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19 pages, no figures
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Weighted Bergman spaces, Berezin-Toeplitz Operators,  
Schroedinger Operators, Feynman-Kac-Ito formula
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%%
%    Berezin-Toeplitz semigroups
%    Bernhard G. Bodmann 
%    v of Sep 18, 1999                                  
%                 
% ToDo:
% analyticity of K (jointly continuous)  
% (terminology) g: weight function            
% (terminology) image function
%
% original Bergman space ZITAT
% Continuity prop richtig zitieren(?)
% state continuity of e-tTf kernel
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\begin{document}

{\noindent\Large\bf A relation of Berezin-Toeplitz operators to\\[0.1cm]
                    Schrödinger operators and the probabilistic\\[0.1cm]
                    representation of Berezin-Toeplitz semigroups}\\[0.25cm]
\bf Bernhard G.\ Bodmann \\[0.1cm]
\normalfont Department of Mathematics, 
            University of Florida, FL 32611, USA\\[0.2cm]
\today 

\hspace*{.3cm}
%\renewcommand\baselinestretch{1.1}

\noindent{\bf Abstract:}

%\noindent 
A class of functions is specified which
give rise to semibounded quadratic forms on weighted Bergman spaces
and thus can 
be interpreted as symbols of self-adjoint Berezin-Toeplitz operators.
A similar class admits a probabilistic expression of the sesqui-analytic 
integral kernel for the associated semigroups.
Both results are the consequence of a relation of Berezin-Toeplitz operators
to Schrödinger operators defined via certain quadratic forms. 
The probabilistic expression is derived in conjunction with the 
Feynman-Kac-It\^o formula.  


\section{Introduction}

The results presented here are inspired by a concept 
of Daubechies and Klauder \cite{DaKl85,DaKl86} which
provides a probabilistic expression for the unitary group generated
by Hamiltonians arising from the so-called anti-Wick quantization prescription.
Based on geometric considerations, 
several works \cite{KlOn89,AKL93,Kla94,AlKl96} advocate natural generalizations
of this expression which can be related \cite{Kla88,Mar92} to Hamiltonians
from the more universal Berezin-Toeplitz quantization scheme \cite{Ber74}.


The generalization presented in this paper evolved from a pattern behind the 
construction in \cite{DaKl85,DaKl86}, according to which certain 
Berezin-Toeplitz
operators can be realized as limits of monotone families of Schr{\"o}\-din\-ger
operators.  Hereby a Berezin-Toeplitz operator $T_f$ is understood as
the compression of a suitable multiplication operator $M_f: \psi
\mapsto f\psi$ from a Hilbert space of square-integrable functions to a
closed subspace of analytic functions.
In this context the real-valued function $f$ is  called a symbol and can be 
interpreted as a classical observable corresponding to the operator $T_f$. 
In quantum mechanics Berezin-Toeplitz operators
model a variety of systems with canonical or other degrees of freedom 
\cite{Ber74,BMM96}.
A Schrödinger operator $H$, on the other hand,
can be thought of as a second-order differential operator
on a subset $\Lambda$ of $d$-dimensional Euclidean space $\R^d$,
formally written $H = \sum_{k=1}^d (i \partial_k + a_k )^2 + v$ with
the vector potential $a: \Lambda \to \R^d$ and the scalar potential
$v:\Lambda \to \R\,$.  Fairly general conditions have been worked out
under which this formal expression characterizes $H$ uniquely as a
self-adjoint operator \cite{LeSi81,CFKS,Hin92,HiSt92,BHL00}.


The first task in this paper is to find conditions on $f$ which guarantee
that $T_f$ can be defined as a self-adjoint operator. Secondly we derive the
analogue of the probabilistic expression in \cite{DaKl85} for
Berezin-Toeplitz semigroups in this general setting. 
Unlike the results \cite{DaKl85,DKP87} derived in analogy of the anti-Wick 
situation 
and the strategies advocated for their generalization  
\cite{KlOn89,AKL93,Kla94,AlKl96}, we stay with a probabilistic representation
which is based on standard Brownian motion \cite{ReYo99}. 
This way one may benefit from the associated repertory of probabilistic 
techniques. 
A minor difference with the original \cite{DaKl85} is that we are concerned 
with
Berezin-Toeplitz semigroups, not with the unitary groups associated
with quantum mechanical time evolution.


The structure and contents of this paper are as follows:
   
After fixing the notation, we start in Section 3 with a review of
Hilbert spaces of analytic functions which are known as weighted
Bergman spaces. In Sections 4 and 5 we construct Berezin-Toeplitz
operators on these spaces and state sufficient conditions on their
symbols which guarantee self-adjointness and semiboundedness.  Hereby
an essential tool is that certain Berezin-Toeplitz operators can be extended to
self-adjoint Schrödinger operators which are defined via quadratic forms.

In Section 6 we continue the spirit of \cite{DaKl85,DaKl86} and derive a
probabilistic expression for Berezin-Toeplitz semigroups which uses
a realization of Berezin-Toeplitz operators as limits of
monotone families of Schrödinger operators and the probabilistic
representation of Schrödinger semigroups with the help of the
Feynman-Kac-It\^o formula \cite{Sim79,Sim82,BHL00}.

This paper is not entirely self-contained. For the 
relevant background information, the reader is referred to
\cite{Sim78} and \cite{BHL00} which neatly comprise 
the essential building blocks for the results presented here.

%
\section{Basic definitions} \label{sec:defs} % and properties

Let $\R$ denote the real numbers.  By convention, $\D$ is always an
open, simply connected set in the plane of complex numbers $\C :=\{z =
z_1 + i z_2 : z_{1,2} \in \R\}$.  The boundary of a set $
A\subset \C$ is written as $\partial A$, its complement in
$\C$ as $\C\setminus A$ or ${ A}^c$.  We will denote the
real and imaginary parts of a complex number $z$ as $z_1$ and $z_2$,
respectively, and the complex conjugate as $\bar z = z_1 - i z_2$.

All functions appearing in this text are tacitly understood to be measurable, 
each one
in its appropriate sense. The positive and negative parts $f^+$ and $f^-$ of 
a real-valued 
function $f$ are defined by $f^\pm:= \max\{\pm f,0\}$, such that 
$f = f^+ - f^-$.
The indicator function of a set $A$ is denoted as $\chi_{A}$.

Several spaces of complex-valued functions are used in the text. 
The space of arbitrarily often differentiable functions with compact support
inside $\D$ is referred to as $C_c^\infty(\D)$. Concerning differentiability, 
$\D$ is hereby regarded as a subset of a real vector space $\C \cong \R^2$.  
The space of Lebesgue-essentially bounded functions on $\D$ 
is denoted as $L^\infty(\D)$.
 

\begin{definition}
  A complex-valued function $\phi : \D \to \C$ defined on $\D$ is called
\bemph{analytic in $\D$} if it is differentiable and satisfies the 
Cauchy-Riemann differential equations, which are stated as
\begin{equation}
  \label{eq:CRDE}
    \left(\partial_1 + i \partial_2 \right) \phi(z_1 + i z_2) = 0,
\end{equation}
with the partial derivatives 
$\partial_{1,2} := \frac{\textstyle \partial}{\textstyle \partial z_{1,2}}$.
\end{definition}

\begin{definition}
  A positive function $g: \D \to \,]0,\infty[$ is said to be 
\bemph{essentially bounded away from zero on compacts inside} $\D$
if for any given compact set $C \subset \D$ there exists a $\delta > 0$ such 
that
\begin{equation}
  \label{eq:BA0}
  \essinf_{z \in C} g(z) := \sup_{A} \inf_{z \in A} g(z) \ge \delta \, .
\end{equation}
Hereby the supremum is taken over all Lebesgue-measurable subsets $A \subset C$
such that the Lebesgue measure $\lambda$ vanishes on the difference, 
$\lambda(C\setminus A) = 0$.  
\end{definition}

\begin{definition}
A real-valued function $f: \C \to \R$ belongs to the
\bemph{Kato class $K$ in two dimensions} \cite{Kat73,Sim82} if the following 
condition is satisfied:
\begin{equation}
  \label{eq:K_def}
  \lim_{r \downto 0} \sup_{z \in \C} 
               \int_{\{\abs{y-z} < r\}} \! \abs{f(y)} 
                   \ln\abs{z-y} \, \dl(y)  = 0 \; .
\end{equation}
Whenever this property only holds locally, \ie
$\chi_C f \in K$ for all compact sets $C$ in $\C$, we write $f \in \Kloc$.
It is useful to know that the local Kato property implies local integrability
with respect to the Lebesgue measure, $\Kloc \subset \Lloc$. 

If a function satisfies $f^+ \in \Kloc$ and $f_- \in K$ then it is called
\bemph{Kato decomposable}, symbolized as $f \in K_\pm$.  

In order to apply these notions to functions which are at first
only defined on a subset $\D \subset \C$,
they are by convention extended to be zero on $\D^c$.
\end{definition}

\section{Hilbert spaces of analytic functions} \label{sec:af}


\begin{definition}
  Given a positive function $g: \D \to \,]0,\infty[$ we  associate with it the
so-called \bemph{weighted Bergman space}
\begin{equation}
  \label{eq:Bdef}
  \Bg := \{ \phi: \D \to \C, \text{ analytic in } \D 
                          \text{ and } (\phi,\phi) < \infty\} \, ,
\end{equation}
a vector space which is endowed with the inner product
\begin{equation}
  \label{eq:BIP}
  (\phi, \psi) := \int_\D  \ol{\phi(z)} \psi(z)  \, g(z) \dz 
\end{equation}
where $\lambda$ stands for the Lebesgue measure on the complex plane.
\end{definition}

\begin{remarks}
%\begin{itemize}
%\item
As a special case, if $\D$ is a bounded domain and the weight function is 
constant, $\Bg$
is the well-known Bergman space \cite{Ber50,Mes62}.

%\item 
The inner product suggests that $\Bg$ can be identified with a
vector-subspace of $L^2(g \lambda)$.  To be precise, we recall that
the latter consists of equivalence classes of
$g\lambda$-square-integrable functions on $\D$ which differ from each
other on a set of Lebesgue measure zero.  Consequently, we identify
each function in $\Bg$ with its equivalence class from $L^2(g
\lambda)$.
% \end{itemize}
\end{remarks}

Now we will state conditions which guarantee that $\Bg$ forms a 
Hilbert-subspace of  
$L^2(g \lambda)$.

\begin{proposition}
  If $g$ is essentially bounded away from zero on compacts inside $\D$ then
$\Bg$ is complete with respect to the norm-topology induced by the inner 
product.
\end{proposition}
\begin{proof}
  Let $(\psi_n)_{n \in \N}$ be a Cauchy sequence in $\Bg$. First we show 
uniform convergence
of $\psi_n$ on compacts inside $\D$. Consider a compact subset $C \subset \D$
and a safety radius $r<\inf_{y \in \partial\D} \abs{z-y}$ for all $z \in C$.
By assumption there is a lower bound $\delta > 0$ such that \eq{BA0} is 
satisfied.
Using the mean value property for analytic functions, Jensen's inequality 
in conjunction
with the convex square-modulus function, and the lower bound for $g$ we 
estimate
\begin{eqnarray}
  \label{eq:unifcvg}
 \sup_{z' \in C} \abs{\psi_n(z') - \psi_m(z')}^2 
      &=& \sup_{z' \in C} \Bigl\vert \frac 1 {\pi r^2} 
           \int_{B(r,z')}  (\psi_n(z) - \psi_m(z)) \, \dz \Bigr\vert^2  \\
   &\le&  \frac 1 {\pi r^2} \sup_{z' \in C} \int_{B(r,z')}  
                              \abs{\psi_n(z) - \psi_m(z)}^2 \dz \\
   &\le&  \frac 1 {\pi r^2 \delta} \int_{B(r,z')}   
                                 \abs{\psi_n(z) - \psi_m(z)}^2 \, g(z) \dz\\
   &\le&  \frac 1 {\pi r^2 \delta} \norm{\psi_n - \psi_m}^2 \, .
\end{eqnarray}
The right-hand side can be made arbitrarily small and thus the sequence 
$(\psi_n)$
converges uniformly on $C$. We can therefore conclude that the pointwise 
limit  
defines a function $\psi: \psi(z) = \lim_{n \to \infty} \psi_n (z)$ which is
also analytic in $\D$.

It remains to show that the convergence $\psi_n \to \psi$ is also in the 
sense of the norm.
Due to pointwise convergence and Fatou's lemma the inequality
\begin{equation}
  \label{eq:FL}
  \norm{\psi - \psi_m} \leq 
  \liminf_{n \to \infty} \norm{\psi_n - \psi_m} 
\end{equation}
follows, therefore the Cauchy property entails norm convergence.
\end{proof}

\begin{lemma}
  Under the same assumption on $g$ as in the preceding lemma, 
the \bemph{point-evaluation functionals} $F_z$ parameterized by $z \in \D$ 
which are defined 
according to
  \begin{eqnarray}
    \label{eq:PEF}
    F_z: \Bg &\longrightarrow& \C \\
     \psi &\longmapsto& \psi(z)
  \end{eqnarray}
  are continuous linear mappings.
\end{lemma}
\begin{proof}
Let $C$ be a compact neighborhood of $z$ and select a convergent sequence 
$(\psi_n)_{n \in \N}$ in $\Bg$. 
Since the sequence has the Cauchy property, we can use the chain of 
inequalities \eq{unifcvg}
to show that $\psi_n(z)$ is also Cauchy, and therefore convergent. 
\end{proof}

\begin{proposition}
  If $g$ is essentially bounded away from zero on compacts inside $\D$, the 
weighted Bergman space
$\Bg$ possesses a \bemph{reproducing kernel}. More explicitly, there is a 
kernel $\kappa: \D \times \D \to \C$ such that any function $\psi$ in $\Bg$ 
satisfies 
the integral equation
\begin{equation}
  \label{eq:RK}
  \psi(z') = \int_\D \kappa(z',z) \psi(z) \, g(z) \dz \, .
\end{equation}
\end{proposition}
\begin{proof}
  To see this, we observe that due to the continuity of the functional $F_z$
and the completeness of $\Bg$ the Riesz representation 
theorem implies that there is a vector $\kappa_z$ in $\Bg$ 
such that 
\begin{equation}
  \label{eq:RKIP}
  (\kappa_z,\psi)=\psi(z) \text{ for all } \psi \in \Bg .
\end{equation}
Inserting the definition of the inner product \eq{BIP}
yields the desired integral equation \eq{RK} with the claimed kernel given by 
$\kappa(z,z')=(\kappa_z,\kappa_{z'})$.  
\end{proof}

The last proposition ensures that all bounded operators have integral kernels.

\begin{corollary} \label{lem:Tfint}
  If $g$ is essentially bounded away from zero on compacts inside $\D$,
then any bounded operator $B$ on $L^2_a(g\lambda)$ possesses an integral kernel
given by $B(z,z') = (\kappa_z, B \kappa_{z'}),$ which means that the image of
$\psi \in \Bg$ is expressed as
\begin{equation}
  \label{eq:Bpsi}
  B \psi(z) = \int_\D \dl(z') g(z') \, B(z,z') \psi(z') \, .
\end{equation}
\end{corollary}
\begin{proof}
  That $B(z,z')$ is indeed an integral kernel results from the 
reproducing property \eq{RK} and Fubini's theorem. The sesqui-analyticity
follows from the point-evaluation property \eq{RKIP} and the analyticity of
the functions in $\Bg$.
\end{proof}

\begin{remark} 
Since \eq{Bpsi} makes sense even for $\psi \in L^2(g \lambda)$, any bounded 
operator extends naturally 
via its integral kernel to $L^2(g\lambda)$. From this point of view, $\kappa$ 
is the integral kernel 
of an \bemph{orthogonal projection operator}, henceforth called $\mathcal K$, 
which maps 
$L^2(g\lambda)$ onto $\Bg$.
\end{remark}

\begin{examples} \label{ex:HW2BG}
  Various examples of Lie group representations 
are realized on specific weighted Bergman spaces. We list a few groups and 
their 
unitary irreducible action which is in all but the last example 
related to Moebius transformations on the associated
representation spaces. For more details, see \cite{Per86} or \cite{Nee96}.
Unless otherwise stated we set $\D = \C$.
\begin{itemize}
\item[1.] {\it Heisenberg-Weyl group.}
  Hereby the Hilbert space is specified by the
weight function $g(z) =  \frac 1 \pi e^{-\abs z^2}$.
The reproducing kernel is $\kappa(z,z')=e^{z \bar z'}$.
This space is also known as Fock-Bargmann space \cite{Bar61}.
The group representation $D(\alpha,\beta)$ with parameters 
$\alpha \in [0,2\pi[, \beta \in \C$ 
acts on a vector $\psi$ by
\begin{equation}
  \label{eq:HWrep}
  D(\alpha,\beta) \psi(z) = e^{i \alpha} \, e^{- \abs \beta^2} 
                               \, e^{\beta z} \, \psi(z - \bar \beta) \, .
\end{equation}

\item[2.] {\it$SU(2)$ group.} For each integer or half-integer 
$j \in \half \mathbb N$ 
a $(2j+1)$-dimensional space is defined by setting
$g(z) = \frac{2j + 1}\pi (1 + \abs z^2)^{-2j -2}$.
The reproducing kernel is $\kappa(z,z') = (1 + z \bar z')^{2 j}$. 
The group parameterized by $\alpha, \beta \in \C$ with 
$\abs \alpha^2 + \abs \beta^2 = 1$
acts as 
\begin{equation}
  \label{eq:su2rep}
  D(\alpha,\beta) \psi(z) = (\beta z + \bar\alpha)^{2j} 
   \psi\Bigl( \frac{\alpha z  - \bar \beta}{\beta z + \bar\alpha}\Bigr)\, .
\end{equation}

\item[3.] {\it $SU(1,1)$ group.} There are two well-known 
ways to represent this group on weighted Bergman spaces. 

\noindent {\it a}) The first one is described in \cite{Per86}.
Unlike the previous examples, here $\D$ is not the whole complex plane, 
but the unit disc $\D = \{z: \abs z < 1\}$, and
$
   g(z) = \frac{2k - 1}{\pi} \, (1 - z \bar z)^{2k - 2} 
$
with a fixed number $k \in \{1,3/2,2,\dots\}$. 
The reproducing kernel is 
$\kappa(z,z') = (1- z \bar z')^{-2k}$.
The group action is given with the parameters 
$\alpha, \beta \in \C, \abs \alpha^2 - \abs \beta^2 = 1$ 
as
\begin{equation}
  \label{eq:su11rep}
  D(\alpha,\beta) \psi(z) =  (\beta z + \bar \alpha)^{-2k} 
    \psi\Bigl(\frac{\alpha z + \bar \beta}{\beta z + \bar \alpha}\Bigr) \, .
\end{equation}

\noindent {\it b}) An alternative to \cite{Per86} comes from the so-called 
Barut-Girardello 
representation \cite{BaGi71} of the $SU(1,1)$ group. We set  
$\D = \C$ again, select $k$ as above and choose
$g(z) = \frac{2 \abs z^{2k-1}}{\pi \Gamma(2k)}  K_{2k-1}(2\abs z) $
with the Gamma-function $\Gamma$ and the modified Bessel function 
$K_\sigma(r) = \frac 1 2 (r/2)^\sigma \int_0^\infty  t^{-\sigma-1} 
\exp(-t-r^2/4t) \, dt$ for 
$r > 0, \sigma \in \R$ \cite[8.432(6)]{GrRy95}. 
The kernel is given as the confluent hypergeometric limit function 
$\kappa(z,z')={}_0F_1(2k,z \bar z')= 
 \sum_{n=0}^\infty (z \bar z')^n / (2k)_n \Gamma(n+1)$.
Hereby $(2k)_n$ denotes the Pochhammer symbol, defined by $(2k)_0:=1$ and 
the recursion
$(2k)_n = (2k)_{n-1} (2k+n-1)$.
The group action, however, does not seem to be related to Moebius 
transformations and will be omitted here.  
\end{itemize}
\end{examples}

\begin{remarks}
In all these examples the kernel can be constructed from the weight function
with the help of the Gram-Schmidt orthogonalization procedure. Due to the 
rotational symmetry $g(z)=g(\abs z)$, monomials $p_n: z \mapsto z^n$ with 
differing degree
$n \in \{0, 1, 2, \dots\}$ are orthogonal, so
the kernel is diagonalized in terms of the basis functions as
$\kappa(z,z') = \sum_n c_n p_n(z) \ol{p_n(z')}$
with suitable normalization constants $c_n$.
Here the summation runs for all the examples over the non-negative integers, 
except for
the $SU(2)$ case, because there, only monomials with maximal degree $2j$ are 
square integrable.
\end{remarks}

In Section \ref{sec:PR} we will present a probabilistic approach to 
construct 
the reproducing kernel, which does not rely on special symmetries of $g$.



\section{Self-adjoint Berezin-Toeplitz operators defined by quadratic forms}

In the remaining text we assume that $\Bg$ is complete.

\begin{definition}
 Given the Hilbert space $\Bg$ and a real-valued function $f: \D \to \R$, 
we consider the \bemph{sesquilinear form}
\begin{eqnarray}
  \label{eq:Qform}
  \t_f: \q( \t_f) \times \q(\t_f)  &\longrightarrow& \C \\
       (\psi, \phi) &\longmapsto& \int_\D f \ol \psi \phi  \, g  \dl 
   \label{eq:tfdef}
\end{eqnarray}
with form domain 
\begin{equation}
  \label{eq:tfdom}
  \q(\t_f) := \{ \psi \in \Bg: \int_\D \abs{ f \psi^2} 
                                   \, g \dl < \infty \} \, .
\end{equation}
When it is interpreted as quadratic form, $\t_f$ is written as 
$\t_f(\psi) := \t_f(\psi,\psi)$. 
\end{definition}

The next concern is a condition to guarantee that $\t_f$ is closed and 
semibounded, which in turn
ensures that there is a self-adjoint operator associated with $\t_f$ 
via the Friedrichs 
representation theorem.

\begin{lemma}
  The sesquilinear form belonging to a non-negative function $f\ge 0$  
is closed. 
\end{lemma}
\begin{proof}
We need to show that $\q(\t_f)$, equipped with the form-norm 
$\norm{\bullet}_{\t_f}$ defined by
  \begin{align}
    \label{eq:tnorm}
       \norm{\psi}_{\t_f} := ( \t_f(\psi) + \norm\psi^2 )^{1/2} 
           & \text{ for } \psi \in \q(\t_f) \, ,
  \end{align}
is complete.

Suppose $(\psi_n)_{n \in \mathbb N}$ is a Cauchy sequence with respect to 
the form-norm. 
Due to the estimate $\norm\psi \leq \norm{\psi}_{\t_f}$
the sequence is convergent in $\Bg$, $\psi_n \to \psi$. Using pointwise 
convergence and Fatou's lemma, 
we obtain
$
   \norm{\psi - \psi_n}_{\t_f} \leq  \liminf_{m \to \infty} 
\norm{\psi_m - \psi_n}_{\t_f}
$
and therefore the sequence $(\psi_n)$ converges with respect to the form-norm. 
\end{proof}

\begin{theorem} \label{thm:KLMN}
  If the form $\t_{f^+}$ belonging to the positive part $f^+\ge 0$ of a 
function 
$f = f^+ - f^-$ is
densely defined and the negative part can be incorporated in $\t_f$
as a form-bounded perturbation
\begin{equation}
  \label{eq:fbd}
  \t_{f^-}(\psi) \leq c_1 \, \t_{f^+}(\psi) + c_2 \norm{\psi}^2 
\end{equation}
with relative form bound $c_1<1$ and a constant $c_2 \ge 0$,
then $\t_f$ is closed on $\q(\t_f) =\q(\t_{f^+})$
and has a (greatest) lower bound $c \in \R$,
such that $\t_f(\psi) \ge c \norm\psi^2$. 
\end{theorem}
\begin{proof}
This is the so-called KLMN theorem.
For the proof, see \cite[Theorem X.17]{ReSiII}.
\end{proof}

\begin{theorem}
If the form $\t_f$ is closed and has the greatest lower bound $c$
as in the preceding theorem, then it belongs to 
a unique self-adjoint operator $T_f$ which is characterized in terms of the 
square-root 
$\sqrt{T_f - c}$ by the domain $\d(\sqrt{T_f - c}) = \q(\t_f)$ and the equality
\begin{equation}
   (\sqrt{T_f - c}\, \phi,\sqrt{T_f - c}\, \psi) + c(\phi,\psi) 
       = \t_f(\phi,\psi)  \text{\ \  for all\ \ }
      \phi, \psi \in \q(\t_f) \, .
\end{equation}
\end{theorem}
\begin{proof}
Again, we refer to the literature \cite[Theorem VIII.15]{ReSiI} or 
\cite[Theorem 5.36]{Wei80}, 
where this result is known as the Friedrichs representation theorem.
\end{proof}

\begin{remark}
We do not adopt the usual term Friedrichs extension for $T_f$, because
here it does not arise from the quadratic form of a given semibounded 
operator, and besides that, $T_f$ is uniquely determined due to the
closedness of the form. In the context of weighted Bergman spaces we 
call $T_f$ a self-adjoint \bemph{Berezin-Toeplitz operator} and 
the function $f$ its \bemph{symbol}.
\end{remark}
 
\section{Relation to Schrödinger operators} \label{sec:BTS}

This section shows how a Berezin-Toeplitz operator can be extended
to a family of Schrödinger operators.
A major benefit is that the knowledge about Schrödinger
operators can be used to find sufficient conditions for the semiboundedness 
of\/ $\t_f$ in terms of $g$ and $f$. These ensure self-adjointness
of the corresponding Berezin-Toeplitz operator.

\subsection{Embedding the weighted Bergman space} \label{sub:embed}
                                \setcounter{subsref}{\value{subsection}}

Consider the unitary mapping 
\begin{eqnarray}
  \label{eq:I}
  U: L^2(g \lambda) & \longrightarrow & L^2(\D) \\
                  \psi & \longmapsto & \sqrt g \psi \, ,
\end{eqnarray}
which simply amounts to a redistribution of the weight function.
This mapping identifies $\Bg$ with a closed subspace of $L^2(\D)$. 

\begin{itremark}
Note that the unitary equivalent 
$\widetilde {\mathcal K} = U {\mathcal K} U^\dagger$ 
of $\mathcal K$ is an orthogonal projection operator
which maps $L^2(\D)$ onto $U(\Bg)$ and has the integral kernel 
$\widetilde \kappa(z,z') = \kappa(z,z') \sqrt{g(z)g(z')}$. However, unlike 
the case 
for $\Bg$, this kernel is continuous if and only if $g$ is continuous.
Following the previous notation we define 
$\widetilde \kappa_{z'} := \sqrt{g(z')} \, U \kappa_{z'}$. 
\end{itremark}


\subsection{A quadratic form and its null space}


The purpose of this subsection is to construct a Schrödinger operator which is 
non-negative and 
has $U(\Bg)$ as its null eigenspace.

\begin{definition}
We define the quadratic form
\begin{eqnarray}
  \label{eq:rmax}  
      \r:      C_c^\infty(\D)  &\longrightarrow& \R \\
             \psi  &\longmapsto& \norm{ (i\partial_1 -\partial_2 + a) \psi}^2 
                \label{eq:da}
\end{eqnarray}
with the function
\begin{equation}
  \label{eq:adef}
  a(z) := (-i \partial_1 + \partial_2) \ln \sqrt{g (z)}
\end{equation}
where $g$ is such that the local integrability condition 
$\abs a^2 \in L^1\loc(\D)$ holds. 
This form is closeable and the resulting form-closure domain $\q(\r)$ 
is contained in the 
set $L^2(\D)$. We say that the corresponding self-adjoint operator $R$ obeys 
\bemph{Dirichlet boundary conditions}.
\end{definition}

\begin{remarks} \label{rem:Rmod} 
  Despite the construction of $R$ via \eq{da},
  in general the null-space $\n(R):=R^{-1}(\{0\})$ is only a closed subspace 
of $U(\Bg)$.
On the other hand,
if instead of $C_c^\infty(\D)$ we start with the maximal form domain,
which means the set of all $\psi \in L^2(\D)$ for which the expression on 
the right-hand side of
\eq{da} makes sense and is finite,
then the resulting self-adjoint operator is said to obey Neumann boundary 
conditions and
possesses the null-space $U(\Bg)$. 
For this reason it seems like an artificial complication to introduce
Dirichlet boundary conditions here. Nevertheless, the preceding definition 
is needed as a preparation 
for the probabilistic expression in Section~\ref{sec:PR}.

If $a$ is absolutely continuous we can formally write $R$
as a Schrödinger-type operator
\begin{equation}
  \label{eq:Rop}
  R = (i\partial_1 + a_1)^2 + (i\partial_2 + a_2)^2 + v 
\end{equation}
with $v := \partial_2 a_1 - \partial_1 a_2$. However, in order to consider $R$ 
as the usual form sum \cite[Remark 2.7]{BHL00} we need additional regularity 
assumptions on $a$ and $v$; 
see also the following subsection. 
\end{remarks}

Next we establish conditions which guarantee the inclusion of $U(\Bg)$ in 
the domain
of $R$.

%\pagebreak[2]
\begin{proposition} \label{thm:nrubg}
  If either $\D=\C$ or
the operator $T_f$ for $f(z)=\inf_{y \in \partial \D} \abs{z-y}^{-2}$ is 
bounded 
on $\Bg$, then the image $U(\Bg)$ is contained in the form-closure 
domain $\q(\r)$ and
consequently $\n(R) = U(\Bg)$.
\end{proposition}
\begin{proof}
In case $\D=\C$ the argument follows a standard procedure, 
compare with \cite{Sim79b} or \cite[Theorem 1.13]{CFKS}. The more general case 
treated here demands an adaptation.

The proof proceeds in two steps:
First we will show that for any function $\psi \in U(\Bg)$, the space of 
compactly supported, 
in $\D$ essentially bounded functions $L^\infty_c(\D)$, provides a sequence 
which converges
to $\psi$ in the sense of the form norm. The second step is a standard 
mollifier argument,
which shows that $C_c^\infty(\D)$ is dense in $L^\infty_c(\D)\cap \q(\r)$.\\

\noindent{\it Step 1:}
Consider a function $\psi \in U(\Bg)$. We perform two alterations: a mollified 
cutoff near $\partial \D$ (if $\D^c$ is nonempty)
and a smooth cutoff towards infinity.

The details are as follows: Let 
$D_n := \{z \in \D: \abs{z-y} \ge  1 / n \, 
 \text{ for all } y \in \partial \D\}$.
Obviously the sequence of
characteristic functions $\chi_{D_n}$ converges pointwise to one on $\D$ 
as $n \to \infty$.
In addition, consider a real-valued function $\eta \in C_0^\infty(\C)$ 
which is non-negative 
and satisfies $\eta(0)=\max_{z \in \C} \eta(z) = 1$ as well as the 
gradient bound 
$\abs{\nabla \eta} \leq 1$. We define a sequence $\eta_n(z)=\eta(z/n)$ 
which also tends to 
one from below.
Furthermore, let the non-negative function
$\delta_1 \in C_c^\infty(\C)$ be an approximate $\delta$-function, which means 
$\int_C  \delta_1(z) \dl = 1$, and $\delta = 0$ for all $\abs z > 1$. 
We define a sequence of approximations $\delta_n(z):= n^2 \delta_1(n z)$ 
to smear out $\chi_{D_n}$ by convolution,
$ (\chi_{D_n} * \delta_n)(z):= \int_\D \dl(y)\, \chi_{D_n}(y) \delta_n(x-y) $. 

Now consider $\phi_n := \psi \eta_n (\chi_{D_n} * \delta_n)$. 
Clearly $\phi_n \to \psi$ in $L^2(\D)$. To show that the convergence is also 
with respect
to the form-norm, we use the triangle inequality,
\begin{eqnarray}
  \lefteqn{\norm{(i\partial_1 - \partial_2 + a)\phi_n} 
          = \norm{ \psi (i\partial_1 - \partial_2) 
           \eta_n (\chi_{D_n} * \delta_n)} } && \nonumber \\
  &\leq&   \norm{ \psi (\chi_{D_n} * \delta_n) (i\partial_1 - \partial_2) 
           \eta_n }
         + \norm{ \psi \eta_n (i\partial_1 - \partial_2) 
           (\chi_{D_n} * \delta_n) } 
          \label{eq:phin}\,.
\end{eqnarray}
The first term on the right-hand side vanishes in the limit $n \to \infty$ 
by dominated convergence;
the second term needs a closer look. Note that
the neigborhood of the boundary contains the support 
$\supp (i\partial_1 - \partial_2) (\delta_n * \chi_{D_n}) 
 \subset \{z \in \D: \abs{z-y}> 2 / n \, \forall y \in \partial\D\}$. 
By Hölder's inequality we can estimate 
$\abs{\nabla \delta_n * \chi_{D_n}} \leq n \lambda(\supp \delta_1) 
\max \abs{\nabla \delta_1}$. 
In consequence there is some constant $c \ge 0$ such that
$\abs{(i\partial_1 - \partial_2)(\chi_{D_n} * \delta_n)} \leq c \sqrt f$ 
with the function $f$ given in the statement of the theorem.
Hence by the assumption on the boundedness of $T_f$ we can apply dominated 
convergence and the 
second term in \eq{phin} also vanishes as $n \to \infty$.\\

\noindent{\it Step 2:}
Let $\psi \in L^\infty_c(\D)\cap \q(\r)$ be given.
We borrow an argument from \cite[Lemma B.3, Assertion 3)]{BHL00}. 
Consider the previously defined $\delta_n$ for only suitably
large $n$ such that $\psi_n := \psi * \delta_n \in C_c^\infty(\D)$.

We have $\lim_{n \to \infty} \norm{\phi * \delta_n - \phi} =0$ for all 
$\phi \in L^2(\D)$. 
Therefore it remains to show that after the estimate
\begin{equation}
  \label{eq:tafcvg}
  \norm{(i\partial_1 - \partial_2 + a) (\psi_n - \psi)} \leq
    \norm{(i\partial_1 - \partial_2) (\psi_n - \psi)}
 +  \norm{a(\psi_n - \psi)}
\end{equation}
each term on the right-hand side converges to zero. 

The first term is taken care of by the identity
\begin{equation}
  \label{eq:dpsin}
  \norm{ (i\partial_1 - \partial_2)(\psi_n - \psi) } 
 = \norm{ ((i\partial_1 - \partial_2) \psi ) * \delta_n - 
          (i\partial_1 - \partial_2) \psi} 
\end{equation}
because the assumption $\abs a ^2 \in \Kloc$  implies $a \in L^2\loc(\D)$ and 
\begin{equation}
  \label{eq:dapsi}
  \norm{ (i\partial_1 - \partial_2) \psi} \leq 
                 \norm{ (i\partial_1 - \partial_2 + a) \psi}
                      + \norm{ a \psi } <\infty \, .
\end{equation}

To ensure that the second term in \eq{tafcvg} vanishes we pass to a
subsequence of $\psi_n$ which is almost everywhere convergent and select a 
compact 
set $C$ which 
contains the support of all but finitely many $\psi_n$. 
The bound $\abs{\psi_n(x)} \leq \chi_C \norm{\psi}_\infty$ for $x \in \D$ 
together with $a \in L^2\loc(\D)$ allows dominated convergence which 
completes the proof.
\end{proof}

\begin{remark}
When $\D$ is a proper subset of the complex numbers, the condition on the 
functions
in $U(\Bg)$ amounts to controlling their decay towards the boundary 
$\partial \D$,
which relates to the intuitive understanding of Dirichlet boundary conditions.
\end{remark}


\subsection{Extending Berezin-Toeplitz operators  
            to Schrödinger operators}


Henceforth, we say $f$ and $g$ are \bemph{admissible} if they are such that
$\abs a^2 \in \Kloc$ and $\partial_2 a_1 - \partial_1 a_2 , f \in \Kpm$.

For such $f$ and $g$ 
we define a family $\{\s_f^{(\nu)}\}_{\nu>0}$ of quadratic forms
\begin{eqnarray}
    \label{eq:snu}
    \s_f^{(\nu)}: \q(\s_f^{(\nu)}) &\longrightarrow& \R \\
            \psi &\longmapsto& \nu \, \r(\psi) + \int_\D \dl f \abs\psi^2 \, . 
                 \label{eq:rf}
\end{eqnarray}
The domain is independent of $\nu$ given by 
$\q(\s^{(\nu)}_f):=\q(\r)\cap \{\psi: \bnorm{\sqrt{f^+}\psi}<\infty\}$, 
because the negative part of the second term in \eq{rf} is, due to the 
assumption, 
infinitesimally form-bounded with respect to $\r$. 

With the above regularity assumptions on $a$, 
$v = \partial_2 a_1 - \partial_1 a_2$ and $f$
the corresponding self-adjoint, semibounded operator $S^{(\nu)}_f$ is 
a Schrödinger operator defined in the usual form sense \cite{BHL00}, 
which is implicitly understood in the expression
\begin{equation}
  \label{eq:Snuf}
  S^{(\nu)}_f = \nu \left[ (i \partial_1 + a_1)^2 + 
                  (i \partial_2 + a_2)^2 +v \right] + f \, .
\end{equation}

\begin{theorem} \label{lem:ext}
  If $f$ and $g$ obey the conditions
$\abs a^2 \in \Kloc$ and $\partial_2 a_1 - \partial_1 a_2 , f \in \Kpm$, and 
the conclusion 
$\n(R) = U(\Bg)$ of Proposition \ref{thm:nrubg} holds, 
then the form $\t_f$ is semibounded and closed on $\q(\t_{f^+})$. Therefore, 
$f$ defines
a self-adjoint semibounded operator $T_f$ on the closure 
$\ol{\q(\t_f)} \subset \Bg$. 
\end{theorem}
\begin{proof}
  First we note $\s^{(\nu)}_f(U\psi) = \t_f(\psi)$ for any $\nu>0$ and 
$\psi \in \q(\t_f)$. 
Thus, we only need to show that
the restriction of $\s^{(\nu)}_f$ to the closed subspace 
${\mathcal N}(\r)=U(L^2_a(g\lambda))$ 
is again a closed and semibounded form. 

  To show closedness, assume a sequence $(\psi_n)_{n \in \mathbb N}$  
in ${\mathcal N}(\r)$
which is Cauchy with respect to the form-norm. Then by the closedness of 
$\s^{(\nu)}_f$
the sequence has a limit $\psi \in \q(\s^{(\nu)}_f)$. However, this limit 
must also be contained
in ${\mathcal N}(\r)$, because ${\mathcal N}(\r)$ is a closed subspace and
the sequence $(\psi_n)_{n \in \mathbb N}$ converges
with respect to the usual norm on $L^2(\D)$.

Semiboundedness follows from the inequality
\begin{equation}
  \label{eq:lbd}
  \inf \{\s^{(\nu)}_f(\psi) : \norm{\psi} = 1 \}
  \leq \inf \{\s^{(\nu)}_f(\psi) : \psi \in {\mathcal N}(\r) \text{ and } 
\norm{\psi} = 1 \} \, .
\end{equation}
\end{proof}

\begin{remarks} As stated, the above theorem does not imply that $\t_f$ 
is densely defined. 
Therefore, $T_f$ might be self-adjoint only on a Hilbert-subspace of $\Bg$.

In analogy with Theorem \ref{thm:KLMN}, it is sufficient for the closedness 
and 
semiboundedness of $\t_f$ when for some $\nu>0$ the negative part
$f^-$ can be incorporated as a form-bounded perturbation of $\s^{(\nu)}_{f^+}$
with relative form bound strictly less than one.
However, this condition is not as easy to characterize in terms of $f$ 
as the stronger assumption in the preceding theorem.
\end{remarks}

\section{Probabilistic representation of Berezin-Toeplitz semigroups} 
\label{sec:PR}

In this section we derive a probabilistic
expression for the sesqui-analytic integral kernel of Berezin-Toeplitz 
semigroups.
The major steps are the reconstruction of $T_f$ via
the monotone convergence of $\s_f^{(\nu)}$ for $\nu \to \infty$ and the
application of the Feynman-Kac-It\^o formula.

\begin{definition}
For a given $z,z' \in \C$ and $t,\nu>0$ we define the integral with respect 
to the 
\bemph{pinned Wiener measure}
\begin{equation}
  \label{eq:mu}
  \int  (\bullet) \, d\mu^{(\nu)}_{z,0;z',t}
       := \frac{1}{4 \pi t \nu} \,\ep{- \abs{z - z'}^2/4t\nu} \,\Ez{\bullet} 
\end{equation}
via the expectation $\Ez{\bullet}$ with respect to the two-dimensional 
Brownian bridge measure with diffusion constant $\nu$.
Both measures are concentrated on the set of continuous paths 
$\{b: [0,t] \to \C, b(0)=z \text{ and } b(t) = z'\}$ 
which are pinned at the start and endpoint. 
As Gaussian stochastic process the Brownian bridge is uniquely determined by 
its mean
\begin{align}
  \label{eq:emean}
  \Ez{b(s)} &= z + (z' - z) \frac st \, , & s \in [0,t] \\ 
\intertext{and covariance}
  \Ez{\ol{b(r)} b(s) } - \Ez{\ol{b(r)}} \Ez{b(s)} &= 4 \nu 
                \left(\min\{r,s\} - \frac{rs}t \right) & \\
  \Ez{b(r) b(s)} - \Ez{b(r)} \Ez{b(s)} &= 0&r, s \in [0,t] \, .
\end{align}
\end{definition}

\begin{definition}
Given $\D$, the random variable 
$\TD:= \inf\{s>0: b(s) \in \D^c\}$
 is called the \bemph{first exit time} of the process. By convention, 
we define $\TD$ to be 
 infinite on the set for which $b$ never leaves $\D$.
\end{definition}


\begin{proposition} \label{prop:sgcv}
If $f$ and $g$ are admissible and the conclusion $\n(R) = U(\Bg)$ of 
Proposition 
\ref{thm:nrubg} holds, then for $\nu \to \infty$ the semigroup generated 
by $S^{(\nu)}_f$
converges strongly, 
  \begin{equation}
    \label{eq:sgconv}
    \lim_{\nu \to \infty} \ep{-t  S_f^{(\nu)}} \psi = \ep{- t S^{(\infty)}_f} 
                                            \widetilde E \psi
               = U \ep{- t T_f} E \, U^\dagger \psi \, ,
  \end{equation}
where $\psi \in L^2(\D)$, $E= E^\dagger E$ projects onto the closure 
$\ol{\q(\t_f)}$, and 
$\widetilde E := U E \, U^\dagger$.
\end{proposition}
\begin{proof}
The limit $\nu \to \infty$ of 
$\s_f^{(\nu)}$ yields a non-densely defined form 
\begin{equation}
  \label{eq:sinfty}
    \s_f^{(\infty)} : \psi \mapsto \lim_{\nu \to \infty}\s_f^{(\nu)}(\psi)  
\end{equation}
with the domain $\q(\s^{(\infty)}_f)=U(\q(\t_f))$. This last equality 
follows from 
Proposition \ref{thm:nrubg} and the definition of the embedding.

The monotone convergence implies that $s^{(\infty)}_f$ is closed \cite{Sim78} 
and semiboundedness follows
from that of $\s^{(\nu)}_f$ for any $\nu>0$.
All these properties hold then for $\t_f$ as well, via
the identity $\s^{(\infty)}_f(U\psi) = \t_f(\psi)$ valid for all 
$\psi \in \q(\t_f)$,
and thus give rise to a semibounded self-adjoint operator 
$T_f= U^\dagger S^{(\infty)}_f U$ on the closure $\ol{\q(\t_f)} \subset \Bg$.

By the monotone convergence of forms 
the self-adjoint operators associated with $\s^{(\nu)}_f$
converge in the strong resolvent sense \cite{Sim78}, 
which in turn implies strong convergence of the semigroups they generate 
\cite[Theorem S.14]{ReSiI}. 
\end{proof}

\begin{theorem}
Provided $f$ and $g$ are such that $\abs a^2 \in \Kloc$ and $v, f \in \Kpm$, 
and the conclusion of Proposition \ref{thm:nrubg} holds, then for $t>0$  
the continuous integral kernel of the semigroup generated by $S^{(\nu)}_f$ 
converges
in the limit $\nu \to \infty$
pointwise to the kernel of the semigroup associated with the generator 
$S^{(\infty)}_f$
on $\widetilde E(L^2(\D))$,
\begin{equation}
  \label{eq:limsnu}
  \lim_{\nu \to \infty} \ep{-t S_f^{(\nu)}}(z,z') 
   = \Bigl(  \ep{-t S_f^{(\infty)}} \widetilde E \Bigr) (z,z') \, .
\end{equation}
\end{theorem}
\begin{proof}
  The proof heavily borrows from the strategy of \cite{BLW99} which is  
accommodated here to the case of unbounded $f$.
The key to the present generalization is the use of monotone convergence.
Fundamental to the proof is the well-known Feynman-Kac-It\^o formula 
\cite{BHL00} which 
expresses the continuous integral kernel of the semigroup 
$e^{-t S_f^{(\nu)}}$ as
\begin{multline}
  \label{eq:FKI}
  \ep{-t S_f^{(\nu)}}(z,z') = \int_{\{\TD > t\}}  \! \exp\Bigl\{
     \half \int_0^t \! \left[a(b(s)) {\ol{db(s)}} - \ol{a(b(s))} d b(s) 
                        \right] \\  
           - \int_0^t \! ds \, (\nu v + f)(b(s))   \Bigr\} \, 
   d\mu^{(\nu)}_{z,0;z',t} \,
\end{multline}
where in this case it does not matter whether the stochastic integral in the 
exponent
is interpreted in the It\^o sense
or according to Fisk and Stratonovich \cite{RoWi87}, because 
$\partial_1 a_1 + \partial_2 a_2 = 0$. 

For notational convenience we fix a reference diffusion constant $\nu_0 > 0$ 
and 
abbreviate for $\alpha \ge 0, w \in \C$
\begin{equation}
  \label{eq:etaw}
  \eta_w^{(\alpha)}(z) := \ep{- tS^{(\nu_0)}_{\alpha f}}(z,w) \, .
\end{equation}
As preparation for the main part of the proof we state some properties of 
$\eta_w^{(\alpha)}$.
Note that each $\eta_w^{(\alpha)}$ is a bounded and continuous function 
\cite[Theorem 4.1]{BHL00}
and lies in $L^2(\D)$, which follows from the inequality 
\begin{equation}
  \label{eq:etawbd}
  \bnorm{\eta_w^{(\alpha)}} \leq \sup_{w \in \D} \bnorm{\eta_w^{(\alpha)}} 
                             = \bnorm{\ep{-tS^{(\nu)}_{\alpha f}}}_{2,\infty} 
\end{equation}
where the last term is the finite operator norm of 
$e^{-tS^{(\nu)}_{\alpha f}}$ considered as
mapping from $L^2(\D)$ to $L^\infty(\D)$ \cite[Estimate (2.39)]{BHL00}.

Also note that $w \mapsto \eta_w^{(\alpha)}$ is a strongly continuous mapping, 
because of the identity 
$(\eta_w^{(\alpha)}, \eta_{w'}^{(\alpha)}) = 
 \exp\bigl(-2 t S^{(\nu)}_{\alpha f}\bigr) (w,w')$ 
and due to the continuity of the kernel \cite[Theorem 6.1]{BHL00}.

In addition, the mapping $\alpha \mapsto  \etwa$ is also strongly continuous 
because the 
integrand in \eq{FKI} can be estimated by 
$\exp\bigl(\int_0^t ds \, (- \nu v + \alpha_0 f^-)\bigr)$ 
with some 
large $\alpha_0$ and then dominated convergence applies. 

As the main part of the proof we show that
\begin{equation}
  \label{eq:etatok}
    \lim_{\nu \to \infty} \ep{-t S^{(\nu)}_f}(z,z') 
       = (\widetilde \kappa_z, \ep{-t S^{(\infty)}_f} \widetilde E \, 
  \widetilde \kappa_{z'})
\end{equation}
which by an analogue of Corollary \ref{lem:Tfint} 
in connection with Subsection \ref{sec:BTS}.\Alph{subsref} 
constitutes the continuous integral kernel for 
$\exp\bigl(-t S^{(\infty)}_f\bigr) \widetilde E$
on $L^2(\D)$.

To see \eq{etatok},
we use the semigroup property and rewrite the integral kernel as 
scalar product 
\begin{equation}
  \label{eq:eee}
  \ep{-t S^{(\nu)}_f}(z,z') 
      = \Bigl(\eta_z^{(\nu_0/\nu)} , 
          \exp\bigl(-t S_{(\nu - 2 \nu_0)f/\nu}^{(\nu - 2 \nu_0)}\bigr)
                        \eta_{z'}^{(\nu_0/\nu)}\Bigr)
\end{equation}
which converges in the limit $\nu \to \infty$ to
\begin{equation}
  \label{eq:eesee}
  \lim_{\nu \to \infty} \ep{-t S^{(\nu)}_f}(z,z')
      = (\eta_z^{(0)}, \ep{-t S^{(\infty)}_f} \widetilde E \, 
                 \eta_{z'}^{(0)}) \, .
\end{equation}
This can be deduced from the strong continuity of $\eta_z^{(\alpha)}$ 
in $\alpha$, the
strong convergence stated in Theorem \ref{prop:sgcv} together with the
uniform boundedness (according to the Banach-Steinhaus theorem)  
of the operators 
$\exp\bigl(-t S_{(\nu - 2 \nu_0)f/\nu}^{(\nu - 2 \nu_0)}\bigr)$.

To finish the proof, we observe that the right-hand side of \eq{eesee} 
is an integral kernel for $\exp(-t S^{(\infty)}_f)$ on $\widetilde E(L^2(\D)$
which is, in addition, continuous in
$z$ and $z'$ and therefore coincides with the right-hand side of \eq{etatok}.
The continuity of \eq{eesee} is clear, and with 
$\widetilde E \exp(-t S^{(\nu_0)}_0) = \widetilde E$
it can be checked that it indeed constitutes an integral kernel.
\end{proof}

\begin{corollary}
Combined with the Feynman-Kac-It\^o formula \eq{FKI}
and the identity
\begin{equation}
  \label{eq:esfetfk}
   \ep{-t T_f} (z,z') = %\bigl(\mbox{$\frac{g(z)}{g(z')}$}\bigr)^{1/2} \, 
                        \bigl( U^\dagger 
                                    \ep{-t S^{(\infty)}_f } 
                               \widetilde E \, U
                        \bigr)(z,z') \, , 
\end{equation}
the result \eq{limsnu} provides a probabilistic expression for the 
continuous integral kernel of the semigroup generated by $T_f$ on 
$\ol{\q(\t_f)}\subset \Bg$. 
This is the
\bemph{generalized Daubechies-Klauder formula}, which states that
for admissible $f$ and $g$ and for $t>0$, 
\begin{multline}
  \label{eq:DK}
  \ep{-t T_f}(z,z') 
    = \lim_{\nu \to \infty} 
       \sqrt{\mbox{$\frac{\textstyle g(z')}{\textstyle g(z)}$}}
       \int_{\!\{\TD > t\}} \! 
       \exp\Bigl\{  
            \half \! \int_0^t \! 
                \left[a(b(s)) {\ol{db(s)}} 
                      - \ol{a(b(s))} d b(s) \right] \\ 
           - \int_0^t \! ds \, (\nu v + f)(b(s))  
       \Bigr\} \, d\mu^{(\nu)}_{z,0;z',t} \, .
\end{multline}  
In particular, the choice $f=0$ yields the reproducing kernel of the 
weighted Bergman space
which $g$ characterizes. 
\end{corollary}

\begin{remarks} 
According to \cite{BLW98,BLW99}, due to the significance of $\nu$
the expression for the integral kernel of the Berezin-Toeplitz semigroup 
in \eq{DK} 
is called an \bemph{ultra-diffusive limit}.
By rescaling the Brownian bridge as in \cite[Equation (11)]{BLW99}, one may 
alternatively
restate \eq{DK} as a long-time limit. Another version of the result pointed 
out there
implies that the integrand in \eq{DK} can be re-expressed in terms of a 
complex It\^o-stochastic integral. 
All these versions will be omitted here.

Reading from the right to the left, \eq{DK} can be interpreted as a 
quantization
formula, which constructs from the functions $f$ and $g$ the semigroup 
generated by 
$T_f$ and thereby specifically selects the relevant Hilbert-space 
$\ol{\q(\t_f)}$ on which 
$T_f$ is properly defined as a self-adjoint operator. In the context of 
quantization, $\ln g$
is interpreted as a K\"ahler potential on $\D$. To our knowledge, without 
additional symmetry 
requirements the setting considered here still lacks the proof of a 
correspondence 
principle \cite{BMS94,Eng95}, for which probabilistic techniques might 
provide helpful tools. 
\end{remarks}


\begin{examples}
Now we revisit the examples from Section \ref{sec:af} again and
list the functions $a$ and $v$ belonging to each
weight function $g$ considered there. 
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
No. & $\D$ & $g(z)$ & $a(z)$ & $v(z)$ \\ \hline
 1. & $\C$ & 
        \teqbox{\frac{1}{\pi}\,e^{- \abs z^2}} & 
        \teqbox{i z}   & 
        $ -2 $ \\[2mm] \hline

2. & $\C$ & \teqbox{\frac {2j+1} \pi  (1 + \abs z^2)^{-2j-2}}   &
         \teqqbox{\frac{2(j+1) i z}{1 + \abs z^2}} &
         \teqqbox{ - \frac{4(j+1)}{(1+\abs z^2)^2}} \\[2mm] \hline

3.$a$) & {\footnotesize $\{\abs z^2<1\}$} &
\teqbox{ \frac{2k-1} \pi (1 - \abs z^2)^{2k-2}} & 
\teqqbox{\frac{2(k-1) i z}{1 - \abs z^2}} &
\teqqbox{- \frac{4(k-1)}{(1-\abs z^2)^2}} \\[2mm] \hline 
3.$b$) & $\C$ &
\teqqqbox{\frac{2 \abs z^{2k-1}}{\pi \Gamma(2k)}  K_{2k-1}(2\abs z)} & 
\teqqbox{\frac{Z_{2k-2} \, i z}{Z_{2k-1}}} &
\parbox{3.6cm}{\small
$ -\frac{2}{Z^2_{2k-1}} \, 
   \bigl( Z_{2k-2} Z_{2k-1} \, +$ 
$\abs z^2 (Z_{2k-2}^2 - Z_{2k-1} Z_{2k-3}) \bigr) $}
\\[2mm] \hline  
\end{tabular}
\end{center}
In the last row we used the abbreviation 
$Z_\sigma := \abs z^\sigma K_\sigma(2 \abs z)$.
\end{examples}

The following remarks examine the validity of equation \eq{DK} for each case.

%\pagebreak[3]
\begin{remarks}
  The primary condition for the validity of \eq{DK} in a specific situation
  is that the weight function leads to admissible $a$ and $v$. If this
  is satisfied, any Kato-decomposable symbol $f$ allows a probabilistic
  expression for the semigroup generated by $T_f$.

  By inspection we decide whether each example can be used in the formula 
\eq{DK}.
  For the examples 1 and 2 the function $a$ is continuous and $v$ is even 
bounded, 
  hence they admit the probabilistic representation according to \eq{DK}. 

  Unfortunately, for the example 3.$a$) either $k=1$ and Proposition 
\ref{thm:nrubg}
  is not satisfied or the scalar potential $v$ is not Kato decomposable
  because of a strong negative singularity towards the boundary of the disc.
  Therefore the probabilistic representation is not valid in either case. 
  However, the desired monotone form convergence can be recovered with the 
modification 
  of the domain suggested in Remark \ref{rem:Rmod} which is associated with  
  Neumann boundary conditions.

  On the other hand, the vector and scalar potentials $a$ and $v$ emerging 
from 
  the Barut-Girardello representation 3.$b$) are admissible for $k>1$, which 
  can be read off from the asymptotics: 
  $\lim_{r \to 0} r^{\abs \sigma} K_\sigma(2r) 
     = \half \int_0^\infty  t^{-\sigma-1} e^{-1/t} \, dt $
  for $\sigma \neq 0$, and  
  $\lim_{r \to 0} K_0(2r) / \ln 2r = -1$.

  Within the setting of example 1 the identity \eq{DK} is in essence a result 
by \cite{DaKl85}. 
  The identity corresponding to example 2 has already been worked out in 
  \cite{BLW99}, where it is also compared to a similar formula derived in 
\cite{DaKl85}.
  These last two alternatives are among the few known ways to obtain a 
mathematically well-founded 
  path integral for spin. The weighted Bergman space of example 3.$b$) 
was not previously known to admit a 
  formula of type \eq{DK}, which can now serve as an alternative to 
path-integral formulas
  that do not contain genuine path measures \cite{IKG92,FuFu97}.
\end{remarks}

\section{Conclusion}

In this paper we have tried to indicate the key principles behind the
construction of Daubechies and Klauder and have thus derived a natural
generalization.  As a major spin-off we developed criteria for
self-adjointness of Berezin-Toeplitz operators. Once the relation to
Schrödinger operators is established, this is an immediate consequence.

As to further ramifications, we point out that with a suitable
analyticity argument one could obtain from \eq{DK} the probabilistic
expression for the unitary group $e^{-i t T_f}$ which was a primary
motivation for \cite{DaKl85,DaKl86}.  It might also be interesting to
investigate random Berezin-Toeplitz operators, for which the
probabilistic representation seems to offer an appropriate analytic
framework.  Finally, it deserves mentioning that the concept of path
transformations is also applicable in the context of \eq{DK} in order
to relate the resolvents of certain Berezin-Toeplitz
operators \cite{Bod}.\\

\begin{acknowledgement} \small
It is a pleasure to thank Hajo Leschke for his 
scientific guidance and Simone Warzel for her valuable criticism and 
participation 
in the struggle for the clear picture. Thanks are extended to John R. Klauder 
for encouragement, inspiration and lots of resourceful advice. 
I am also indebted to Kazuyuki Fujii for drawing my attention to the 
Barut-Girardello 
representation. The Studienstiftung des deutschen Volkes is acknowledged 
for financial 
support. \end{acknowledgement}




%%%%%%%%%%% bibliography %%%%%%%%%%

\renewcommand\baselinestretch{1}
\small

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\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   THE END
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Berezin-Toeplitz semigroups}

\begin{lemma}
  If $\q(\r)$ contains $U(\Bg)$, then 
   and $\lim_{\nu \to \infty} e^{t\nu R} \psi = U K U^\dagger \psi$. 
\end{lemma}
\begin{proof}
  Spectral theorem, dominated convergence.
\end{proof}

\begin{appendix}
\section{Calculations for the Barut-Girardello representation}

Throughout this section we will use polar coordinates $r=\abs z$
and abbreviate $\nu := 2k -1$. Define
\begin{equation}
  \label{eq:Ldef}
  L_\nu (r) := 2 r^\nu K_\nu(2r) = \int_0^\infty dt \, t^{-\nu - 1} e^{-r^2t - 1/t} 
\end{equation}
then by the recursion $\partial_r L\nu(r) = - 2 r L_{\nu - 1}(r)$ we obtain for
\begin{equation}
  \label{eq:DeltalnL}
  v = \Delta \ln \sqrt g =
  \frac 1 {2r} \partial_r r \partial_r \ln L_\nu 
  = 2 L_\nu^{-2} (r^2 L_{\nu - 2} - L_\nu L_{\nu - 1} - r^2 L_{\nu-1})                          
\end{equation}
>From the asymptotics

we conclude that $v$ is well-behaved if $k,\nu > 1$.
\end{appendix}

\begin{lemma} \label{lem:Tfintkern}
Under the assumptions of the preceding theorem and supposing $\t_f(\kappa_z)$ is well-defined for each $z$, 
then the self-adjoint, semibounded operator $T_f$ possesses an integral kernel
\begin{equation}
  T_f(z,z') = \int_\D \dl f \ol{\kappa_z} \kappa_{z'}
\end{equation}
which means $T_f \psi(z') = \int_\D \dl(z) g(z) \, T_f(z',z) \psi(z)$
whenever $\psi$ is in the domain of $T_f$.
\end{lemma}
\begin{proof}
  First we define the shifted quadratic form 
$\tilde \t$ by $\tilde \t(\psi) = \t(\psi) - c \norm{\psi}^2$
which is positive semidefinite. By the Cauchy-Schwarz inequality we obtain the estimate 
\begin{equation}
  \label{eq:CS}
  \abs{\tilde \t(\kappa_{z'},\kappa_z)}^2 \leq \tilde \t(\kappa_z) \tilde \t(\kappa_{z'})
\end{equation}
and the right-hand side is finite by assumption. As a consequence, $\t(\kappa_{z'},\kappa_z)$ 
is also finite.

To show that in fact $\t(\kappa_z, \kappa_{z'})$ is a kernel for $T_f$ we first observe that if $\psi$
is in the domain of ${T_f}$ then it is also in the form domain. 

Using the point evaluation property \eq{RKIP} and the fact that $\kappa_z$ is in the form domain yields
\begin{equation}
  (T_f \psi)(z) = (\kappa_z, T_f \psi) = \t(\kappa_z, \psi) \, ,
\end{equation}
which is explicitly written as
\begin{align}
  \label{eq:TfK}
  \lefteqn{\t(\kappa_z, \psi) =  \int_\D \dl(z') g(z')\,  \ol{\kappa_z(z')} f(z') \psi(z') } &\\
 &=  \int_\D \dl (z') g(z') \, 
      \int_\D \dl(w) g(w) \, \ol{\kappa_z(z')} f(z') \kappa_{z'}(w) \psi(w) \label{eq:itint}\\
 &=  \int_\D \dl(w) g(w) T_f(z,w) \psi(w) \, .
\end{align}
The last step uses Fubini's theorem
which is justified by the existence of the iterated integrals in \eq{itint}.
\end{proof}
---------------9910311459165--
