on $\cA(\ch_0).$ Let  $A $ be a bounded and non-negative operator on $\ch$. Recall that $\varphi \in \ch$ is an analytic vector for an operator $B$ on $\ch$  if $\varphi \in D(B^n)$, $n \in \N$,  and if
$$
 \sum_{n=0}^\infty \frac{\| B^n \varphi\|}{n!} t^n < \infty
$$
for some $t > 0$. In the rest of this paper, we assume that $A$ satisfies the following properties.

\vspace{0.2cm}
\noindent {\bf Assumption 3.1} \quad (a) There exists $\alpha \in (0,1) $ such that
$$
0 < A \le \alpha \1 < \1.
$$

(b) The inverse $A^{-1}$ of $A$ exists as a (unbounded) self-adjoint and positive operator on $\ch$.

(c) For any $z \in \C$,  $A^{z}$ leaves $\ch_0$ invariant, i.e., $A^{z} \ch_0 \subset \ch_0$. Moreover, $z \mapsto A^z \varphi$ is entire analytic for any $\varphi \in \ch_0$.

(d) Any $\varphi \in \ch_0$ is an analytic vector for $A^{-1/2}$.

\vspace{0.2cm}
We remark that a dense submanifold $\ch_0$ of $\ch$ satisfying the assumption exists by the spectral theorem.

\vspace{0.2cm} \noindent {\bf Example 3.1.} {\bf (Ideal Bose
gases) }  Let  $\ch$  be $L^2(\R^d,dx)$ and $\Delta$ the Laplacian
operator on $L^2(\R^d,dx)$. Let $A$ be given by
$$
A=\exp\{-\beta (-\frac{1}{2}\Delta +\mu \1)\},
$$
where $\beta>0$ and $\mu>0$. Then $\alpha= \exp(-\beta\mu)$. For
$f \in L^2(\R^d,dx), $ denote by $\hat{f}$ the Fourier transform
of $f$. Choose $\ch_0=\{f \in L^2(\R^d,dx) : \hat{f} \in
C_c(\R^d)\}.$ Clearly Assumption 3.1 satisfied.

\vspace{0.2cm}
For given $ A \in \cL(\ch)$ satisfying Assumption 3.1, the gauge invariant quasi-free state $\omega$ on $\cA(\ch_0) $ is defined by
\begin{equation} \label{3.2}
\omega(W(f)) = \exp \{-( f,\frac1 4(\1+A)(\1-A)^{-1}f)\}, \quad
f \in \ch_0.
\end{equation}
Let$ (\cH_\omega,\pi_\omega,\Omega_\omega)$ be the $GNS$ representation~\cite{BR} of $(\cA(\ch_0),\omega)$, and let $\cM=\pi_\omega(\cA(\ch_0))''$. Notice that the representation $(\cH_\omega,\pi_\omega,\Omega_\omega)$ of the CCR algebra $\cA(\ch_0)$ is regular~\cite{BR} in the sense that the unitary groups $t \in \R \longmapsto \pi_\omega(W(tf))$ are strongly continuous for all $f \in \ch_0$. We use the notation $\Phi_\omega(f)$ to denote the infinitesimal generator of the unitary group
$\pi_\omega(W(tf)):$
\begin{equation}\label{3.3}
\pi_\omega(W(tf)) = \exp(it\Phi_\omega(f)).
\end{equation}
Notice that, since $\omega$ is an entire analytic state\cite{BR} by (\ref{3.2}) and Assumption 3.1, the cyclic vector $\Omega_\omega$ is an entire analytic vector for all $\Phi_\omega (f)$, $f \in \ch_0$.
The annihilation and creation operators defined for each $f \in \ch_0$ by
$$
D(a_\omega(f))=D(\Phi_\omega(f))\cap D(\Phi_\omega(if)) = D(a^*_\omega(f))
$$
and
\begin{eqnarray}
\label{3.4}
a_\omega(f) &:= &2^{-1/2}(\Phi_\omega(f)+i \Phi _\omega(if)), \\
a_\omega ^*(f) &:=& 2^{-1/2}(\Phi_\omega(f)-i \Phi_\omega(if))
\nonumber
\end{eqnarray}
are densely defined, closed and $a_\omega(f)^* = a^*_\omega(f)$, and satisfy the canonical commutation relations( CCRs): for $\forall f,g \in \ch_0$
$$
[ a_\omega (f), a_\omega (g) ] =  0 ,
$$
\begin{equation}
\label{3.5}
[ a_\omega (f), a_\omega ^* (g) ] = ( f,g ) \1.
\end{equation}
See Section 5.2.3 of \cite{BR}  for the details.

In the rest of this paper, we suppress $\omega$ and $\pi_\omega$ from the notations. Thus $ \cH = \cH_\omega,\,W(f) = \pi_\omega(W(f)),\,\Phi(f) = \Phi_\omega(f),a\,(f) = a_\omega(f)$ and $a^*(f) = a^*_\omega(f)$ for any $f \in \ch_0$. We also use the notation $\xi_0 = \Omega_\omega. $ Let $\sigma_t :\cM \rightarrow \cM$ be the group of automorphisms on $\cM$ defined by
$$
\sigma_t(W(f)) = W(A^{it} f),\quad f \in \ch_0.
$$
It can be checked that the state $\omega$ defined by (\ref{3.2}) is a $\sigma$-KMS state~\cite{BR}. Thus $\sigma_t, \,t \in \R,$ is the modular automorphism by Theorem 5.3.10 of \cite{BR}.  Let $\Delta$ and $J$ be the modular operator and modular conjugation respectively. Then $(\cM,\cH,\cP,J) $ is the natural standard form associated to $(\cM, \xi_0)$.

Next, we introduce several dense manifolds of $\cH$. For any $B \in \cM$, define
\begin{equation}
\label{3.6}
B_n = \left(\frac n \pi\right)^\frac12 \int \sigma_t(B)e^{-nt^2}dt,\quad n \in \N.
\end{equation}
Then $B_n$ is an entire analytic element for $\sigma_t$, $\|B_n\|\le \|B\|$ for all $n \in \N$ and $B_n \to B$ strongly. See the proof of Proposition 2.5.22 of \cite{BR}. Put
\begin{eqnarray}
\cM_0 &:=& \text{ the algebra generated by }  B_n, \quad B \in \cM, n \in \N
\nonumber \\
\label{3.7}
\cW_0 &:= & \text{the algebra generated by }  W_n(f),\quad  f \in \ch_0, n \in \N, \\
\cW &:=& \text{ the algebra generated by } W(f),\quad f \in \ch_0.
\nonumber
\end{eqnarray}
Denote by $\cH_{fin}$ the subset of finite linear combinations of the vectors of the following type:
$$
\psi_n = \left[\prod^n_{j=1}\Phi(f_j)\right]\xi_0, \quad \text{for}\, f_j \in \ch_0,\, j=1,\cdots,n, \, n \in\N .
$$
Clearly $\cM_0 \xi_0, \,\cW_0 \xi_0, \,\cW \xi_0$ and $\cH_{fin}$ are dense in $\cH$ (See Lemma \ref{lem4.1}).

We denote by $a^\# (f)$ either $a(f)$ or $a^*(f)$, for any  $f \in\ch_0$. Notice that $a^\# (f)$ and $j(a^\# (f))$, $f\in \ch_0$, are affiliated with $\cM$ and $\cM'$ respectively. For any $f \in \ch_0$ and $z \in \C$, we write
\begin{eqnarray}
\label{3.8}
\sigma_z (a(f)) &:=  &a(A^{i\bar{z}} f), \\
\sigma_z (a^*(f)) &:=  & a^*(A^{iz} f). \nonumber
\end{eqnarray}
In fact, one may be able to show that for any $f \in \ch_0$ and $\xi \in \cM_0 \xi_0$ the function $t \mapsto \sigma_t(a^\#(f)) \xi$ has an analytic extension on $\C$, which is denoted by $\sigma_z(a^\# (f)) \xi$, and that $\sigma_z(a^\# (f))$ is equal to that in the right hand side of (\ref{3.8}) on $\cM_0 \xi_0$. See the proof of Lemma \ref{lem4.3}.

We are ready to describe Dirichlet forms. An admissible function $f$ is said to be {\it normalized} if $\int f(t)dt = 1$. For given normalized admissible function $f$ and a complete orthonormal system(CONS) $\{g_n\}^\infty_{n=1} \subset \ch_0$ of $\ch, $ define a sesquilinear form $\cE: D(\cE)\times D(\cE)\to \C$
as follows:
$$
D(\cE) = \cW_0 \xi_0 \,(\,\text{or}\,\cW\xi_0,\,\cM_0 \xi_0 ),
% : \sum_{n=1}^\infty \cE^{(n)}(\xi,\xi) < \infty \}
$$
\begin{equation}\label{3.9}
\cE(\eta,\xi) = \sum^\infty_{n=1} \cE^{(n)} (\eta,\xi),\quad\eta,\xi \in D(\cE),
\end{equation}
where for each $n \in \N$, $\eta, \xi \in D(\cE^{(n)} ) = \cW_0 \xi_0\,$ ( or $ \cW \xi_0, \, \cM_0 \xi_0)$
\begin{eqnarray}
\label{3.10}
\cE^{(n)}(\eta, \xi)
&= &\int \langle\big(\sigma_{t-i/4} (a(g_n))-j\,
(\sigma_{t-i/4}(a^*(g_n)))\big)\eta,   \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(a(g_n))-j\,(\sigma_{t-i/4}(a^*(g_n)))\big)\xi\ \rangle f(t) \,dt
\\[12pt]
 &&\nonumber +\int \langle\big(\sigma_{t-i/4} (a^*(g_n))-j\,
(\sigma_{t-i/4}(a(g_n)))\big)\eta,  \\
  & & \nonumber \quad \quad   \big(\sigma_{t-i/4}(a^*(g_n))-j\,(\sigma_{t-i/4}(a(g_n)))\big)\xi\rangle f(t) \,dt .
\end{eqnarray}
We also define the associated quadratic forms by
\begin{eqnarray*}
\cE^{(n)} [\xi]& = & \cE^{(n)} (\xi, \xi), \quad \xi \in D(\cE^{(n)}
),\,n \in \N \\
\cE[\xi] &=& \sum_{n=1}^\infty \cE^{(n)} [\xi], \quad \xi \in D(\cE).
\end{eqnarray*}
We remark that the expression $\cE^{(n)} (\eta,\xi)$ in (\ref{3.10}) can be obtained from $\cE(\eta,\xi)$ in (\ref{2.2}) by replacing $x$ by $a(g_n)$.

We state main results. It turns out that the form defined in (\ref{3.9}) and (\ref{3.10}) is independent of the normalized admissible function $f$ and the CONS $\{g_n\}\subset \ch_0$ we have chosen:

\begin{prop}\label{prop3.1}
  Let $D(\cE)$ be either $\cW_0 \xi_0$ or else $\cW\xi_0$ and let $(\cE,D(\cE)) $ be defined as in (\ref{3.9}) and (\ref{3.10}). The form $(\cE,D(\cE))$ is closable. Moreover,$(\cE,D(\cE))$ is independent of the normalized admissible function $f$ and the CONS $\{g_n\}\subset \ch_0$ we have chosen.
\end{prop}

\begin{prop}\label{prop3.2}
 For each $ n \in \N,\,\,(\cE^{(n)}, \cM_0 \xi_0)$ is closable. The closure $(\overline{\cE}^{(n)},D(\overline{\cE}^{(n)}))$ is a Dirichlet form.
\end{prop}

\begin{thm}\label{thm3.1}
 Let us consider the following form :
\begin{eqnarray*}
D(\overline{\cE}) & = & \{ \xi \in \bigcap_{n=1}^\infty D(\overline{\cE}^{(n)} ) : \, \sum_{n=1}^\infty \overline{\cE}^{(n)} [\xi] < \infty \}, \\
\overline{\cE} [\xi]&= &\sum_{n=1}^\infty \overline{\cE}^{(n)} [\xi],
\end{eqnarray*}
where for each $n \in \N, \,({\overline
\cE}^{(n)},D(\overline{\cE}^{(n)}))$ is the Dirichlet form
obtained in Proposition \ref{prop3.2}. Then
$(\overline{\cE},D(\overline{\cE}))$ is a densely defined
Dirichlet form.
\end{thm}

\begin{thm} \label{thm3.2}
Let $ \{T_t\}$ be the symmetric Markovian semigroup associated to the Dirichlet form $(\overline{\cE},D(\overline{\cE}))$, and let  $H$ be the Dirichlet operator, i.e.,  $T_t=e^{-tH},t \ge 0.$ Then the following results hold:
\begin{enumerate}
\item[(a)]   $H$ is essentially self-adjoint on $\cH_{fin}$.
\item[(b)]   $H$ is independent of the normalized admissible function $f$ and the CONS $\{g_n\}\subset \ch_0$ we have chosen.
\item[(c)]    The zero is a simple eigenvalue of $H$ with eigenvector $\xi_0$.  Moreover $(0,\alpha^{-1/2}-\alpha^{1/2})\cap \sigma(H)=\emptyset.$ Thus $\{T_t\}$ is ergodic.
\end{enumerate}
\end{thm}

By the spectral theorem, Theorem \ref{thm3.2} (c) implies that for any $\xi \in \cH$ and $t\ge 0$,
\begin{equation} \label{3.11}
\| T_t \xi - \langle \xi_0, \xi \rangle \xi_0 \|_{\cH} \le e^{-mt} \| \xi - \langle \xi_0, \xi \rangle \xi_0 \|_{\cH}
\end{equation}
where $m = \alpha^{-1/2} - \alpha^{1/2}$. Thus $\{ T_t \}_{t\ge 0}$ converges to the equilibrium exponentially fast. Before closing this section, we would like to make the following remark:

\begin{rema} \label{rema3.1} The main results in this paper can be generalized in several ways. For an instance, let $B$ be the operator on $\ch$ defined in (1.4). If one replace $g_n$ by $B^\lambda g_n$, $n \in \N$, for some $\lambda \in [-\frac12, \infty)$ in the definition  of $\cE(\eta, \xi)$ in (\ref{3.9}) and (\ref{3.10}), and modify Assumption 3.1 (d) appropriately, then all of the results in this section still hold with a modified spectral gap in Theorem \ref{thm3.2} (c), i.e. $(0, (\alpha^{-1/2}-\alpha^{1/2}) ^{1+2\lambda}) \cap \sigma(H) =\emptyset$.
\end{rema}

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

\section{Markovianity of Forms : Proofs of Proposition \ref{prop3.1} - Theorem \ref{thm3.1}}

In this section, we produce the proofs of Proposition \ref{prop3.1}, Proposition \ref{prop3.2} and Theorem \ref{thm3.1}. The most difficult part is the proof of Proposition \ref{prop3.2} which states the Dirichlet property of each component $\cE^{(n)}$, $n \in \N$, of $\cE$ in (\ref{3.10}). In Theorem \ref{thm2.1}, we established the Dirichlet property for any $\sigma_t$-analytic element $x\in \cM$. However $a(g_n)$ in the definition of $\cE^{(n)}$ in (\ref{3.10}) is an unbounded operator affiliated with $\cM$ for any $g_n \in \ch_0$. Thus we have to employ several limiting processes which make the paper lengthy.

Recall the definitions of $\cW, \, \cW_0$, and $ \cM_0$ in (\ref{3.7}) and $\cH_{fin}$ in the below of (\ref{3.7}). In the rest of this paper, we denote by $a^\# (f)$ either $a(f)$ or $ a^*(f)$, $f\in \ch_0$. We first state elementary facts.

\begin{lemm} \label{lem4.1} $\quad$ (a) $\cW \xi_0, \, \cW_0 \xi_0, \, \cM_0 \xi_0 $ and $ \cH_{fin}$ are dense in $\cH$.

(b) The inclusions
$$ \cW \xi_0 \subset D(a^\# (f)), \quad \cW_0 \xi_0 \subset D(a^\#(f)) \,\, \text{and} \quad \cH_{fin} \subset D(a^\#(f))
$$
hold for any $f \in \ch_0$.

(c) The inclusion
$$
\cM_0 \xi_0 \subset D(a^\# (f))
$$
holds for any $f \in \ch_0$. Moreover the relation
$$
a^\# (f) B\xi_0 = j(\sigma_{-i/2} (B^*)) a^\# (f) \xi_0
$$
holds for any $f \in \ch_0$ and $B\in \cM_0$.
\end{lemm}

\begin{pf}
(a) Since $\cW$ is norm-dense in $\pi_\omega(\cA(\ch_0))$, $\cW \xi_0$ is dense in $\cH$. Let $f\in \ch_0$ be given. Using (\ref{3.6}) and (\ref{3.2}) it is easy to show that the sequence $\{W_n (f) \xi_0 \}$ converges to $W(f) \xi_0$. This implies that $\cW_0 \xi_0$ is dense in $\cH$. Since $\cW_0 \xi_0 \subset \cM_0 \xi_0$, $\cM_0 \xi_0$ is dense in $\cH$. It follows from (\ref{3.2}) and Assumption 3.1 that $\omega$ is an entire analytic state \cite{BR} and so the cyclic vector $\xi_0$ is an entire analytic vector for $\Phi(f)$, $ f\in \ch_0$. See also Section 5.2.3 of \cite{BR}. Thus the sequence of the vectors
$$
\sum_{k=0}^n \frac{i^k} {k!} \Phi (f)^k \xi_0
$$
converge to $W(f) \xi_0$, which implies that $\cH_{fin}$ is dense in $\cH$.

(b) For any $f \in \ch_0$ and $n \in \N$, we write that
$$
\Phi (f,n) = - in\Big( W(\frac1n f) -1\Big).
$$
For given $g \in \ch_0$, consider the sequence $\{ \Phi (f,n) W(g) \xi_0 \}$. Using (\ref{3.1}) and (\ref{3.2}), it can be checked that the sequence is a Cauchy sequence. Thus $W(g) \xi_0 \in D(\Phi (f))$ for any $f,g \in \ch_0$, and so $\cW\xi_0 \subset D(\Phi(f))$, $f\in \ch_0$. By (\ref{3.4}), this implies that $\cW \xi_0 \subset D(a^\# (f))$, $f\in \ch_0$. The method similar to that used in the above implies that $\cW_0 \xi_0  \subset D(a^\# (f))$, $f\in \ch_0$. Since $\xi_0$ is an entire analytic vector for $\Phi(f)$, $f\in \ch_0$, $\cH_{fin} \subset D(a^\#(f))$, $f \in \ch_0$.

(c) Notice that for any $B \in \cM_0$, $j(\sigma_{-i/2} (B^*)) \xi_0 = B\xi_0$. Thus we have that
\begin{eqnarray*}
\Phi (f,n) B\xi_0 &=& \Phi (f,n) j(\sigma_{-i/2}(B^*)) \xi_0 \\
&=& j(\sigma_{-i/2}(B^*)) \Phi(f,n) \xi_0.
\end{eqnarray*}
Since the sequence $\{ \Phi (f,n) \xi_0 \}$ converges to $\Phi(f) \xi_0$, it follows that $B\xi_0 \in D(\Phi(f))$ and the relation
$$
\Phi (f) B\xi_0 =j(\sigma_{-i/2} (B^*)) \Phi (f) \xi_0
$$
hold for any $f \in \ch_0$ and $B \in \cM_0$. By (\ref{3.4}), this proved the part (c).
\end{pf}


We next state a well-known formula ((\ref{4.2})) on quasi-free expectations which we will use repeatedly in the sequel.

\begin{lemm} \label{lem4.2}
Let $\omega$ be the quasi-free state given in (\ref{3.2}). The equalities
\begin{equation} \label{4.1}
\omega(\Phi(f) \Phi(g)) = \frac12 \Big( ( f, (\1-A)^{-1} g) + ( g, A(\1-A)^{-1} f) \Big)
\end{equation}
and
\begin{eqnarray}
\label{4.2}
&&\omega \left(\Phi(f) \left( \prod^m_{j=1} \Phi(g_j)\right) \left( \prod^n_{l=1} W(h_l)\right)\right) \\
\nonumber
&& \qquad = \sum_{k=1}^m \omega (\Phi(f)\Phi(g_k))\omega \left( \Big(\prod^{k-1}_{j=1} \Phi(g_j)\Big) \Big(\prod^{m}_{j=k+1} \Phi(g_j)\Big) \Big(\prod^{n}_{l=1} W(h_l)\Big)\right) \\
\nonumber
&& \qquad \quad + i  \sum_{k=1}^n \omega (\Phi(f)\Phi(h_k))\omega \left( \Big(\prod^{m}_{j=1} \Phi(g_j)\Big) \Big(\prod^{n}_{l=1} W(h_l)\Big)\right)
\end{eqnarray}
hold for any $ f,g,g_j,h_l \in \ch_0$, $j=1,\cdots, m$, $l=1,\cdots,n$.
\end{lemm}

\begin{pf}
The Weyl relations in (\ref{3.1}) and the quasi-free expectation in (\ref{3.2}) yield
$$
\omega(W(f)W(g)) =\exp \{ - \frac{i}2  \text{Im} ( f,g ) -\frac14 ( f+g, D(f+g))\}
$$
for any $f,g \in \ch_0$, where $D= (\1+A)(\1-A)^{-1}$. Replacing $f$ and $g$ by $tf$ and $sg$ respectively in the above, and differentiating both sides with respect to $t \in \R$ and $s\in \R$ at the zero, one obtains (\ref{4.1}).

Again, replacing $f$  by $tf$ in the above relation and differentiating with respect to $t\in \R$ at the zero, one can deduce that
$$
\omega(\Phi(f)W(g)) = i \omega (\Phi(f) \Phi(g)) \omega (W(g)).
$$
Setting $g = g_1 + \cdots + g_p$ in the above and using the relation in (\ref{3.1}), one obtains
$$
\omega(\Phi(f) (\prod^p_{j=1} W(g_j))) = i (\sum^p_{k=1} \omega(\Phi(f) \Phi(g_k))) \omega(\prod^p_{j=1} W(g_j)).
$$
Choose $p=m+n$. Replacing $g_j$ by $t_j g_j$, $j=1,\cdots, m$ and
$g_{m+l}$ by $h_l$, $ l=1,\cdots, n $, in the above, and
differentiating with respect to $t_j$, $j=1,\cdots,m$, at the
zero, we obtain (\ref{4.2}).
\end{pf}

We remark that $f \mapsto a(f)$ is conjugate linear and $f \mapsto a^*(f)$ is linear. See (\ref{3.4}), (\ref{3.1}) and (\ref{3.2}). Recall the definitions of $\sigma_z(a^\# (f))$, $f \in \ch_0$, $z\in \C$ in (\ref{3.8}), i.e.,
\begin{eqnarray*}
       \sigma_z (a(f))&:=&a(A^{i\overline{z}}f),\\
       \sigma_z (a^*(f))&:=&a^* (A^{iz}f).
 \end{eqnarray*}
We also write that
\begin{equation} \label{4.3}
\sigma_z(\Phi(f)):=\frac{1}{\sqrt2}\{a(A^{i\overline{z}}f)+a^*(A^{iz}f)\}
\end{equation}
for any $z \in \C$ and $f\in \ch_0$. Using CCRs in (\ref{3.5}), one can check that for any $f,g \in \ch_0$ the relations
\begin{eqnarray}
\label{4.4}
{[a(f),W(g)]}&=&\frac{i}{\sqrt2} ( f, g ) W(g),\\
  {[a^* (f),W(g)]}&=&-\frac{i}{\sqrt2}( g,f ) W(g),\nonumber
\end{eqnarray}
hold on $\cW \xi_0$ ( also on $\cW_0 \xi_0$ ). See also the proof of Proposition 5.2.4.(1) of \cite{BR}.

For any $f\in \ch_0$ and $m,n \in \N$, we write that
\begin{eqnarray}
\nonumber
  \Phi(f,n)&:=&-in\{\exp(\frac{i}{n}\Phi(f))-1\},\\
\label{4.5}
  \Phi_m(f)&:=&\sqrt{\frac{m}{\pi}}\int\sigma_t
  (\Phi(f))e^{-mt^2}\,dt,\\
  \Phi_m(f,n)&:=&\sqrt{\frac{m}{\pi}}\int\sigma_t
  (\Phi(f,n))e^{-mt^2}\,dt.\nonumber
 \end{eqnarray}
Notice that for any $f\in\ch_0$ and $m,n\,\in\N,$ $\Phi_m(f,n)$ is
an entire analytic element for $\sigma_t$ and
$\sigma_z(\Phi_m(f,n))$ is given by
\begin{equation}\label{4.6}
\sigma_z(\Phi_m(f,n))=\sqrt{\frac{m}{\pi}}\int\sigma_t
  (\Phi(f,n))e^{-m(t-z)^2}\,dt
 \end{equation}
 for any $\, z\in\mathbb{C}.$ See the proof of Proposition 2.5.22 of
 \cite{BR}.

We establish technical lemmas which will be used later.

\begin{lemm} \label{lem4.3} $\quad$
(a) For any $f\in \ch_0$, $m,n \in \N$ and $z=t+is \in \C$, there exist constants $M_1(f,s)$ and $M_2(f,s)$ depending only on $f \in \ch_0$ and $s= $ Im $z$ such that the bounds
\begin{eqnarray*}
\| \sigma_z(\Phi_m (f,n)) \xi_0 \| &\le& M_1 (f,s), \\
\| \sigma_z (\Phi (f)) \xi_0 \| &\le&  M_2 (f,s)
\end{eqnarray*}
hold (uniformly on $m,n \in \N$).

(b) For any $f \in \ch_0$, $n \in \N$ and $z\in \C$, the equality
$$
   \sigma_z(\Phi(f))\xi_0 =\lim_{n\rightarrow\infty}
               \sigma_z(\Phi_n (f,n))\xi_0
  $$
holds.
\end{lemm}
The proof of the above lemma will be given in Appendix.

 We define that for any $f \in \ch_0$ and $m,n \in \N$
\begin{eqnarray}\label{4.13}
 &&a_m(f,n):=\frac{1}{\sqrt2}\{\Phi_m(f,n)+i\Phi_m(if,n)\},\\
 &&a^*_m(f,n):=\frac{1}{\sqrt2}\{-\Phi_m(-f,n)+i\Phi_m(-if,n)\}. \nonumber
 \end{eqnarray}
We denote by $a^\#_m (f,n)$ either $a_m (f,n)$ or $a_m^* (f,n)$. The following is a consequence of Lemma \ref{lem4.3}.

\begin{cor} \label{cor4.1}
$\quad$ (a) For any $f \in \ch_0$, $m,n \in \N$ and $z =t+is \in \C$, there exist constants $M_3(f,s)$ and $M_4(f,s)$ depending only on $f \in \ch_0$ and $ s =$ Im $z$ such that the bounds
\begin{eqnarray*}
\| \sigma_z(a^\#_m (f,n) )\xi_0 \| &\le& M_3 (f,s), \\
\| \sigma_z(a^\# (f)) \xi_0 \| &\le& M_4 (f,s)
\end{eqnarray*}
hold (uniformly on $m,n \in \N$).

(b) For any $f\in \ch_0$, $n \in \N$ and $z \in \C$, the equality
$$
\sigma_z(a^\#(f)) \xi_0 = \lim_{n \to \infty} \sigma_z( a^\#_n (f,n)) \xi_0
$$
holds.
\end{cor}

\begin{pf} The corollary follows from Lemma \ref{lem4.3} and the definition of $a^\#_m (f,n)$ in (\ref{4.13}).
\end{pf}

\begin{lemm} \label{lem4.4}
$\quad$ (a) For any $f \in \ch_0$, $m,n \in \N$, $z=t+is \in \C$
and $B\in \cM_0$, there exist constants $M_5 (B,f,s)$ and
$M_6(B,f,s)$ such that the bounds
\begin{eqnarray*}
\| \sigma_z(a^\#_m (f,n) )B\xi_0 \| &\le& M_5 (B,f,s), \\
\| \sigma_z(a^\# (f)) B\xi_0 \| &\le& M_6 (B,f,s)
\end{eqnarray*}
hold $($uniformly on $m,n \in \N).$

 (b) The equality $$
\sigma_z(a^\#(f)) B \xi_0 = \lim_{n \to \infty} \sigma_z( a^\#_n (f,n)) B\xi_0
$$
holds for any $f \in \ch_0$, $z\in \C$ and $B \in \cM_0$.
\end{lemm}

\begin{pf} (a) As in the proof of Lemma \ref{lem4.1} (c), one has that for any $m,n \in \N$, $B \in \cM_0$ and $ z \in \C$,
\begin{equation} \label{4.14}
\sigma_z(a^\#_m (f,n)) B\xi_0 = j(\sigma_{-i/2} (B^*)) \sigma_z(a^\#_m (f,n))\xi_0.
\end{equation}
Thus the bounds follow from Corollary \ref{cor4.1} (a), Lemma \ref{lem4.1} (c) and the above relations.

(b) This follows from Corollary \ref{cor4.1} (b), (\ref{4.14})
and Lemma \ref{lem4.1} (c).
\end{pf}

In order to show the closability of the form $(\cE^{(n)},\cM_0\xi_0)$, $ n \in \N$, in Proposition \ref{prop3.2}, we will
      use the proposition listed below. The proposition is probably well known to
      the experts.

\begin{prop} \label{prop4.1}
Let $(\Omega,\mu)$ be a probability space, $\cH$ a
        separable Hilbert space and $D$ a dense subset in $\cH.$
        For any $t\in\Omega$, let $A(t)$ be an operator defined on $D$ satisfying
         the following properties:

(a) $(A(t),D)$ is closable for each $t\in\Omega$,

(b) for any $\xi\in D,\, \Omega \ni t\mapsto\|A(t)\xi\|^2 $
         is an integrable function on $\Omega.$

\noindent Then the quadratic form defined by
         \begin{eqnarray*}
             & &\quad D(\cE)=D,\\
             & &\cE[\xi]=\int_{\Omega} \|A(t)\xi\|^2
             d\mu(t), \quad \xi \in D,
          \end{eqnarray*}
        is closable.
\end{prop}

\begin{pf}
Let $\{\xi_n\}$ be a sequence in $D$ such that $\xi_n \rightarrow 0$           as $n\rightarrow\infty$ and  $\cE[\xi_n -\xi_m ]\rightarrow 0$
          as $n,m\,\rightarrow \infty$. One has to show that $\cE[\xi_n]\rightarrow
          0$ as $n\rightarrow\infty.$ Since $\{\xi_n \}$ is a
          $\cE$-Cauchy sequence, one can choose a subsequence $\{\xi_{n_k}\}$ of
          $\{\xi_n \}$ such that $\cE[\xi_{n_{k+1}}-\xi_{n_k}]<\frac{1}{2^{2k}}$
           for any $k\in\mathbb{N}$, which implies that
           $$
\sum_{k=1}^{\infty}\cE[\xi_{n_{k+1}}-\xi_{n_k}]^{1/2}<\infty.
$$
 It follows from the Schwarz inequality and the above
           bound that
           $$
\sum_{k=1}^{\infty}\int_{\Omega}\|A(t)(\xi_{n_{k+1}}-\xi_{n_k})\|
           \,d \mu(t)<\infty.
$$
The monotone convergence theorem and the above bound imply that
    $$
\sum_{k=1}^{\infty}\|A(t)(\xi_{n_{k+1}}-\xi_{n_k})\|
          <\infty, \quad\mu-a.e..
$$
          This implies that $\{A(t)\xi_{n_k}\}$ is a Cauchy sequence $\mu-a.e..$
           Since $A(t)$ is closable on $D$ for any $t\in\Omega$,
           we conclude that
  $$
A(t)\xi_{n_k}\rightarrow 0 \quad as\,k\rightarrow \infty, \mu-a.e..
$$
Using Fatou's lemma and the above result, we obtain that
            \begin{eqnarray*}
     \cE[\xi_n ]&=&\int \underline{\lim}_{k \rightarrow\infty}
                \|A(t)(\xi_n -\xi_{n_k}) \|^2 \,d\mu(t)\\
            &\le& \underline{\lim}_{k \rightarrow\infty}\cE[\xi_{n}-\xi_{n_k}],
            \end{eqnarray*}
            which implies that $\cE[\xi_n]\rightarrow 0$ as $n\rightarrow\infty.$
            This proved the proposition.
\end{pf}

We write that
\begin{equation}\label{4.15}
    \delta(a^\#(f)):=a^\#(f)-j(\sigma_{-i/2}((a^\#(f))^*))
\end{equation}
for $f \in \ch_0$.
By Lemma \ref{lem4.1}, $\delta(a^\# (f))$ is well defined on
$\mathcal{W}\xi_0$ and also on $\mathcal{M}_0\xi_0$ for any $f \in \ch_0$. Notice that for any $B\in\cM_0$, $j(\sigma_{-i/2}(B^*)) \xi_0 =B\xi_0$. Thus it follows from Lemma \ref{lem4.1} (b) and Corollary \ref{cor4.1} (b) that for any $f,h \in \ch_0$
\begin{equation}\label{4.16}
\delta(a^\# (f))W(h)\xi_0 =[a^\#(f),W(h)]\xi_0 .
\end{equation}
Since
$(\sigma_{-i/4}(a^\#(f)))^*=\sigma_{i/4}((a^\#(f))^*)$ by (\ref{3.8}),
we have that
\begin{equation}\label{4.17}
\delta(\sigma_{-i/4}(a^\#
(f)))=\sigma_{-i/4}(a^\#(f))-j(\sigma_{-i/4}((a^\#(f))^*))
\end{equation}
for any $f\in\ch_0$.


We are ready to prove Proposition \ref{prop3.1}, Proposition \ref{prop3.2} and Theorem \ref{thm3.1}. We first
produce the proof of Proposition \ref{prop3.1}.

\vspace{0.2cm} \noindent{\bf Proof of
Proposition\ref{prop3.1}.}\quad Let us first consider $(\cE, \cW
\xi_0)$. Recall the definition of $\cE(\eta,\xi)$ in
(\ref{3.9})-(\ref{3.10}). Let $f$ be a normalized admissible
function and let $\{g_n \}\subset\ch_0$ be a CONS for $\ch$. We
first note that by (\ref{4.16}) and (\ref{4.4})
\begin{eqnarray*}\delta(\sigma_{t-\frac{i}{4}}(a(g_n )))W(h)\xi_0
                 &=&\delta(a(A^{it-\frac{1}{4}}g_n ))W(h)\xi_0 \\
                 &=&\frac{i}{\sqrt2}( A^{it-\frac14}g_n
                 ,h) W(h)\xi_0 ,\\
\delta(\sigma_{t-\frac{i}{4}}(a^*(g_n )))W(h)\xi_0
                &=&-\frac{i}{\sqrt2} ( h,A^{it+\frac14}g_n ) W(h)\xi_0 .
\end{eqnarray*}
It follows from (\ref{3.9}), the above relations, the dominated
convergence theorem and the Parserval relations that for
$g,\,h\in\ch_0$
\begin{eqnarray}
&&\cE(W(g)\xi_0 ,W(h)\xi_0 )\nonumber\\
 &=&\Big\{\frac12
\sum^\infty_{n=1} \int ( A^{-it-\frac{1}{4}}g, g_n)  ( g_n,
A^{-it-\frac{1}{4}}h) f(t)\,dt\Big\}\langle W(g)\xi_0 ,W(h)\xi_0\rangle\nonumber\\
&& + \Big\{\frac12 \sum^\infty_{n=1} \int ( A^{-it+\frac{1}{4}}h, g_n) ( g_n, A^{-it+\frac{1}{4}}g) f(t)\,dt \Big\}\langle W(g)\xi_0
,W(h)\xi_0\rangle\nonumber\\
\label{4.21}
&=&\frac{1}{2}\{ ( g,A^{-\frac12}h )+ ( h,A^{\frac{1}{2}}g )
\}\langle W(g)\xi_0 ,W(h)\xi_0\rangle.
\end{eqnarray}
Here we have used the fact that by the Schwarz inequality and the
Bessel inequality
$$
\left|\sum_{n=1}^{m} ( A^{-it}h_1 ,g_n ) ( g_n
,A^{-it}h_2 ) \right|\le \|h_1 \|\|h_2 \|
$$
for any
$m\in\mathbb{N},\,t\in\mathbb{R}$ and $h_1 ,\,h_2 \in\ch_0.$ Thus
$\cE$ is well defined on $\mathcal{W}\xi_0$ and independent of the
normalized admissible function $f$ and  the CONS $\{g_n\}$ chosen.

In order to show that the closability of $(\cE, \cW \xi_0)$, we
introduce the operator $(H, \cW\xi_0)$ defined by
\begin{eqnarray}
  D(H)&=&\mathcal{W}\xi_0,\nonumber\\
\label{4.18}
 H W(h)\xi_0 &=&\frac{i}{\sqrt
 2}(\delta(a(A^{-\frac{1}{2}}h)))^*W(h)\xi_0 \\
 &-&\frac{i}{\sqrt
 2}(\delta(a^*(A^{\frac{1}{2}}h)))^*W(h)\xi_0 .\nonumber
\end{eqnarray}
By (\ref{4.15}) and Lemma \ref{lem4.1} (b), $H$ is well-defined on $\cW \xi_0$. It follows from (\ref{4.16}) and (\ref{4.4}) that for any $g, h \in \ch_0$
\begin{eqnarray}
&& \langle W(g)\xi_0 ,HW(h)\xi_0 \rangle\nonumber \\
                 &&=\frac{i}{\sqrt2}
                 \langle\delta(a(A^{-\frac12}h))W(g)\xi_0 ,W(h)\xi_0 \rangle\nonumber\\
                 &&-\frac{i}{\sqrt2}
                 \langle\delta(a^*(A^{\frac12}h))W(g)\xi_0 ,W(h)\xi_0 \rangle\nonumber\\
\label{4.20}
               &&=\frac{1}{2}\{ ( g,A^{-\frac12}h )+ ( h,A^{\frac12}g) \}\langle W(g)\xi_0,W(h)\xi_0 \rangle.
\end{eqnarray}
By the method used in the above, we get that for any
$g,h\in\ch_0$,\,\, $\langle H W(g)\xi_0 ,W(h)\xi_0 \rangle$ is
equal to the right hand side of (\ref{4.20}). Thus $H$ is
symmetric on $\mathcal{W}\xi_0$. It follows from (\ref{4.21}) and
(\ref{4.20}) that
$$
\langle \eta ,H \xi \rangle = \cE( \eta , \xi)
$$
for $\eta, \xi \in \cW \xi_0$. Since $(H, \cW\xi_0)$ is a positive symmetric operator, the form $(\cE, \cW \xi_0)$ is closable (Theorem X.23 of [RS]).

Next, we consider $(\cE, \cW_0 \xi_0)$. Employing the method similar to that used to derive (\ref{4.21}), one can check that $(\cE, \cW_0 \xi_0)$ is independent of admissible function $f$ and the CONS $\{g_n\} \subset \ch_0$ chosen. To prove the closability, one may introduce the Dirichlet operator $(H, \cW_0 \xi_0)$ similar to that in (\ref{4.18}) and then use the argument in the below of (\ref{4.18}). We leave the details to the reader.  $\quad \square $

\begin{rema}\label{rem4.1}
For any CONS $\{ g_n \} \subset \ch_0$, the relation
\begin{eqnarray} \label{4.15-1}
\cE(\eta, \xi) & = & \sum_{n=1}^\infty \langle \delta (\sigma_{-i/4} (a(g_n))) \eta, \delta (\sigma_{-i/4} (a(g_n))) \xi \rangle \\
\nonumber && + \sum_{n=1}^\infty \langle \delta (\sigma_{-i/4} (a^*(g_n))) \eta, \delta (\sigma_{-i/4} (a^*(g_n))) \xi \rangle
\end{eqnarray}
holds for any $\eta, \xi \in \cW \xi_0$. In fact, the relation follows from the method used to derive (\ref{4.21}).
\end{rema}

In order to show Proposition \ref{prop3.2}, we introduce the following forms: For given
normalized admissible function $f$ and $g\in\ch_0$, let $(\widetilde{\cE}, \cM_0 \xi_0)$ be  a
sesquilinear form defined by
\begin{eqnarray}
&&\quad D(\widetilde{\cE})=\cM_0 \xi_0,\nonumber \\
\label{4.23}
          &&\widetilde{\cE}(\eta,\xi)=\int\langle \delta(\sigma_{t-\frac{i}{4}}(a(g)))\eta,
          \delta(\sigma_{t-\frac{i}{4}}(a(g)))\xi\rangle f(t)dt \\
          &&\quad\quad+\int\langle \delta(\sigma_{t-\frac{i}{4}}(a^*(g)))\eta,
          \delta(\sigma_{t-\frac{i}{4}}(a^*(g)))\xi\rangle f(t)dt\nonumber
\end{eqnarray}
where for any $h\in\ch_0,$ $\delta(a^\# (h))$ has been defined in
(\ref{4.15}). For any $n\in\mathbb{N}$, let $(\widetilde{\cE}_n ,\cH)$ be
the form given by
\begin{eqnarray}
\label{4.24}
          \widetilde{\cE}_n (\eta,\xi)&=&\int\langle \delta(\sigma_{t-\frac{i}{4}}(a_n(g,n)))\eta,
          \delta(\sigma_{t-\frac{i}{4}}(a_n(g,n)))\xi\rangle f(t)dt \\
          &&+\int\langle \delta(\sigma_{t-\frac{i}{4}}(a_n^*(g,n)))\eta,
          \delta(\sigma_{t-\frac{i}{4}}(a_n^*(g,n)))\xi\rangle f(t)dt\nonumber
\end{eqnarray}
where for $n\in\mathbb{N}$ and $g\in\ch_0$, $a^\#_n(g,n)$ has
been defined in (\ref{4.13}). Since $a^\#_n(g,n)\in\cM_0$, it follows from Theorem \ref{thm2.1} that $(\widetilde{\cE}_n , \cH)$ is a Dirichlet form for each $n \in \N$.

\begin{lemm}\label{lem4.5}
For any $\eta,\,\xi\in\cM_0 \xi_0$,
$$
\widetilde{\cE}(\eta,\xi)=\lim_{n\rightarrow\infty}
                              \widetilde{\cE}_n(\eta,\xi).
 $$
\end{lemm}

\begin{pf}
 The lemma follows from Lemma \ref{lem4.4} and the dominated convergence theorem.
\end{pf}

\begin{prop} \label{prop4.2}
The form $(\widetilde{\mathcal{E}}, \cM_0 \xi_0 ) $ defined in (\ref{4.23}) is closable.
            Denote by $(\widetilde{\mathcal{E}},D(\widetilde{\mathcal{E}}))$ the closure
            of $(\widetilde{\cE},\cM_0 \xi_0 )$ and by $\widetilde{H}$ the positive self
            -adjoint operator associated to $(\widetilde{\mathcal{E}},D(\widetilde{\mathcal{E}}))$.
            Then the following properties hold :
          \begin{enumerate}
             \item[(a)] $\xi_0 \in D(\widetilde{H})$ and $\widetilde{H}\xi_0 =0$,
              \item[(b)]  $ \widetilde{\mathcal{E}}$ is $J$-real,
              \item[(c)]  $ \widetilde{\mathcal{E}}(\xi_+ , \xi_{-} )\leq 0$ for any $\xi \in D(\widetilde{\cE})^J$.
            \end{enumerate}
 Furthermore the form $(\widetilde{\mathcal{E}}, D( \widetilde{\mathcal{E}}))$ is a Dirichlet form.
 \end{prop}

\vspace{0.2cm} \noindent {\bf Proof of Proposition \ref{prop3.2}.}
$\quad$ By setting $g=g_n$, $n \in \N$, the proposition follows
from Proposition \ref{prop4.2}. $\quad \square$

\vspace{0.3cm} \noindent {\bf Proof of Proposition \ref{prop4.2}.}
$\quad$ The closability of $(\widetilde{\cE}, \cM_0\xi_0)$ follows
from Lemma \ref{lem4.1} (c), Lemma \ref{lem4.4} and Proposition
\ref{prop4.1}.

(a)  It follows from (\ref{4.23}) that for $m,n \in \N$ and $h_1, h_2 \in \ch_0$
$$
\widetilde{\cE}(W_m (h_1) \xi_0, W_n (h_2) \xi_0) = \langle W_m (h_1) \xi_0 , \widetilde{H} W_n(h_2) \xi_0\rangle,
$$
where
\begin{eqnarray*}
\widetilde{H}& = & \int \big(\delta (\sigma_{t-i/4}(a(g)))\big)^* \delta ( \sigma_{t-i/4}(a(g))) \, f(t)\,dt \\
&& + \int \big(\delta (\sigma_{t-i/4}(a^*(g)))\big)^* \delta ( \sigma_{t-i/4}(a^*(g))) \, f(t)\,dt.
\end{eqnarray*}
Using (\ref{4.16}) and Lemma \ref{lem4.4} (a), it is not hard to show that $\widetilde{H}W_n (h_2) \xi_0$ is defined as a vector in $\cH$. Since $\cW_0 \xi_0 \subset \cM_0 \xi_0$, we conclude that $\cW_0 \xi_0 \subset D(\widetilde{H})$. By (\ref{4.16}), we get that
$$
\delta(\sigma_{t-\frac{i}{4}}(a^\#(g)))\xi_0 =0
$$
for any $t\in\mathbb{R}$ and $g\in\ch_0$, and so $\widetilde{H}\xi_0 =0$.

(b) This follows from the method in the proof of the property (b) in Theorem
              \ref{thm2.1} and the fact that $\cM_0 \xi_0$ is a form core.

(c) The proof of the property (c) is the hardest part in the proof.
              We have to employ several limiting processes. We assert that
              \begin{equation} \label{4.25}
              \xi \in \mathcal{M}_0 \xi_0 \cap \mathcal{H}^J
             \Rightarrow \xi_+,\xi_{-}
              \in D(\tilde{\mathcal{E}})\quad and \quad\tilde{\mathcal{E}}[\xi_\pm]=
               \lim_{m\rightarrow\infty}\tilde{\cE}_m [\xi_\pm].
             \end{equation}
Let us prove our assertion. Let $s_{+}$ and $s_{-} $ be the projections
               onto  the closure of $\mathcal{M}'\xi_+$ and $\mathcal{M}'\xi_- $, where $s_+ ,\,s_- \in\cM.$
               See \cite{Ara}. For $\xi = A \xi_0 ,\,A\in\mathcal{M}_0
               $, we write that
  $$
\xi_{n,\pm}=(s_\pm A)_n \xi_0
$$
               where
 $$
(s_{\pm}A)_n =\sqrt{\frac{n}{\pi}}\int \sigma_t
               (s_{\pm}A)e^{-nt^2}\,dt.
$$
Notice that $\| (s_{\pm} A)_n \| \le \|A\|$, $n \in \N$, and that for any $\eta \in \cH$, $(s_{\pm} A)_n \eta \to s_{\pm} A\eta$ as $ n \to \infty$. See, i.e., the proof of Proposition 2.5.22 of \cite{BR}.
Since $\Delta^{it}\mathcal{P} \subset \mathcal{P}$ for
               $t\in\mathbb{R}$,
                $\xi_{n,\pm}=(s_\pm A)_n \xi_0 =j((s_\pm A)_n
                )\xi_0$, and so
$$
\sigma_z(a^\# (g))\xi_{n,\pm}=j((s_{\pm}A)_n )
                \sigma_z (a^\#(g))\xi_0
$$
                for any $z\in\mathbb{C}$. Since $\sigma_z
                (a^\#(g))$ is a closed operator for any
                $g\in\ch_0,\,z\in\mathbb{C}$,
             we take $n$ to infinity to conclude that $\xi_\pm \in D(\sigma_z (a^\# (g)))$,
                 and
$$
\sigma_z (a^\# (g))\xi_\pm =j(s_\pm A)\sigma_z (a^\#
                 (g))\xi_0 ,
$$
                 and so
\begin{eqnarray}
\label{4.27}
                &&\delta(\sigma_z (a^\# (g)))\xi_\pm\\
                &=&j(s_\pm A)\sigma_z (a^\# (g))\xi_0 -(s_\pm A)j(\sigma_{-\frac{i}{2}}
                 (\sigma_z (a^\#(g)))^* )\xi_0 .\nonumber
                 \end{eqnarray}
 Thus, it follows from Corollary \ref{cor4.1} (a) and the dominated convergence theorem that $ \widetilde{\cE} [\xi_{m, \pm} -\xi _{n, \pm} ] \to 0$ as $ m,n \to \infty$. Since $(\widetilde{\cE}, D(\widetilde{\cE})) $ is closed, $\xi_\pm \in D(\widetilde{\cE})$.
 Next we will prove
               that $\widetilde{\cE}_n [\xi_\pm ]$ converges to $\widetilde{\cE}[\xi_\pm ]$
               as $n$ tends to infinity. Using (\ref{4.27}), the  analogous relation for $\sigma_z (a^\#_n(g,n))$,
               and Corollary \ref{cor4.1} (a), we conclude that for  $\xi=A\xi_0 $,
               there exists a constant $M>0$ such that the bounds
$$
\|\delta(\sigma_{t-\frac{i}{4}}(a^\#_n(g,n)))\xi_\pm
               \|+\|\delta(\sigma_{t-\frac{i}{4}}(a^\#(g)))\xi_\pm  \|\le M
$$
               hold uniformly in $n\in\mathbb{N}$ and
               $t\in\mathbb{R}$.
Using the above bounds, (\ref{4.27}) and Corollary \ref{cor4.1},
               we get that
               \begin{eqnarray*}
&&|\widetilde{\cE}_n [\xi_\pm ]-\widetilde{\cE}[\xi_\pm
               ]|\\
&&\le 2M\int \|\{\delta(\sigma_{t-\frac{i}{4}}(a^\#_n(g,n)))-
               \delta(\sigma_{t-\frac{i}{4}}(a^\#(g)))\}\xi_\pm
               \|f(t)dt\\
 &&\le M'\int \|\sigma_{t-\frac{i}{4}}(a^\#_n(g,n))\xi_0 -
         \sigma_{t-\frac{i}{4}}(a^\#(g))\xi_0 \|f(t)dt\\
               &&\rightarrow 0\,\,\,\text{as}\,\,\,n\rightarrow\infty.
               \end{eqnarray*}
Here we have used Corollary \ref{cor4.1} (b) and the dominated convergence theorem to obtain
     the last conclusion in the above. This completes the proof of  our assertion.

We turn to the proof of the property (c). A direct computation shows that
     \begin{equation}\label{4.28}
     \|\xi_\pm -\eta_\pm \|\le \|\xi-\eta\|,\quad
     \forall\xi,\,\eta\in\cH^J .
     \end{equation}
     See also the proof of Proposition 1.2 of \cite{DL}. Since $|\xi|=\xi_+ +\xi_- ,$ the assertion
     (\ref{4.25}) implies that
      \begin{equation}\label{4.29}
     \xi\in \cM_0 \xi_0 \cap \cH^J \Rightarrow |\xi|\in D(\widetilde{\cE})\quad
     \text{and}\quad\widetilde{\cE}[|\xi|]=\lim_{n\rightarrow\infty}\widetilde{\cE}_n [|\xi|].
     \end{equation}
Let $\xi\in D(\widetilde{\cE})\cap\cH^J$ be given. Choose
   $\{\xi_k\}$ in $\cM_0 \xi_0 \cap \cH^J$ such that $\xi_k
   \rightarrow\xi$ and $\widetilde{\cE}[\xi_k
   ]\rightarrow\widetilde{\cE}[\xi]$ as $k\rightarrow\infty$. By
   (\ref{4.28}), $|\xi_k |\rightarrow|\xi|$ as $k\rightarrow\infty$.
   Notice that $\widetilde{\cE}[|\xi|]\le\widetilde{\cE}[\xi]$ is equivalent to
   $\widetilde{\cE}(\xi_+ ,\xi_- )\le 0$ and that each $\widetilde{\cE}_m ,\,
   m\in\mathbb{N},$ satisfies the property (c) by Theorem \ref{thm2.1}.
By the lower
   semi-continuity of $\widetilde{\cE}$ and (\ref{4.29}),
 we obtain that
   \begin{eqnarray*}
     \widetilde{\cE}[|\xi|]&\le&\underline{\lim}_{k\rightarrow\infty}\widetilde{\cE}[|\xi_k |]\\
          &=&\underline{\lim}_{k\rightarrow\infty}(\lim_{n\rightarrow\infty}
          \widetilde{\cE}_n[|\xi_k |])\\
          &\le&\underline{\lim}_{k\rightarrow\infty}(\lim_{n\rightarrow\infty}
          \widetilde{\cE}_n[\xi_k ])\\
          &=&\underline{\lim}_{k\rightarrow\infty}\widetilde{\cE}[\xi_k ]\\
          &=&\widetilde{\cE}[\xi].
   \end{eqnarray*}
Thus $|\xi| \in D(\widetilde{\cE})$ and $\widetilde{\cE}[|\xi|] \le \widetilde{\cE}[\xi]$.   This completes the proof of the property (c).

 Since $\widetilde{\cE}(\xi, \xi_0)=0$ for any $\xi \in D(\widetilde{\cE})$, the properties (b) and (c) imply that $\widetilde{\cE}$ is a Dirichlet form. $\quad \square$

\vspace{0.2cm} \noindent{\bf Proof of Theorem \ref{thm3.1}.} By
Proposition \ref{prop3.1}, $\cW_0 \xi_0 \subset
D(\overline{\cE}).$
     Since each component $\overline{\cE}_{n}$ is a Dirichlet form, it follows
     from Theorem 5.2 of \cite{Cip} that $(\overline{\cE},D(\overline{\cE}))$ is
      a Dirichlet form. $\quad \square$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decomposition of Quasi-free Hilbert Space : Ergodicity}

  For given quasi-free state $\omega$, we will decompose the Hilbert space
$\cH=\cH_{\omega}$ into direct sum of $\cH^{(m,n)}, m,n\in
\mathbb{N} \cup \{0\},$ where $\cH^{(m,n)}$ is the Hilbert space of
$m$ quasi-particles and $n$ anti quasi-particles. We then use the
result to show that the symmetric Markovian semigroup is ergodic.

Recall the definition of $\delta(a^\# (g)), g\in \ch_0$ in
(\ref{4.15}). Denote by $B$  the operator given by
\begin{equation}\label{5.1}
  B=A^{-\frac12}-A^{\frac12}.
\end{equation}
It follows from (\ref{4.15}) that for $g\in\ch_0$
\begin{eqnarray}\label{5.2}
\delta(a(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g))&=&
          a(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)-j(\sigma_{-i/2}(a^*(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)))\\
          &=&a(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)-j(a^*(B^{-\frac{1}{2}}A^{\frac{1}{4}}g)),\nonumber
\end{eqnarray}
\begin{eqnarray*}\delta(a^* (B^{-\frac{1}{2}}A^{\frac{1}{4}}g))
                        &=&a^* ( B^{-\frac{1}{2}}A^{\frac{1}{4}}g)
                       -j(\sigma_{-i/2}(a(B^{-\frac{1}{2}}A^{\frac{1}{4}}g)))
                        \\ &=&a^* ( B^{-\frac{1}{2}}A^{\frac{1}{4}}g)
                       -j(a(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)).
\end{eqnarray*}
The above operators are well defined on $\cM_0 \xi_0$ and also on
$\cH_{fin} .$ Since $(j(a(g)))^* =j(a^* (g)),$ etc, a computation
shows that
\begin{eqnarray}
(\delta(a(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)))^*
              &=&a^*(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)-j(a(B^{-\frac12}A^{\frac14}g))
              \nonumber\\&=& a^*(B^{-\frac{1}{2}}A^{-\frac{1}{4}}g)-j(\sigma_{-\frac{i}{2}}(a(B^{-\frac12}A^{\frac34}g)))
              \nonumber\\
\label{5.3}
&=&a^*(B^{\frac{1}{2}}A^{\frac{1}{4}}g)+\delta(a^*
              (B^{-\frac12}A^{\frac34}g)).
\end{eqnarray}
Here we have used the fact that
$B^{-\frac12}(A^{-\frac14}-A^{\frac34})=B^{\frac12}A^{\frac14}.$
Using the method similar to that used in the above, we get that
\begin{equation}\label{5.4}
(\delta(a^*(B^{-\frac{1}{2}}A^{\frac{1}{4}}g)))^*=-a(B^{\frac{1}{2}}A^{-\frac{1}{4}}g)
                       +\delta((a(B^{-\frac{1}{2}}A^{-\frac{3}{4}}g)).
\end{equation}
From notational brevity, we write that for $g\in\ch_0$
            \begin{eqnarray}\label{5.5}
                   D_1 (g)&:=& \delta(a(B^{-\frac12}A^{-\frac14}g)),\\
                   D_2 (g)&:=&
                   \delta(a^*(B^{-\frac12}A^{\frac14}g)).\nonumber
            \end{eqnarray}
   Then it follows from (\ref{5.3}) and (\ref{5.4}) that
              \begin{eqnarray} \label{5.6}
                   D_1 (g)^* &=& a^* (B^{\frac12}A^{\frac14}g)+D_2 (A^{\frac12}g),\\
                   D_2 (g)^* &=& -a(B^{\frac12}A^{-\frac14}g)+D_1 (A^{-\frac12}g).\nonumber
            \end{eqnarray}

We first collect some properties of $D_i (g)$ for $g\in\ch_0$ and
  $i=1,\,2.$

\begin{lemm}\label{lem5.1}
         $D_i (g)\xi_0 =0$ for any $g\in\ch_0$ and $i=1,\,2$.
\end{lemm}
\begin{pf}
This follows from (\ref{5.5}) and (\ref{4.16}).
\end{pf}

\begin{lemm} \label{lem5.2}
As operators defined on $\mathcal{W}\xi_0$ and $\mathcal{W}_0
\xi_0$, the following relations hold for any $g,\,h\in\ch_0$ :
\begin{enumerate}
\item[(a)] $[D_i (g),D_j (h)] =0,\,\,i=1,2,\,j=1,2$
\item[(b)] $[D_1 (g),a(h)] =0,$
\item[(c)] $[D_2 (g),a^*(h)] =0,$
\item[(d)] $[D_1 (g),a^*(B^{\frac{1}{2}}A^{\frac{1}{4}}h)]
                = ( g,h )\1,$
\item[(e)] $[D_2 (g),a(B^{\frac{1}{2}}A^{-\frac{1}{4}}h)]
        =- ( h,g ) \1.$
\end{enumerate}
\end{lemm}

\begin{pf}
          We first remark that each operator in the commutators in the
          lemma is defined on $\cW\xi_0$ and also $\cW_0 \xi_0 .$
   (a) This follows from (\ref{4.16}), (\ref{4.4}) and the definition
          of $D_i (g),\,i=1,2,$ in (\ref{5.5}). (b) and (c) follow from (\ref{5.5}) and (\ref{4.15}). (d) and (e) follow from
   (\ref{5.5}), (\ref{4.15}) and the CCRs in (\ref{3.5}).
          \end{pf}

\begin{prop} \label{prop5.1}
As operators defined on $\mathcal{W}\xi_0 ,\,\mathcal{W}\xi_0 $
and also on $\cH_{fin},$ the following canonical commutation
relations (CCRs) hold for any $g,\, h\in\ch_0$:
 \begin{enumerate}
\item[(a)] $[D_1 (g),D_1 (h)^*] = (g,h ) \1$,\\
            $[D_1 (g),D_1 (h)]=0,\quad [D_1 (g)^* , D_2 (h)^* ]=0,$
\item[(b)] $[D_2 (g),D_2 (h)^*] =( h,g) \1,$\\
           $[D_2 (g),D_2 (h)]=0,\quad[D_2 (g)^* ,D_2 (h)^* ]=0$
\item [(c)]$[D_1 (g)_, D_2 (h)]=0,\quad [D_1 (g),D_2 (h)^* ]=0$,\\
           $[D_1 (g)^*,D_2 (h) ]=0,\quad [D_1 (g)^* ,D_2 (h)^* ]=0.$
\end{enumerate}
\end{prop}

\begin{pf} The commutation relations on $\mathcal{W}\xi_0$and $\mathcal{W}_0\xi_0$ in the
proposition follow from Lemma \ref{lem5.2} and (\ref{5.6}). Thus we need to
extend the relations to $\cH_{fin}$. Recall the definition of
$\Phi (f,n),\,f\in\ch_0,n\in\mathbb{N},$ in (\ref{4.5}). Using Lemma \ref{lem4.2} and  (\ref{3.2}), it is not hard to check that for any
$g,h,f_j ,\in\ch_0$, $n_j \in \N$, $j=1,...,m,$
\begin{equation}\label{5.7-1}
  \Phi(g)\Phi(h)(\prod_{j=1}^m \Phi(f_j ,n_j ))\xi_0 \rightarrow
  \Phi(g)\Phi(h)(\prod_{j=1}^m \Phi(f_j ))\xi_0
\end{equation}
as $n_j \rightarrow \infty,\,j=1,...,m.$ This implies that the
relations in the proposition extend to $\cH_{fin}.$
 \end{pf}

We are ready to decompose the Hilbert space $\cH=\cH_{\omega} ,$
 called quasi-free Hilbert space. According to Lemma \ref{lem5.1} and the
 CCRs in Proposition \ref{prop5.1}, $D_i (g)$ and $D_i (h)^* ,\,g,h\in\ch_0,\,i=1,2,$ can
 be thought as annihilation and creation operators respectively.
 We remark that $h\mapsto D_1  (h)^*$ is linear, but $g\rightarrow D_2
 (g)^*$ is conjugate linear. With an abuse of terminology, we call $D_1 (h)^*$
 and $D_2 (h)^*$ the creation operators for quasi-particles and anti
 quasi-particles respectively for $h\in\ch_0$. The following is the
 decomposition of $\cH$:

\begin{thm}\label{thm5.1}
 The following decomposition holds:
            $$\cH =\bigoplus_{m,n=0}^{\infty}\cH^{(m,n)}$$
   where for each $m,n\in\mathbb{N}\cup \{0\},$ $\cH^{(m,n)}$
    is the closure of the subspace spanned by the vectors of the form
    $$
(\prod_{j=1}^{m} D_1 (g_j )^*)(\prod_{l=1}^{n}
    D_2 (h_l )^* )\xi_0 ,\,g_j ,\,h_l \in\ch_0.
$$
    In the case in which $m=0$ $(n=0),$ we replace the operator in
    the first(second) parenthesis in the above by the identity.
 \end{thm}

 \begin{pf}
  It follows from (\ref{5.6}) that any $\Phi(g),\,g\in\ch_0$, can be written
  as sum of four $D_i (h)^\# ,h\in\ch_0,\,i=1,2 .$ Thus any
  $(\prod_{l=1}^m \Phi(g_l ))\xi_0 , \,g_l \in\ch_0,\,l=1,\cdots,m$, can
  be expressed as a finite linear combination of the vectors of the form
  $$
(\prod_{j=1}^p D_1 (g_j' )^\# )
  (\prod_{l=1}^q D_2 (h_l' )^\# )\xi_0 ,
\,g_j' ,\,h_l' \in \ch_0,
$$
where $D_i^\# (g)$ is either $D_i(g)$ or $D_i^*(g)$, $i=1,2$.
  As a consequence of Lemma \ref{lem5.1} and the CCRs in Proposition \ref{prop5.1},
  the above vector can be expressed as a finite linear combination
  of the vector of the form
    $$
 (\prod_{j=1}^{m'} D_1 (g_j' )^* )
  (\prod_{l=1}^{n'} D_2 (h_l' )^* )\xi_0 ,\,g_j' ,\,h_l' \in \ch_0,\,
  m',n'\in\mathbb{N}\cup\{0\}.
$$
  Since $\cH_{fin}$ is dense in $\cH$, we conclude that the set of
  finite linear combinations of the vectors of the above form is
  dense in $\cH$. Thus the decomposition follows from Lemma \ref{lem5.1}  and the CCRs in Proposition \ref{prop5.1}.
  \end{pf}

Recall that $\ch_0$ is a dense subspace of a complex Hilbert space $\ch.$
Let $\mathcal{F}=\mathcal{F}(\ch)$ be the symmetric Fock space
over $\ch$, and $a(g)$ and $a^* (g),\,g\in\ch_0,$ the annihilation
and creation operator respectively. Denote by $\Omega$ the vacuum
vector in $\mathcal{F}.$ Let $C : \ch\rightarrow\ch$ be an
anti-unitary operator. If $\ch$ is a $L^2$-space, one may consider
that $C$ is the complex conjugation. Denote by $\Gamma(C)$ the
second quantization of $C.$ See Section 5.2.1 of \cite{BR}. Let
$\mathcal{F}_1,\,\Omega_1 ,\,a_1 (g) $ and $a_1^* (g) ,\,g\in\ch_0$
 be the identical copies of $\mathcal{F},\,\Omega,\,a(g)$ and $a^*
(g),\,g\in\ch_0$ respectively. Notice that $\Gamma(C)a^\#
(g)\Gamma(C)^{-1} =a^\# (Cg).$ We write that $\mathcal{F}_2
=\Gamma(C)\mathcal{F}(=\mathcal{F})$, $\Omega_2 =\Omega , \,a_2
(g)=a(Cg)$, and $a_2^* (g)=a^* (Cg),\,g\in\ch_0$. Then the
following commutation relations hold : for $g,\,h\in\ch_0,$
\begin{eqnarray} \label{5.7}
     &&[a_2 (g),a^*_2 (h)]= ( h,g ) \text{\bf 1},\\
     &&[a_2 (g),a_2 (h)]=0.\nonumber
\end{eqnarray}
One may compare the above relations to those in Proposition \ref{prop5.1} (b).

\begin{prop} \label{prop5.2}
      Let $U$ be the operator defined by
      $$
U : \cH \rightarrow\mathcal{F}_1 \otimes\mathcal{F}_2
$$
      $$(\prod_{j=1}^{m} D_1 (g_j )^* )(\prod_{l=1}^{n} D_2 (h_l )^* )\xi_0 \mapsto
      (\prod_{j=1}^{m} a^*_1 (g_j ))\Omega_1 \otimes (\prod_{l=1}^{n} a^*_2 (h_l ))\Omega_2 $$
      for $g_j ,\,h_l \in\ch_0,\,j=1,...,m,l=1,...,n.$ Then $U$ is
      unitary.
\end{prop}

\begin{pf}
Since $D^\#_1 (g)$ and $a^\#_1 (g)$, and $D^\#_2 (g)$ and
$a^\#_2 (g)$ for $g\,\in\ch_0$ satisfy the same commutation
relations respectively by Proposition \ref{prop5.1} and (\ref{5.7}), the unitarity
of $U$ follows from  the fact that $a(g)\Omega=0$ for
any $g\in\ch_0$.
\end{pf}

We next turn to the spectral analysis of $\overline{H}$, where
$\overline{H}$ is the generator of the symmetric Markovian
semigroup $\{T_t\}_{t\ge0}$ associated to the Dirichlet form
$(\overline{\cE},D(\overline{\cE})).$

Let us first describe the basic idea of the proof Theorem \ref{thm3.2}.
Recall the definitions of $D_1 (g)$ and $D_2 (g)$ in (\ref{5.5}). Let
$\{f_n \}\subset\ch_0$ be a CONS for $\ch$. By Remark \ref{rem4.1} and (\ref{5.5}), we have that
for any $g,\,h\in \ch_0$
 \begin{eqnarray*}
            &&\overline{\cE}(W(g)\xi_0 ,W(h)\xi_0 )\\
            &=&\sum_{n=1}^{\infty}\langle D_1 (B^{\frac12}f_n)W(g)\xi_0 ,D_1
            (B^{\frac12}f_n)W(h)\xi_0 \rangle \\
            &&+\sum_{n=1}^{\infty}\langle D_2 (B^{\frac12}f_n)W(g)\xi_0 ,D_2
            (B^{\frac12}f_n)W(h)\xi_0 \rangle \\
            &=&\langle W(g)\xi_0 ,H W(h)\xi_0 \rangle ,
 \end{eqnarray*}
 where
\begin{equation}\label{5.8}
  H=\sum_{n=1}^{\infty}\{D^*_1 (B^{\frac12}f_n )D_1
  (B^{\frac12}f_n )+D^*_2 (B^{\frac12}f_n )D_2
  (B^{\frac12}f_n )\}
\end{equation}
 as a bilinear form on $\mathcal{W}\xi_0 \times \mathcal{W}\xi_0 $. By Proposition
 \ref{prop5.2}, one can see that $H$ is unitary equivalent to sum of two
 second quantizations of $B$. If one can show that
 $\cH_{fin}\subset D(\overline{H}),\, \overline{H}=H$ on
   $\cH_{fin},$ and that $\cH_{fin}$ is a core for $\overline{H},$
then one expect that spectrum of  $\overline{H}$ can be
analyzed completely.

We first establish technical lemmas. Recall
that $\mathcal{W}_0 \xi_0 \subset D(\overline{\cE}).$

\begin{lemm} \label{lem5.3}
          $\mathcal{W}\xi_0 \subset D(\overline{\cE})$
\end{lemm}

\begin{pf}
    Since $(\overline{\cE},D(\overline{\cE}))$ is closed and $\mathcal{W}_0 \xi_0 \subset D(\cE),$
    it is sufficient to show that for any $g\in\ch_0$
    $$
\overline{\cE}[W_n (g)\xi_0 -W(g)\xi_0 ]\rightarrow 0 \quad \text{as}\quad n\rightarrow\infty.
$$
Notice that by Remark \ref{rem4.1}
\begin{eqnarray*}
\overline{\cE} [(W_n (g)-W(g))\xi_0] & =& \sum_{m=1}^\infty \int \| \delta(\sigma_{\tau-i/4} (a(g_m))) (W_n(g)-W(g)) \xi_0 \|^2 f(\tau) d\tau \\
&&+ \sum_{m=1}^\infty \int \| \delta(\sigma_{\tau-i/4} (a^*(g_m))) (W_n(g)-W(g)) \xi_0 \|^2 f(\tau) d\tau \\
&\equiv& A_n^{(1)} + A_n^{(2)}.
\end{eqnarray*}
Employing the method similar to that used to derive (\ref{4.21}), it is easy to show that
$$
A_n^{(1)}  = \frac n {2\pi} \int\int K(t_1, t_2, g) e^{-nt_1^2}
e^{-nt_2^2} \,dt_1 dt_2
$$
where
\begin{eqnarray*}
K(t_1,t_2,g) &=& ( A^{it_1}g, (A^{it_2} - \1) A^{-1/2} g ) \,\langle W(A^{it_1}g)\xi_0, W(A^{it_2}g) \xi_0 \rangle \\
&& + ( A^{it_1}g, A^{-1/2}g)\, \langle W(A^{it_1}g)\xi_0, (W(A^{it_2}g)-W(g))\xi_0 \rangle \\
&&+ (g, A^{-1/2}A^{it_2}g) \,\langle W(g) \xi_0, (W(g) -W(A^{it_2}g))\xi_0 \rangle \\
&&+ ( g, A^{-1/2}(\1-A^{it_2})g)\, \langle W(g) \xi_0 , W(g) \xi_0 \rangle.
\end{eqnarray*}
Notice that $K(t_1,t_2,g)$ is bounded uniformly with respect to $(t_1,t_2) \in \R^2$. Changing the variables ($t_1'=n^{1/2} t_1$, $t_2'=n^{1/2}t_2$) and using the dominated convergence theorem, we conclude that
$$
A_n^{(1)} \to 0  \quad \text{as} \quad n \to \infty.
$$
By the similar calculation, we get that
$$
A_n^{(2)} \to 0  \quad \text{as} \quad n \to \infty.
$$
This proved the lemma completely.
\end{pf}

\begin{lemm} \label{lem5.4}
(a) $\cH_{fin}\subset D(\overline{\cE}).$

(b)Let $\{g_n \}\subset \ch_0$ be a CONS for $\ch$. Then for $\xi\in\cH_{fin}$ the equality
\begin{equation}\label{5.9}
  \overline{\cE}[\xi]=\sum_{n=1}^{\infty}\{\|D_1 (B^{\frac12}g_n )\xi\|^2 +\|D_2 (B^{\frac12}g_n )\xi\|^2 \}
\end{equation}
holds.
\end{lemm}
The proof of the above lemma will be given in Appendix.

 The following is one of main results in this section:

\begin{thm} \label{thm5.2} $\quad$
(a) Let $H : \cH_{fin} \rightarrow\cH$ be the operator
defined by
\begin{eqnarray}
\label{5.15}
&&H \prod_{p=1}^m D_1 (g_p )^* \prod_{q=1}^n D_2 (h_q )^* \xi_0 \\
& =&\sum_{k=1}^m \prod_{p=1}^{k-1} D_1 (g_p )^* D_1 (Bg_k )^*
                \prod_{p=k+1}^{m} D_1 (g_p )^* \prod_{q=1}^n D_2 (h_q
                )^* \xi_0 \nonumber \\
&& +\sum_{k=1}^n \prod_{p=1}^{m} D_1 (g_p )^* \prod_{q=1}^{k-1}
D_2 (h_q )^* D_2 (Bh_k )^* \prod_{q=k+1}^{n} D_2 (h_q )^* \xi_0
\nonumber
\end{eqnarray}
for any $m,n\in\mathbb{N}\cup\{0\}$ and $g_p ,h_q \in\ch_0
,\,p=1,...,m,q=1,...,n.$ Then the relation
$$
\overline{\cE}(\eta,\xi)=\langle \eta,H\xi\rangle
$$
holds for any $\eta,\xi\in\cH_{fin} .$

(b) $H$ is  essential self-adjoint and the self-adjoint
extension denoted by $H$ again is equal to the Dirichlet operator
$\overline{H}$.
\end{thm}

\begin{pf}
$\,\,$ (a) Let $(\cE_1 ,\cH_{fin} )$ be the form given by
          $$
\cE_1 [\xi]=\sum_{k=1}^{\infty}\|D_1 (B^{\frac12}f_k
          )\xi\|^2,
$$
          where $\{f_k \},\,f_k \in\ch_0$ is a CONS for $\ch$. We write $H=H_1 +H_2$, where the image under $H_1$(resp.$H_2$) is defined by
          the first(resp. second) vector in the right hand side of (\ref{5.15}).
    The CCRs in Proposition \ref{prop5.1} (a) and Lemma \ref{lem5.1} imply that
    \begin{eqnarray*}
    &&\cE_1 [(\prod_{j=1}^m D_1 (g_j )^* )\xi_0
               ]\\ &&=\sum_{k=1}^{\infty} \sum_{p=1}^m
               \sum_{q=1}^m ( B^{\frac12}g_p ,f_k ) ( f_k , B^{\frac12}g_q ) G(g_1 ,....,g_m
               ;p,q)
         \end{eqnarray*}
where
         \begin{eqnarray*}
         &&G(g_1 ,...,g_m ;p,q)\\
          &&:=\Big\langle \prod_{\tau=1}^{p-1} D_1 (g_{\tau})^* \prod_{\tau=p+1}^m D_1 (g_{\tau} )^* \xi_0 \,,\,
         \prod_{\tau=1}^{q-1} D_1 (g_{\tau})^* \prod_{\tau=q+1}^m D_1 (g_{\tau})^* \xi_0 \Big\rangle.
         \end{eqnarray*}
Using the Parserval relations and the fact that
         \begin{eqnarray*}
         &&\sum_{p=1}^{m} ( g_p ,B g_q ) G(g_1 ,...,g_m ;p,q)\\
         &&=\Big\langle D_1 (Bg_q )\prod_{k=1}^m D_1 (g_k )^* \xi_0 \, ,\, \prod_{k=1}^{q-1}D_1 (g_q )^*
         \prod_{k=q+1 }^{m} D_1 (g_k )^* \xi_0 \Big\rangle,
        \end{eqnarray*}
  we have that
        \begin{eqnarray*}
         &&\cE_1 \left[\prod_{j=1}^{m}D_1 (g_j )^* )\xi_0
         \right]\\
&&=\sum_{q=1}^m \Big\langle \prod_{j=1}^{m}D_1 (g_j )^* \xi_0 ,
         \prod_{j=1}^{q-1} D_1 (g_j )^* D_1 (Bg_{q} )^*
          \prod_{j=q+1}^{m} D_1 (g_j )^* \xi_0 \Big\rangle \\
          &&=\Big\langle \prod_{j=1}^{m} D_1 (g_j )^* \xi_0 \, ,\,H_1 \prod_{j=1}^{m} D_1 (g_j )^*\xi_0\Big \rangle.
        \end{eqnarray*}
Notice that $H_2 =H-H_1$ commutes with $D_1 (g)^* $ for any
$g\in\ch_0$ by (\ref{5.15}). Thus by the polarization identity, we proved
that
$$
\cE_1 (\eta,\xi)=\langle \eta,H_1\xi\rangle
$$
for any
$\eta,\,\xi\in\cH_{fin}$. The method similar to that used in the above implies that
$$
\cE_2 (\eta, \xi) =\langle \eta, H_2 \xi\rangle
$$
for any $\eta, \xi \in \cH_{fin}$.This proved the part (a) of the theorem.

(b) By Proposition \ref{prop5.2}, we have that
\begin{equation}\label{5.16}
 UHU^{-1} =d\Gamma_1 (B)\otimes\text{\bf 1} +\text{\bf
 1}\otimes d\Gamma_2 (B),
\end{equation}
where each $i,\,i=1,2,$ $d\Gamma_i (B)$ is the second
quantization of $B$ on $\mathcal{F}_i .$ We remark that $d\Gamma_2
(B)$ is anti-unitary equivalent to $d\Gamma_1 (B)$. By Assumption
3.1, any $g\in\ch_0$ is an analytic vector for $B$, and so it is
easy to check that $(\prod_{j=1}^m a^* (g_j ))\Omega$ is an
analytic vector for $d\Gamma (B)$ for any $g_j
\in\ch_0,\,j=1,...,m.$ Thus it follows that any $\xi\in\cH_{fin}$
is an analytic vector for $H$. Since $H\cH_{fin} \subset
\cH_{fin}$ by (\ref{5.15}) and since $H=\overline{H}$ on $\cH_{fin}$ by the
part (a) of the theorem, it follows from Corollary 2 of Theorem X. 39
in \cite{RS} that $H$ and $\overline{H}$ are essentially
self-adjoint on $\cH_{fin}$, and  so $H=\overline{H}.$
\end{pf}

Finally we are able to produce the proof of Theorem \ref{thm3.2}.

\vspace{0.2cm} \noindent{\bf Proof of Theorem \ref{thm3.2}.}
$\quad$ (a) and (b) follow from Theorem 5.2.

To show (c), recall $0<
A\le\alpha\text{\bf 1},\, 0<\alpha<1.$ Since
$B=A^{-\frac12}-A^{\frac12},$
$$
\inf \sigma(B)\ge
\alpha^{-1/2}-\alpha^{1/2}.
$$
It follows from the above lower bound and (\ref{5.16}) that zero is a
simple eigenvalue with eigenvector $\xi_0$ and
$$
\inf(\sigma(H)-\{0\})\ge \alpha^{-1/2}-\alpha^{1/2}.
$$
Thus the Markovian semigroup $\{T_t \}_{t\ge0}$ is ergodic. This
completes the proof of Theorem \ref{thm3.2}.
$\quad \square$

\vspace{0.5cm} \noindent {\bf Acknowledgments} \\ This work was
supported in Part by Korea Research Foundation(99-005-D00010),
Korean Ministry of Education.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{Appendix : Proofs of Lemma \ref{lem4.3} and Lemma
\ref{lem5.4}} \noindent{\bf Proof of Lemma \ref{lem4.3}.} (a)  Let
us prove the first bound. Recall the definition of $\Phi(f,n)$,
$f\in \ch_0$, $n \in \N$, in (\ref{4.5}). Notice that
\begin{equation} \label{4.7}
\Phi(f,n) = \int_0^1 W( \frac {s} n f) \Phi (f) ds.
\end{equation}
It follows from (\ref{4.6}) and (\ref{4.7}) that
\begin{eqnarray*}
&&\| \sigma_z (\Phi_m (f,n)) \xi_0 \|^2  \\
&& \qquad = (\frac{m} \pi) \int \int \left( \int_0^1 \int_0^1 F(t_1, t_2, s_1, s_2 ; f,n)\,ds_1 ds_2 \right) \\
&& \qquad \quad \cdot e^{-m (t_1-\overline{z})^2} e^{-m (t_2 -z)^2 } \, dt_1 dt_2,
\end{eqnarray*}
where
\begin{eqnarray*}
&& F(t_1, t_2, s_1, s_2 ; f,n) \\
&& \qquad := \omega \Big(\sigma_{t_1}(\Phi(f)) \sigma_{t_1}
(W(- \frac {s_1} n f)) \sigma_{t_2}(W( \frac {s_2} n f))  \sigma_{t_2}(\Phi(f))\Big).
\end{eqnarray*}
We use (\ref{4.4}) and the fact that $\sigma_t(\Phi(f)) =\Phi (A^{it} f)$, etc, to obtain that
\begin{eqnarray*}
&& F(t_1, t_2, s_1, s_2 ; f,n) \\
&& \qquad = \omega \Big( \Phi(A^{it_1} f) \Phi (A^{it_2} f)
W(-\frac {s_1} n A^{it_1} f) W(\frac {s_2} n A^{it_2} f)\Big)\\
&& \qquad \quad +  \frac {s_1}n  \text{Im}\, (A^{it_1} f, A^{it_2} f)  \omega \Big( \Phi(A^{it_1} f)
W(-\frac {s_1} n A^{it_1} f) W(\frac {s_2} n A^{it_2} f)\Big).
\end{eqnarray*}
Iterating (\ref{4.2}) and then using (\ref{4.1}), (\ref{3.1}) and (\ref{3.2}), the above function can be calculated explicitely. More precisely, let us use (\ref{4.2}) iteratively  in the above  to obtain that
$$
F(t_1, t_2, s_1, s_2 ; f,n)=\sum_{k=1}^7 F^{(k)}(t_1, t_2, s_1, s_2 ; f,n),
$$
where
\begin{eqnarray*}
&& F^{(1)}(t_1, t_2, s_1, s_2 ; f,n) \\
&& \qquad = \omega \Big( \Phi(A^{it_1} f) \Phi (A^{it_2} f)\Big) \omega \Big(
W(-\frac {s_1} n A^{it_1} f) W(\frac {s_2} n A^{it_2} f)\Big),
\end{eqnarray*}
and appropriate expressions for $F^{(k)} (t_1, t_2, s_1, s_2 ; f, n) $, $k=2, \cdots , 7$. It follows from (\ref{4.1}), (\ref{3.1}) and (\ref{3.2}) that
\begin{eqnarray*}
&& F^{(1)} (t_1,t_2, s_1,s_2 ; f,n) \\
&& \qquad = \frac12 \left( ( f, (\1-A)^{-1} A^{-it_1} A^{it_2} f ) + ( f, A(\1-A)^{-1} A^{it_1} A^{-it_2} f ) \right) \\
&& \qquad \quad \cdot \exp \{ \frac{i}2 \text{Im}\, ( f, \frac {s_1} n \frac {s_2} n A^{-it_1} A^{it_2} f) \} \\
&& \qquad \quad \cdot \exp \{ -\frac14 ((\frac {s_2} n A^{it_2} - \frac {s_1} n A^{it_1})f, D(\frac {s_2} n A^{it_2} - \frac {s_1} n A^{it_1})f ) \},
\end{eqnarray*}
where $D=(\1+A) (\1-A)^{-1}$.  By Assumption 3.1, $F^{(1)}$ has an analytic extension on $\C\times \C$ in $t_1 $ and $t_2$ variables. From the above expression, it is easy to see that there exists a constant $C_1(f, $ Im $z$) independent of $n$ such that
$$
\sup _{s_1, s_2 \in [0,1]} | F^{(1)} (t_1+\overline{z}, t_2+z, s_1, s_2 ; f, n) | \le C_1(f,  \text{Im}\, z)
$$
for any $n \in \N$. Using the Cauchy integral theorem and the above bound, we conclude that the contribution of $F^{(1)}$ in $F$ is bounded by $C_1(f$, Im $z$). Now, it is  obvious that the function $F(t_1, t_2, s_1, s_2 ; f,n)$ has an analytic extension on $\C \times \C$ in $t_1$ and $t_2$ variables and there exists a constant $M_1(f,$ Im $z$) depending only on $f\in \ch_0$ and Im $z$ such that the bound similar to that in the above holds. We use the Cauchy integral theorem to conclude that
 \begin{eqnarray*}
&&\| \sigma_z (\Phi_m (f,n)) \xi_0 \|^2  \\
&& \qquad = (\frac{m} \pi) \int \int \left( \int_0^1 \int_0^1 F(t_1+ \overline{z}, t_2+z, s_1, s_2 ; f,n)\,ds_1 ds_2 \right) \\
&& \qquad \quad \cdot e^{-m t_1^2} e^{-m t_2^2 } \, dt_1 dt_2\\
&& \qquad  \le M_1 (f, \text{Im}\,z)^2.
\end{eqnarray*}
We leave the details to the reader.

We next consider the second bound. It follows from (\ref{4.1}) that
\begin{eqnarray} \label{4.8}
\omega( a^*(f) a(g)) &=& ( g, A(\1-A)^{-1} f ) , \\
\omega( a(g) a^*(f)) &=& ( g, (\1-A)^{-1} f ).
\nonumber
\end{eqnarray}
A direct computation yields that
\begin{eqnarray*}
&& \| \sigma_z(\Phi(f))\xi_0 \|^2 \\
&& \qquad = \frac 12 \Big( \omega(a^*(A^{i \overline{z}} f) a(A^{i\overline{z}} f)) +\omega(a(A^{i {z}} f) a^*(A^{iz} f))\Big).
\end{eqnarray*}
Thus the bound follows from (\ref{4.8}) and the above result.

(b) Notice that
\begin{eqnarray}
\nonumber
&&\| \sigma_z(\Phi_n(f,n))\xi_0 -\sigma_z(\Phi(f))\xi_0 \| \\
\nonumber
&& \qquad \le  \|\sigma_z(\Phi_n(f,n))\xi_0-\sigma_z(\Phi_n(f))\xi_0 \| + \| \sigma_z(\Phi_n(f))\xi_0 -\sigma_z(\Phi(f))\xi_0 \| \\
&& \qquad \equiv A^{(1)}_n + A^{(2)}_n.
\label{4.9}
\end{eqnarray}
It follows from (\ref{4.5}) and (\ref{4.7}) that
\begin{eqnarray*}
(A^{(1)}_n )^2 &=& (\frac{n} \pi) \int \int \left( \int_0^1 \int_0^1 G(t_1, t_2, s_1, s_2 ; f,n)\,ds_1 ds_2 \right) \\
&&  \quad \cdot e^{-n (t_1-\overline{z})^2} e^{-n (t_2 -z)^2 } \, dt_1 dt_2,
\end{eqnarray*}
where
\begin{eqnarray}
\label{4.10}
&& G(t_1, t_2, s_1, s_2 ; f, n) \\
&& \qquad := \omega (\Phi (A^{it_1}f) (W(-\frac {s_1} n A^{it_1}f) -\1) (W(\frac {s_2} n A^{it_2}f) -\1)
\Phi (A^{it_2} f)).
\nonumber
\end{eqnarray}
Notice that for $s, t \in \R$
\begin{equation} \label{4.11}
W(\frac {s} n A^{it}f) -\1 = i \frac s n \Phi( A^{it} f) \int_0^1 W(\frac {s \tau} n A^{it}f) d\tau.
\end{equation}
We substitute (\ref{4.11}) into (\ref{4.10}). Iterating (\ref{4.2}), and then using (\ref{4.1}) and (\ref{3.2}), the function given in (\ref{4.10}) can be calculated explicitely. Using Assumption 3.1, it can be proved that $G(t_1, t_2, s_1, s_2; f,n)$ has an analytic extension on $\C\times \C$ in the $t_1$ and $t_2 $ variables. By using the method similar to that used in the proof of the part (a) of the lemma, it is not hard to check that there exists constant $C_2(f$, Im $z)$ such that
\begin{eqnarray*}
&& \sup _{s_1, s_2 \in [0,1]} | G(t_1+ \overline{z}, t_2 +z, s_1, s_2 ; f, n) | \\
&& \qquad \le \frac 1{n^2} C_2 (f ,\,\text{Im}\, z).
\end{eqnarray*}
It follows from the Cauchy integral theorem and the above bound that
$$
(A^{(1)}_n )^2 \to 0 \quad \text{as}\,\,\, n \to \infty.
$$
We leave again the details to the reader.

Next, consider $A^{(2)}_n $ defined in (\ref{4.9}). Notice that
$$
(A_n^{(2)} )^2 = (\frac{n} \pi) \int \int \widetilde{ G}(t_1, t_2 ; f) e^{-n (t_1-\overline{z})^2} e^{-n (t_2 -z)^2 } \, dt_1 dt_2,
$$
where
$$
 \widetilde{G}(t_1, t_2 ; f) = \omega \Big((\Phi (A^{it_1}f) -\sigma_{\overline{z}}(\Phi(f)))  (\Phi (A^{it_2}f) -\sigma_{{z}}(\Phi(f)))\Big).
$$
We use (\ref{4.8}) to obtain that
\begin{eqnarray*}
\widetilde{G}(t_1, t_2 ; f) &=& \frac12 \omega (\{ a(A^{it_1} f)-a(A^{iz}f)\}\{a^*(A^{it_2}f)- a^*(A^{iz}f) \} ) \\
&& + \frac 12 \omega( \{ a^*(A^{it_1}f) -a^*(A^{i\overline{z}}f)\} \{ a(A^{it_2}f) -a(A^{i\overline{z}}f) \} )\\
&=& \frac12 ( ( A^{it_1} - A^{iz})f, (\1-A)^{-1}
( A^{it_2} - A^{iz})f ) \\
&& + \frac12 (( A^{it_2} - A^{i\overline{z}})f, A(\1-A)^{-1} ( A^{it_1} - A^{i\overline{z}})f ).
\end{eqnarray*}
Thus $\widetilde{G} (t_1, t_2; f)$ has an analytic extension on $\C \times \C$ in the $t_1$ and $t_2$ variables and
\begin{eqnarray}
\label{4.12}
&&\widetilde{G} (t_1+ \overline{z}, t_2+z; f) \\
\nonumber
&& \qquad =\frac12 ( f, (\1-A)^{-1} (A^{-it_1}-\1)(A^{it_2}-\1) A^{-i\overline{z}}A^{iz} f ) \\
\nonumber
&& \qquad \quad + \frac12 ( f, A(\1-A)^{-1} (A^{it_1}-\1)(A^{-it_2}-\1) A^{-i{z}}A^{i\overline{z}} f ).
\end{eqnarray}
We use the Cauchy integral theorem to get
$$
(A_n^{(2)} )^2 = (\frac{n} \pi) \int \int \widetilde{ G}(t_1+\overline{z}, t_2+z ; f) e^{-n t_1^2} e^{-n t_2 ^2 } \, dt_1 dt_2.
$$
By (\ref{4.12}), $\widetilde{G}(t_1+\overline{z}, t_2+z; f)$ is bounded uniformly with respect to $(t_1, t_2) \in \R^2$. Changing the variables $(t_1' = n^{1/2} t_1$ and $ t_2' =n^{1/2}t_2 $) and using the dominated convergence theorem, we conclude that
$$
A_n^{(2)} \to 0 \quad \text{as} \,\,\,n \to \infty.
$$
This proved the part (b) of the lemma completely. $\quad \square$

\vspace{0.4cm}\noindent{\bf Proof of Lemma \ref{lem5.4}.} (a)
Recall the definition of $\Phi(f,n)$ in (\ref{4.5}) for any
$f\in\ch_0$ and $n\in\mathbb{N}.$ For given(fixed)
$m\in\mathbb{N}\cup\{0\}$ and $h_l \in\ch_0,\,l=1,...,m$ we will
use the following abbreviated notations:
\begin{eqnarray}
&&\Phi_l (n):=\Phi(h_l ,n),\,l=1,...,m,\nonumber\\
\label{5.10}
&&\Phi_l :=\Phi(h_l ),\,l=1,...,m,\\
&\xi(n)&:=(\prod_{l=1}^m \Phi_l (n))\xi_0 ,\,\text{and}\,\,
\,\xi:=(\prod_{l=1}^m \Phi_l )\xi_0 .\nonumber
\end{eqnarray}
Notice that $\xi(n)\in \mathcal{W}\xi_0,\,n\in\mathbb{N}$ and
$\xi\in\cH_{fin}.$ It is not hard to show that
$\xi(n)\rightarrow\xi$ as $n\rightarrow \infty.$ See the method
used in the below. We will show that
$\overline{\cE}[\xi(n)-\xi]\rightarrow0$ as $n\rightarrow \infty.$
Since $\overline{\cE}$ is closed, this implies that $\cH_{fin}
\subset D(\overline{\cE})$ and
\begin{equation}\label{5.11}
\overline{\cE}[\xi]=\lim_{n\rightarrow \infty}\overline{\cE}[\xi(n)].
\end{equation}
By Remark 4.1 and (5.5), the relation (\ref{5.9}) holds for
$\xi\in \cW\xi_0 .$ Let $\xi$ and $\xi(n),\,n\in\mathbb{N}$ be
defined as in (\ref{5.10}). Notice that
$$
\xi(n)-\xi=\sum_{p=1}^m (\prod_{l=1}^{p-1}\Phi_l (n))
           (\Phi_p (n) -\Phi_p )(\prod_{l=p+1}^m \Phi_l )\xi_0 .
$$
 We use the Schwarz inequality twice to obtain that
           \begin{eqnarray}
\nonumber
 &&\|D_1 (B^{\frac12}g_k )(\xi(n)-\xi)\|^2 \\
\label{5.12}
       &&\le m\sum_{p=1}^m \|D_1 (B^{\frac12}g_k )
        (\prod_{l=1}^{p-1}\Phi_l (n))(\Phi_p (n) -\Phi_p )
             (\prod_{l=p+1}^m \Phi_l )\xi_0\|^2 .
           \end{eqnarray}

Recall that $D_1 (B^{\frac12}g_k )=a(A^{-\frac14}g_k
)-j(\sigma_{-\frac{i}{2}}(a(A^{-\frac14}g_k))).$ It follows from
the CCRs and Lemma \ref{lem5.1} that
             \begin{eqnarray}
\label{5.13}
     && D_1 (B^{\frac12}g_k )
     (\prod_{l=1}^{p-1}\Phi_l (n))(\Phi_p (n) -\Phi_p )(\prod_{l=p+1}^m \Phi_l
      )\xi_0\\
     &&\quad =\frac{1}{\sqrt2}\sum_{q=1}^m ( g_k ,A^{-\frac14}h_q
      ) \Psi^{(m)}(p,q;n)\nonumber
           \end{eqnarray}
where
  \begin{eqnarray*}
  \Psi^{(m)}(p,q;n)&:=&(\prod_{l=1}^{q-1}\Phi_l (n))W(\frac1n h_q )
                       (\prod_{l=q+1}^{p-1}\Phi_l (n))\\
                      &&\cdot (\Phi_p (n)-\Phi_p )(\prod_{l=p+1}^m \Phi_l )\xi_0
                    ,\quad 1\le q\le p-1,\nonumber\\
 \Psi^{(m)}(p,q;n)&:=&(\prod_{l=1}^{p-1}\Phi_l (n))(W(\frac1n h_p
                               )-\text{\bf 1})(\prod_{l=p+1}^m \Phi_l )\xi_0
                               ,\quad q=p,\nonumber\\
\Psi^{(m)}(p,q;n)&:=&(\prod_{l=1}^{p-1}\Phi_l (n))(\Phi_p (n)
                     -\Phi_p )(\prod_{l=p+1}^{q-1}\Phi_l (n))\nonumber\\
                     &&\cdot (\prod_{l=q+1}^{m} \Phi_l )\xi_0
                       ,\quad p+1\le q\le m.\nonumber
  \end{eqnarray*}
We use the Schwarz inequality twice again to (\ref{5.13}) and substitute
the result into (\ref{5.12}) to conclude that
$$
\|D_1 (B^{\frac12}g_k )(\xi(n)-\xi)\|^2 \le \frac12 m^2
\sum_{p=1}^m \sum_{q=1}^m |( g_k ,A^{-\frac14}h_q ) |^2
\|\Psi^{(m)}(p,q;n)\|^2 .
$$
Using the Parserval relation, we obtain that
\begin{eqnarray}
\label{5.14}
     &&\sum_{k=1}^{\infty}\|D_1 (B^{\frac12}g_k )(\xi(n)-\xi)\|^2 \\
     &&\quad \le \frac12 m^2 \sum_{p=1}^m \sum_{q=1}^m \|A^{-\frac14}h_q \|^2
     \|\Psi^{(m)}(p,q;n)\|^2 . \nonumber
\end{eqnarray}
Iterating (\ref{4.2}), $\|\Psi^{(m)}(p,q;n)\|^2$ can be calculated
explicitly for any $p,q$ and $n$. One notes that each
$\Psi^{(m)}(p,q;n)$ contains either $(\Phi_p (n)-\Phi_p )$ or
else $(W(\frac1n h_p )-\text{\bf 1}),$ which implies that
$$
\|\Psi^{(m)}(p,q;n)\|^2\rightarrow 0 \,\,\,\text{as}\,\,\,n\rightarrow\infty
$$
for any $p,\,q.$ Thus it follows from (\ref{5.14}) that
$$
\sum_{k=1}^{\infty} \|D_1 (B^{\frac12}g_k )(\xi(n)-\xi)\|^2
\rightarrow 0\,\,\,as\,\,\,n\rightarrow\infty.
$$

The method similar to that used in the above implies that
$$
\sum_{k=1}^{\infty} \|D_2 (B^{\frac12}g_k )(\xi(n)-\xi)\|^2
\rightarrow 0\,\quad \text{as}\,\,\,n\rightarrow\infty.
$$
This proved the part (a) of this lemma.

(b) This follows from (\ref{5.8})  and  (\ref{5.11}). $\quad
\square$\\

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

\begin{thebibliography}{99}
\bibitem[Acc]{Acc}  L.  Accardi, Topics in quantum probability, {\it Phys. Rep.} {\bf 77}, 169-192 (1981).
\bibitem[AC]{AC}  L.  Accardi and Cecchini, C.: Conditional expectations  in von Neumann algebras and a theorem of Takesaki, {\it J. Funct. Anal.}
{\bf 45}, 245-273 (1982).
\bibitem[AFL]{AFL}  L.  Accardi, A. Frigerio, and J.T. Lewis,  Quantum stochastic processes, {\it Publ. RIMS} {\bf 18}, 97-133 (1982).
\bibitem[AH-K]{AH-K}  S. Albeverio and R.  H\o egh-Krohn,  Dirichlet forms and Markovian semigroups on $C^*$-algebras,  {\it Commun. Math. Phys.}
{\bf 56}, 173-187 (1977).
\bibitem[Ara]{Ara}  H.  Araki,  Some  properties of modular  conjugation operator  of von Neumann  algebras and  noncommutative Radon-Nikodym
theorem with chain rule, {\it Pacific J. Math.} {\bf 50} (2), 309-354 (1974).
\bibitem[BKP]{BKP} C. Bahn, C. K. Ko and Y. M. Park, Dirichlet forms and symmertic Markovian semigroups on $\Z_2$-graded von Neumann algebras : preprint preparation.
\bibitem[BR]{BR}  O.  Bratteli and D.W. Robinson,
 {\it  Operator algebras  and quantum  statistical mechanics},
 Springer-Verlag, New  York-Heidelberg-Berlin, vol I (1979), vol. II (1981).
\bibitem[Cip]{Cip}  F.  Cipriani,  Dirichlet forms  and Markovian semigroups on standard forms of von Neumann algebras,  {\it J. Funct. Anal.} {\bf
147}, 259-300 (1997).
\bibitem[CFL]{CFL} F. Cipriani, F. Fagnola and J. M. Lindsay, Spectral Analysis and Feller Properties for Quantum Ornstein-Uhlenbeck Semigroups, {\it  Comm. Math. Phys.} {\bf 210},  85-105 (2000).
\bibitem[Dav]{Dav}  E.B.  Davies,  {\it Quantum theory of open systems},
 Academic Press, London-New York-San Francisco (1976).
\bibitem[DL]{DL}  E.B.  Davies and J.M. Lindsay,
Non commutative symmetric Markov semigroups, {\it Math. Zeit.}
 {\bf 210}, 379-411 (1992).
\bibitem[FNW]{FNW}  M.  Fanes and  B. Nachtergaele, R.F. Werner,  Finitely correlated states on  quantum spin chains, {\it Commun. Math. Phys.}
{\bf 144}, 443-490 (1992).
\bibitem[Fuk]{Fuk} M. Fukushima, {\it Dirichlet forms and Markov processes}, North-Holland,  Kodansha (1980).
\bibitem[GIS]{GIS}  D.  Guido, T. Isola and  S. Scarlatti,  Non-commutative
Dirichlet forms on semifinite von Neumann algebras, {\it J. Funct.
Anal.} {\bf 135}, 50-75 (1996)
\bibitem[GL1]{GL1}  S.  Goldstein and J.M. Lindsay,  Beuring-Deny  conditions for KMS-symmetric  dynamical semigroups, {\it C.  R. Acad. Sci.
Paris. Ser. I} {\bf 317}, 1053-1057 (1993).
\bibitem[GL2]{GL2}  S.  Goldstein and  J.M.  Lindsay,
KMS-symmetric Markov semigroups, {\it Math. Zeit.} {\bf 219}, 590-608 (1995).
\bibitem[MZ1]{MZ1}  A.W.   Majewski and B. Zegarlinski,  Quantum stochastic  dynamics I: Spin systems on a  lattice, {\it MPEJ} {\bf 1}, Paper  2
(1995).
\bibitem[MZ2]{MZ2}  A.W.  Majewski and B. Zegarlinski, Quantum stochastic dynamics II, {\it Rev. Math. Phys.} {\bf 8} (5), 689-713 (1996).
\bibitem[Mat]{Mat}  T.  Matsui,  Markov semigroups on UHF algebras,
 {\it Rev. Mod. Phys.} {\bf 5}, 587-600 (1993).
\bibitem[Na]{Na}  B.  Nachtergaele,  The spectral gap for some spin chains with discrete symmetry breaking, {\it Commun. Math. Phys.} {\bf 175}, 565-606 (1996).
\bibitem[Pa]{Pa} Y. M. Park, Construction of Dirichlet forms on standard
forms of von Neumann algebras,  {\it Infinite Dimensional Analysis, Quantum Probability and Related Topics}, Vol. 3, No. 1, 1-14 (2000).
\bibitem[Part]{Part}  K.R.  Parthasarathy,  {\it An Introduction to Quantum
Stochastic Calculus}, Birkh\"auser, Basel (1992).
\bibitem[RS]{RS}  M.  Reed and B. Simon,  {\it Method of modern mathematical
physics I, II: Fourier analysis, Self-adjointness}, Academic Press, New York (1975).
\bibitem[Sau1]{Sau1}  J.L.  Sauvageot,  Semi-groupe de la chaleur transverse sur la $C^*$ algebre d'un feuilletage riemannien,  {\it C. R. Acad. Sci.
Paris. Ser. I} {\bf 310}, 531-536 (1990).
\bibitem[Sau2]{Sau2}  J.L. Sauvageot, Tangent  bimodule and locality for dissipative operators on $C^*$-algebras, in  {\it Quantum probability and
applications IV} (L.  Accardi and W. von  Waldenfels, Eds.),
Lecture Notes  in Math. vol. 1396,  Springer-Verlag,
Berlin-Heidelberg-New York (1987).
\bibitem[Sua3]{Sau3}  J.L.  Sauvageot,  Quantum Dirichlet forms, differential calculus and semigroups, in {\it Quantum probability and applications V}
(L. Accardi and W. von Waldenfels, Eds.),
Lecture Notes in Math. vol. 1442, Springer-Verlag,
Berlin-Heidelberg-New York (1988).
\bibitem[Tak]{Tak} M. Takesaki, {\it Tomita's Theory of Modular Hilbert
Algebras and its applications}, Lecture Notes in Mathematics, vol. 128,
Springer-verlag, Berlin-Heidelberg-New York (1970).
\end{thebibliography}

\end{document}
---------------0102142116445--
