\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 71, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/71\hfil Eigenfunction expansions]
{Convergence of generalized eigenfunction expansions}

\author[M. Sakata\hfil EJDE-2007/71\hfilneg]
{Mayumi Sakata}  

\address{Mayumi Sakata \newline
William Jewell College, 
500 College Hill, Box 1108,
Liberty, MO 64068-1896, USA}
\email{sakatam@william.jewell.edu}

\thanks{Submitted March 6, 2007. Published May 15, 2007.}
\subjclass[2000]{46L10, 47E05, 47F05, 47B25, 11F25, 11F03}
\keywords{Generalized eigenfunction expansion;
 Generalized eigenprojection; \hfill\break\indent
 Fourier transform; differential operators,
 Hecke operators; modular group}

\begin{abstract}
 We present a simplified theory of generalized eigenfunction expansions
 for a commuting family of bounded operators and with finitely many
 unbounded operators. We also study the convergence of these expansions,
 giving an abstract type of uniform convergence result, and illustrate
 the theory by giving two examples:  The Fourier transform on Hecke
 operators, and the Laplacian operators in hyperbolic spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

A generalized eigenfunction expansion is a generalization of the
Fourier transform. Just as the Fourier transform in higher
dimensions may be regarded as an expansion for the functions in
the domain of the self-adjoint operators associated with
\[
\big\{  i\frac{\partial}{\partial x_{j}}\big\} _{j=1}^{n},
\]
it is possible to study generalized eigenfunction expansions for
families of commuting operators, not just a single operator. In
this paper, we develop  an alternative approach to such expansions
for a commuting family of operators concentrating on questions of
convergence. Instead of using the spectral projections for
families of commuting operators, which appear in the spectral
theorem for such families arising ultimately from the
Gelfand-Naimark representation theorem, we use limits (in a
topological vector sense) of such projections to produce what we
call generalized eigenprojections; hence spectral properties of
the operators are automatically inherited by these generalized
eigenprojections.

Generalized eigenfunction expansions are widely used in
mathematical physics.  Also, many integral expansions, including
some occurring in analytic number theory, are generalized
eigenfunction expansions. There is much literature on this
subject. To name a few, the classical literature is anchored by
Gelfand and Vilenkin in \cite{Gel}, Berezhanskii in \cite{Ber},
and Maurin in \cite{Maurin}. The first modern paper on the
foundations of generalized eigenfunction expansions with an
application in mathematical physics was given by Simon in
\cite{Simon} in 1982. Also, Poerschke, Stolz, and Weidmann in
\cite{PSW} gave a simplified version of generalized eigenfunction
expansions for a single self-adjoint operator with an application
in mathematical physics. One difference between the classical
literature and ours is that we define a generalized
eigenprojection as a limit of the products of spectral projections
and some real numbers in a specific space (Definition
\ref{geprojection}) and use it to expand the operators, thus it
gives some sort of uniform convergence of the expansion in the
specific space. The easiest way to illustrate is to use the
Fourier series. The Fourier series of any $L_2$ function converges
in $L_2$, which is the type of convergence studied by other
authors. However, if the function being expanded lies in the
Sobolev space $W_2^1$ for example, then the series converges
uniformly and error estimates may be given, which apply on the
entire unit ball of $W_2^1$. The convergence results of this paper
are obtained in the same way as differentiability hypothesis are
needed to guarantee uniform convergence of usual Fourier series.
As a continuation and with adding more conditions, we are also
able to extend some result in analytic number theory, which is
given in Section \ref{sec:nt} (\cite{paper2}). Now a difference
between a modern approach, namely Poerschke, Stolz, and Weidmann's
approach, and ours is that we expand a commuting family of
operators rather than a single self-adjoint operator. Although
their approach also gives some asymptotic behavior of the
eigenfunctions, which does not appear to have been considered by
earlier authors, their approach is not extended to a family of
operators. Since our approach is for a family of operators, we
concentrate more on absolute convergence of the integral involved,
and as a result, we obtained the same sort of uniform convergence
of the integrals as by the Fourier transform on a set of functions
lying in $L_1$. We shall compare the previous approach and ours
more in Section \ref{sec:comp}.

In order to consider the convergence of the integral, and because
the  formalism surrounding these expansions can be confusing, we
develop in this paper a formalism for generalized eigenfunction
expansions which proceeds from the spectral theorem in exactly the
way that the usual Fourier transform may be derived from the
spectral decomposition of the operator
\begin{equation} \label{e1}
i\frac{d}{dx}
\end{equation}
in $L_{2} (  \mathbb{R}) $. We obtain a theory analogue to the
theory of the inverse Fourier transform. The purpose of this is to
simplify the construction of the expansion and also to obtain new
results about its convergence.

For the above operator \eqref{e1}, the spectral theorem gives a
projection valued measure
\[
\Delta\to E(\Delta)
\]
in $L_{2}(\mathbb{R})$, where $\Delta$ is a Borel set. Here it is
possible to calculate the projections $E( \Delta)$; they are the
inverse images of characteristic functions under the Fourier
transform. Clearly, for any $\lambda$,
\[
\lim_{n\to\infty}E(\lambda+\frac{1}{n},\lambda-\frac{1} {n})  =0
\]
in the strong operator topology on $L_{2}(\mathbb{R})  $. However,
if
\[
\chi(\lambda-\frac{1}{n},\lambda+\frac{1}{n})
\]
is the characteristic function, it is clear that in the tempered
distributions
\[
n\chi(\lambda-\frac{1}{n},\lambda+\frac{1}{n})  \to
2\delta(\lambda)  ,
\]
the point measure at $\lambda$. It follows that
\[
n\chi(\lambda-\frac{1}{n},\lambda+\frac{1}{n})
\]
converges in the distributions to the function $e^{-i\lambda x};$ the complex
conjugate is because the embedding of functions into distributions contains a
conjugation. In other words, the associated eigenfunction is actually
$e^{-i\lambda x}$.

In this fashion, our approach derives the Fourier transform from a
realization of the Gelfand transform of the smallest closed
subalgebra of $B(\mathfrak{h})  $ (the set of bounded operators
from $\mathfrak{h}$ to $\mathfrak{h}$) containing the translation
operators. The inverse Fourier transform is a generalized
eigenfunction expansion.

One of the most important questions we study is ``how does the
expansion converge?''. Although a variety of approaches exist for
the derivation of such expansions, our results on convergence
appear to be new. We need,  in general, more hypothesis than in
Poerschke, Stolz, and Weidmann to obtain convergent integrals and
the eigenprojections. This is to be expected because, for the
Fourier transform, the theory must guarantee that the transform is
in $L_1$.

As in previous approaches, we also take the point of view that it
expands operators instead of functions. As we mentioned above, the
inverse Fourier transform is an example of an expansion of the
identity operator (we shall explain in Section 4.1). If one takes
this point of view, the obvious question is ``how closely does the
expansion approximate the operator?''. This question is the basis
for the theory of approximation numbers of operators. In the case
of Sobolev spaces of functions on compact sets, the operator in
question is often an embedding map from the Sobolev space into
$L_{2}$. The approximation numbers are often calculated using
Fourier series. For this purpose, it is necessary to have uniform
estimates of the form ``any element of the unit ball in the
Sobolev space may be approximated in $L_{2}$ to within an accuracy
of $\varepsilon$ by $n(\varepsilon)  $ terms of its Fourier
series''. This will be given in Corollary \ref{maincor} in Section
3.3.

To illustrate the theory, we give two examples in the last
section;  one is the Fourier transform, i.e., we apply Theorem
\ref{mainforjointsp} to the Laplacian on
$C_{0}^{\infty}(\mathbb{R})$, and the other is an application to
analytic number theory.


\section{Background}



The basic idea of a generalized eigenfunction expansion follows from the next
theorem.

\begin{theorem}\label{base}
Let $\mathfrak{X}$ be a compact Hausdorff space. Let $\mu$ be a positive
Borel measure on $\mathfrak{X}$. Let $f \in C(\mathfrak{X})$. Let $T_f :L_2
(\mathfrak{X}, \mu ) \to L_2 (\mathfrak{X}, \mu )$ be defined by
$$
T_f (g) = f\cdot g.
$$
Then there exists a projection valued measure $E$ on the Borel subsets of $\mathfrak{X}$
such that
\begin{equation}\label{e2}
T_f =\int_\mathfrak{X} f \, dE.
\end{equation}
Recall that this equation  is an abbreviation for
$$
(T_f g, h) = \int_\mathfrak{X} f  \, dE_{g, h}
$$
where $E_{g, h}(\Delta ) = (E(\Delta)g, h)$ for Borel set $\Delta$
(see \cite{RudinF}).
Also, if $\mathcal{B} = \{ T_f : f \in C(\mathfrak{X})\}$, then, $E(\Delta )$
commutes with
the set $\mathcal{B}'$ of all bounded linear transformations $Q$
taking $L_2 (\mathfrak{X}, \mu )$
into itself such that $Q$ commutes with $\mathcal{B}$.
\end{theorem}

\begin{proof}
Let $\hat{\Delta}$ be a Borel set in the range of $f$. Then let
$\Delta = f^{-1} \hat{\Delta})$, and define
$E(\Delta) =\chi_{\Delta}$ where $\chi_{\Delta}$ is a characteristic function
on $\Delta$. Then $E$ is a projection valued measure and
$$
\int_{\mathfrak{X}} f \, dE_{g, h} = \int_{\mathfrak{X}} fg\overline{h} \, d\mu,
$$
hence the theorem follows.
\end{proof}

The above theorem is very elementary and shows the existence of
the expansion (\ref{e2}) for a multiplication operator; and thus
for a family of multiplication operators (see Theorem
\ref{multi}). Our aim is to get the expansion for a commuting
family of normal operators. The idea of the process is to map
operators into a compact Hausdorff space (so that the operators
turn into multiplication operators), get the expansion there, and
pull it back to its original space.

Before we move on, let us state two corollaries follow from the above theorem.

\begin{corollary}\label{bcor1}
$\mathcal{B}' = \{ T_f : f \in L_{\infty} \}$.
\end{corollary}

\begin{corollary}\label{bcor2}
$\{ T_f : f\in L_{\infty} \}$ is a von Neumann algebra.
\end{corollary}

\section{Generalized Eigenfamily}

We first give some definitions and then begin the process of obtaining
generalized eigenfunction expansions from Theorem \ref{base}.

%\subsection{Definitions}

\begin{definition}\label{topology} \rm
Let $W$ be a locally convex topological vector space and $W'$ be its dual space with
the weak*-topology. Denote $C(W, W')$ as the set of continuous conjugate
lineartransforms from $W$ into $W'$. We shall topologize $C(W, W')$ using
sub-base open sets about $0$, which are of the form
$\Theta_{xV} := \{ A : A(x) \subset V \}$,
where $x \in W$ and $V$ is a neighborhood in $W'$. With this topology,
$C(W, W')$ is a locally convex topological vector space.
\end{definition}

\begin{definition}\label{geprojection} \rm
Let $W, W'$, and $C(W, W')$ be defined as in Definition \ref{topology}.
Let $\mathcal{H}$ be a Hilbert space such that $W \subset \mathcal{H} \subset W'$ all dense.
Denote $\mathcal{D}$ as a commuting family of normal operators $\{ A
\}$ on $\mathcal{H}$ such that the restriction of $A$ to $W$ takes $W$ into $W$
continuously.
Suppose there is a commutative von Neumann algebra $\mathcal{A}$ with which
$\mathcal{D}$ is affiliated, i.e., for every $A \in \mathcal{D}$, the
spectral projections of $A$ are in $\mathcal{A}$.
Then a \emph{generalized eigenprojection} for
$\mathcal{D}$ is an operator $Q \in C(W, W')$ ($Q \not\equiv 0$), with
the following properties.
\begin{enumerate}
\item There exist a sequence $\{ P_n \}$ of projections in
$\mathcal{A}$ and a sequence $\{ r_n \}$ of real numbers such
that $r_n P_n$ converges to $Q$ in $C(W, W' )$;
\item For each $A \in \mathcal{D}$, there exists $\lambda_A \in \mathbb{C}$
with the property that for every $\epsilon >0$, there exists
$N \in \mathbb{N}$ such that $E(A, \lambda_A ,
\epsilon ) P_n = P_n$ for any $n >N$, where $E(A, \lambda_A , \epsilon )$
 is the spectral projection for $A$ corresponding to
$\{ y : |y-\lambda_A |<\epsilon \}$.
Note that we fix one $\lambda_A$ for each $A$ to get a
generalized eigenprojection.
\end{enumerate}
\end{definition}

\begin{remark} \label{remk3.3} \rm
The properties of a generalized eigenprojection in the above definition
indicate that a generalized eigenprojection is basically the limit
of spectral projections for all $A \in \mathcal{D}$.
\end{remark}

\begin{definition}\label{gefunction} \rm
An element of the range of a generalized eigenprojection is called a
\emph{generalized eigenfunction} of $\mathcal{D}$, and $\lambda_A$
is called a \emph{generalized eigenvalue} of $A$.
\end{definition}

The above definitions of a generalized eigenfunction and eigenvalue
follow from the next theorem.

\begin{theorem} \label{thm3.5}
If $Q$ is a generalized eigenprojection for $\mathcal{D}$, then for
every $A \in \mathcal{D}$ and any element $\psi$ in the range
of $Q$,
$$
\hat{A}^t \psi = \lambda_A \psi
$$
where $\hat{A} = A|_W$ and $\hat{A}^t$ is the transpose of $\hat{A}$.
\end{theorem}

\begin{proof}
By the definition of a transpose, we have
$\hat{A}^t \psi (\theta) = \psi (\hat{A}(\theta ))$ where $\theta \in W$.
Since $Q \in C(W, W' )$ and $\psi$ is in the range of $Q$, there exists
$\phi\in W$ such that $Q(\phi) = \psi$. Hence
 $ \hat{A}^t \psi (\theta ) = Q(\phi ) (\hat{A} (\theta ))$.
Since $r_n P_n \to Q$ by the
definition of $Q$, we have
\begin{equation}\label{eqn:limit}
(\hat{A} (\theta ),  r_n P_n (\phi)) \to Q(\phi )(\hat{A}
(\theta)).
\end{equation}
Let $\epsilon >0$. Then there exists $N \in \mathbb{N}$ such that $E(A, \lambda_A , \epsilon )
P_n = P_n$ for $n >N$. Hence
$$
(\hat{A} (\theta ),  r_n P_n (\phi )) = (\hat{A}(\theta ),\, r_n E(A, \lambda_A, \epsilon
)P_n (\phi )) = (\hat{A} E(A,\lambda_A , \epsilon) (\theta ),  r_n P_n (\phi ))
$$
since $\hat{A}, P_n, E(A, \lambda_A, \epsilon )$ commutes (because they are all in $\mathcal{A}$). By
the spectral theorem (\cite{RudinF}), we have
$$
(\hat{A} E(A, \lambda_A , \epsilon) (\theta ),  r_n P_n (\phi )) = \int t\,
dF_{E(\hat{A}, \lambda_{\hat{A}}, \epsilon) (\theta), r_n P_n (\phi )} = \int_{|t-\lambda_A | < \epsilon}
t \, dF_{\theta, r_n P_n (\phi )}.
$$
We now use the integration by parts and then let $n$ goes to infinity. Since $\epsilon$
is arbitrary, we get
$$
(\hat{A} (\theta ),  r_n P_n (\phi )) = (\hat{A} E(A, \lambda_A
, \epsilon) (\theta ),  r_n P_n (\phi )) \to \lambda_A (Q(\phi
))(\theta ).
$$
Hence by (\ref{eqn:limit}), we get $
\hat{A}^t \psi (\theta ) = Q(\phi )(\hat{A}(\theta )) = \lambda_A (Q(\phi ))(\theta
)= \lambda_A \psi (\theta )$.
\end{proof}

\begin{definition} \label{gefamily} \rm
Let $\mathcal{H}, W, \mathcal{D}, \mathcal{A}$ be given as above. Let $\mathfrak{X}$ be a
locally compact Hausdorff space with a positive measure $\mu$.
Denote $H$ as an isometric isomorphism from $\mathcal{A}$ to $L_\infty
(\mathfrak{X}, \mu)$ such that $H(A) = f \in C(\mathfrak{X} ) \cap L_\infty
(\mathfrak{X}, \mu)$ if $A \in \mathcal{D}$. Then a
\emph{generalized eigenfamily for $W, \mathcal{D}, \mathfrak{X}$} is a
function $g : \mathfrak{X} \to C(W, W')$ ($g(x) = Q_x$) such that
\begin{enumerate}
\item $g$ is continuous;
\item for any $A \in \mathcal{D}$,
$A = \int \overline{H(A)} Q_x \, d\mu (x)$;
where the integral converges in $C(W, W')$;
\item $Q_x$ is a generalized eigenprojection of $\mathcal{D}$ with the generalized
eigenvalue $H(A)(x)$ (note that this is for each fixed $x \in \mathfrak{X}$).
\end{enumerate}
The integral expansion in 2 is called the
\emph{generalized eigenfunction expansion}.
\end{definition}

\begin{remark} \label{remark} \rm
If $A \in \mathcal{D}$ is bounded and affiliated with $\mathcal{A}$, then $A \in \mathcal{A}$ hence
$\mathcal{D} \subset \mathcal{A}$, because $A$ is a limit of spectral projections by
the spectral theorem. Also, if $\mathcal{D} \subset \mathcal{A}$ and $\mathfrak{X}$ is a maximal ideal
space of $\mathcal{A}$ in Definition \ref{gefamily}, then $H$ is the Gelfand transform and
thus $H(A)$ is continuous.
\end{remark}

We must now show the existence of a generalized eigenfamily in order to get
some result in convergence of a generalized eigenfunction expansion.
The key point is to construct the space $W$ so that the expansion converges
in $C(W, W')$. In the next section, the existence for a commuting family
of bounded operators will be given, where $\mathfrak{X}$ in this case will
be a certain subset of the maximal ideal
space of the family of bounded operators.

\subsection{Existence of a Generalized Eigenfamily for Bounded Operators}

Let us first show the existence of a generalized eigenfamily for a family of
multiplication operators (analogue to Theorem \ref{base}).

\begin{theorem}\label{multi}
Let $\mathfrak{X}$ be a locally compact metric space and $\mu$ be a
positive  measure on $\mathfrak{X}$. Let $\mathcal{D} = \{ T_f : f \in
C(\mathfrak{X}) \cap L_\infty (\mathfrak{X}, \mu)\}$ where $T_f (g) = f \cdot g$
for any $g \in L_2 (\mathfrak{X}, \mu)$. Let $W = C(\mathfrak{X})\cap L_2
(\mathfrak{X}, \mu)$. Then there exists a generalized eigenfamily for $W,
\mathcal{D}, \mathfrak{X}$.
\end{theorem}

\begin{proof}
In this case, we have a Hilbert space $\mathcal{H} := L_2 (\mathfrak{X}, \mu)$
and the von Neumann algebra $\mathcal{A} := \mathcal{D} '$ by Corollary
\ref{bcor2}. Define $H:\mathcal{A} \to L_\infty (\mathfrak{X}, \mu)$ such that
$H(T_f) = f \in L_\infty (\mathfrak{X}, \mu)$. Then $H$ is an isometric
isomorphism. Notice that, if $T_f \in \mathcal{D}$, then $H(T_f) = f
\in C(\mathfrak{X} ) \cap L_\infty (\mathfrak{X}, \mu)$. Now, let $x \in \mathfrak{X}$.
Define $\hat{x} : W \to \mathbb{C}$ such that $\hat{x} (\phi) = \phi( x
)$. Define also $g :\mathfrak{X} \to C(W, W')$ such that $g (x ) = Q_{x}$
where $Q_{x } (\phi) = \overline{\phi(x )} \hat{x}$, for any $\phi
\in W$. Notice $Q_{x} (\phi) \in W'$, i.e., for any $\psi \in W$,
$(Q_{x} (\phi))(\psi) = \overline{\phi(x )} \hat{x } (\psi ) =
\overline{\phi(x )} \psi(x )$. We claim that $g$ is a generalized
eigenfamily for $W, \mathcal{D}, \mathfrak{X}$. In order to show the claim,
we must show
\begin{itemize}
\item[(a)] $g$ is continuous;
\item[(b)] for $A \in \mathcal{D}$,
\begin{equation}\label{eqn:A}
A = \int \overline{H(A)} Q_{x} \, d\mu ;
\end{equation}
\item[(c)] each $Q_{x}$ is a generalized eigenprojection of $\mathcal{D}$,
corresponding to the eigenvalue $H(A)(x)$.
\end{itemize}
(a) Let $\epsilon >0$. Let $V$ be a neighborhood of $0$ in $W'$,
i.e., for $h \in W$,
$$
V=\{ F \in W' : |F(h)|<\epsilon \}.
$$
Let $f \in W$. Let $Z:=\theta_{fV}$ be the sub-base open set about $0$ in
$C(W, W')$, i.e.,
$$
Z  =\{ P \in C(W,W') : P(f) \subset V \}
  = \{ P \in C(W,W') : |P(f)(h)|<\epsilon \}.
$$
Let $x \in \mathfrak{X}$. Since $f, h \in W = C(\mathfrak{X}) \cap L_2 (\mathfrak{X},\mu )$, there exists
$Y \subset \mathfrak{X}$ such that, for any $y \in Y$,
$$
|\overline{f(y)}h(y) - \overline{f(x)}h(x)|<\epsilon
\Rightarrow |g (y)(f)(h) - g
(x)(f)(h)|< \epsilon \Rightarrow g (y) - g (x) \in Z.
$$
Hence $g$ is continuous.

(b) First recall that (\ref{eqn:A}) is an abbreviation for
$$
(\psi,\, A(\phi) ) = \int \overline{H(A)} (Q_x (\phi ))(\psi ) \,
d\mu \quad \text{for } \phi, \psi \in W.
$$
Since $A\in \mathcal{D} \subset \mathcal{A}$, there exists
$f \in C(\mathfrak{X}) \cap L_{\infty} (\mathfrak{X}, \mu )$
such that $A=T_f$, i.e., $A(\phi)= T_f (\phi) = f\cdot \phi$
and $H(A)=f$.
Thus we get
$$
(\psi,\, A(\phi )) = \int \psi (x ) \overline{A(\phi)(x )} \, d\mu
= \int \psi (x ) \overline{f(x )} \overline{\phi(x )} \, d\mu .
$$
Also,
$$
\int \overline{H(A)} (Q_{x}(\phi))(\psi )\, d\mu = \int
\overline{f(x)}\overline{\phi(x)}\psi (x)\, d\mu
$$
Hence the result follows.

(c) We must show, for each fixed $x \in \mathfrak{X}$,
\begin{itemize}
\item[(i)] there exists $\{ P_n \}$ of projections in $\mathcal{A}$ and $\{ r_n \}$
of real numbers such that $r_n P_n \to Q_{x }$ in $C(W,W')$,
\item[(ii)] for any $\epsilon >0$, and for each $A \in \mathcal{D}$, there
is a spectral
projection $E(A, \lambda_A , \epsilon)$ such that there exists $N \in \mathbb{N}$ with
$E(A, \lambda_A , \epsilon)P_n = P_n$ for any $n > N$, where $P_n$ is the projection
from i and $\lambda_A = H(A)(x)$.
\end{itemize}
(i) By Corollary \ref{bcor1} and \ref{bcor2}, we have
$\mathcal{A} = \{T_f : f \in L_{\infty} (\mathfrak{X}, \mu ) \}$. Let $\{ \Delta_n \}$ be
the sets containing $x$ such that $\Delta_{n+1}\subset \Delta_n$
for any $n \in N$. (Such sets exist since $\mathfrak{X}$ is a metric
space.) Let $P_n = T_{\chi_{\Delta_n}}$. Then $P_n \in \mathcal{A}$ and
$P_n (\phi) = T_{\chi_{\Delta_n}} (\phi) = \chi_{\Delta_n}\cdot
\phi$. Let $r_n = \frac{1}{\mu (\Delta_n )}$. Then we have
\begin{align*}
(\psi,\, r_n P_n (\phi))
 &= (\psi,\, r_n \chi_{\Delta_n} (\phi)) \\
 &= \frac{1}{\mu (\Delta_n )} \int_{\Delta_n} \psi (y) \overline{\phi(y)} \, d\mu \\
 &= \frac{1}{\mu (\Delta_n )} \int_{\Delta_n} \psi (x ) \overline{\phi(x )} \,
 d\mu + \frac{1}{\mu (\Delta_n )} \int_{\Delta_n} (\psi (y) \overline{\phi(y)}
- \psi (x )
\overline{\phi(x )} )\, d\mu.
\end{align*}
The first integral is
$$
\frac{\psi (x )\overline{\phi(x )}}{\mu (\Delta_n )} \int_{\Delta_n }
\, d\mu = \frac{\psi (x )\overline{\phi(x )}}{\mu (\Delta_n)} \mu
(\Delta_n ) = \psi (x ) \overline{\phi(x )}.
$$
Since $\mathfrak{X}$ is a metric space, for any $\epsilon >0$, there
exists $N \in \mathbb{N}$ such that $|\psi (s) \overline{\phi(s)} - \psi
(x ) \overline{\phi(x )}|< \epsilon$ for any $s \in \Delta_n$ for
$n >N$. Hence the second integral goes to zero as $n \to \infty$.
Thus $(\psi,  r_n P_n (\phi) ) \to \psi (x ) \overline{\phi(x )}
= [Q_x(\phi)](\psi)$, i.e., $r_n P_n \to Q_{x }$ as $n \to
\infty$.

(ii) Again, if $A \in \mathcal{D}$, then there exists $f \in C(\mathfrak{X}) \cap L_{\infty}
(\mathfrak{X}, \mu )$ such that $A=T_f$ and $\lambda_A:= H(A)(x) = f(x)$. Let $\epsilon >0$
arbitrary. Let $E(A, \lambda_A ,\epsilon )$ be the spectral projection of $A$ to $\{ y :
|y-\lambda_A |<\epsilon \}$. We need to show $\exists N$ such that $E(A, \lambda_A ,
\epsilon )P_n = P_n$ for any $n>N$. We have
$$
E(A, \lambda_A, \epsilon )P_n (\phi) = E(A,
\lambda_A, \epsilon )T_{\chi_{\Delta_n}} (\phi) = E(A, \lambda_A , \epsilon )
\chi_{\Delta_n} \phi .
$$
 Pick the smallest integer $N$ such that $f(\Delta_N ) \subset
\{ y : |y-\lambda_A | <\epsilon \}$.
Then we get $E(A, \lambda_A ,\epsilon ) P_n = P_n$ for
any $n >N$.
\end{proof}

\begin{remark} \label{rmk3.9} \rm
Instead of using a locally compact metric space $\mathfrak{X}$ in
Theorem \ref{multi}, we can use a locally compact Hausdorff space.
However, in that case, we must assume
some additional properties for $\mathfrak{X}$ and $\mathcal{D}$ as follows:
\begin{enumerate}
\item for any $x \in \mathfrak{X}$, there exists $\{ \Delta_n \}$ of sets containing $x$
such that $\Delta_{n+1} \subset \Delta_n$ for any $n \in \mathbb{N}$;
\item for any $\epsilon > 0$ and for any $\phi, g \in W$,  there exists $N$
such that $|\phi (s) \overline{g(s)} - \phi (\lambda )
\overline{g(\lambda )}| < \epsilon$ for any $s \in \Delta_n$ for $n>N$;
\item for the sets $\{ \Delta_n \}$ from $1$, $\sup_{x, y \in \Delta_n } |f(x) - f(y)|<
\frac{1}{n}$ for any $f$ such that $T_f \in \mathcal{D}$.
\end{enumerate}
\end{remark}

To show the existence for more general bounded
operators, we first define the property of the locally convex topological
vector space $W$ that is required.

\begin{definition} \label{acpdef} \rm
Let $\mathcal{H}$ and $W$ be given as in Definition \ref{geprojection}
and $\mathfrak{X}$ as in Definition \ref{gefamily}. Let $\mathcal{C}$ be a
commutative $C^*$-algebra of bounded operators on $\mathcal{H}$. Define
$\mathcal{C}_e := \{ Ae | A \in \mathcal{C} \}$ for $e \in \mathcal{H}$, $\mathcal{S}_e$ as
the smallest closed subspace such that $\mathcal{C}_e \subset \mathcal{S}_e
\subset \mathcal{H}$, and $ P[\mathcal{S}_e] : \mathcal{H} \to \mathcal{S}_e$ as the
projection onto $\mathcal{S}_e$. Assume there is an isometric isomorphism
$\hat{G}_e: \mathcal{S}_e \to L_2 (\mathfrak{X}, \mu)$ such that $\hat{G}_e
|_{\mathcal{C}_e}$ is from $\mathcal{C}_e$ onto $C(\mathfrak{X})$. Then $W$ has
\emph{the almost continuity property with respect to $\mathcal{S}_e$
and $\mathfrak{X}$} if, for any $\epsilon > 0$, there exists a compact
set $K \subset \mathfrak{X}$ such that $\mu (\mathfrak{X} \backslash K) <
\epsilon$, $\hat{G}_e ( P[\mathcal{S}_e] (\phi))$ is continuous on $K$
for any $\phi \in W$, and every open set in $K$ has a positive
measure.
\end{definition}

We now investigate when $W$ has the almost continuity property.

\begin{definition} \label{knuc} \rm
Let $W_1$ and $W_2$ be locally convex topological vector spaces. A
map $E:W_1 \to W_2$ is called \emph{$k$-nuclear} if there exist
$\{ \alpha_i \} \subset \ell_k$, $\{ F_i | F_i \in W_1' \}$
equicontinuous, and $\{ g_i \} \subset W_2$ uniformly bounded in
$W_2$ such that, for any $f \in W_1$,
$$
E(f) = \sum_{i=1}^{\infty} \alpha_i \cdot F_i(f)\cdot g_i.
$$
Note here that, in Hilbert space setting, $1$-nuclear is called the \emph{trace class} and
$2$-nuclear is called the \emph{Hilbert-Schmidt}.
\end{definition}

\begin{theorem}\label{bob}
Let $\mathcal{H}$ be a Hilbert space and $\mathcal{A}$ be a commutative von
Neumann algebra. Suppose we have a Banach space $V$ such that $V \subset
\mathcal{H} \subset V'$, $V$ is dense in $\mathcal{H}$, and the embedding of
$V$ into $\mathcal{H}$ is continuous. If the embedding of $V$ into
$\mathcal{H}$ is $1$-nuclear, then there exists a constant $\beta$ such that
for any finite set $\{ \theta_r \}_{r=1}^s$ of elements of the unit ball of
$V$, and for any $e \in \mathcal{H}$,
$$
\sum_{r=1}^s | (P(\xi_r )\theta_r ,  e )| \leq \beta \| e \|_{\mathcal{H}}
$$
for any disjoint family $\{ \xi_r \}_{r=1}^s$ of Borel subsets of the
maximal ideal space $\mathfrak{A}$ of $\mathcal{A}$.
In case that $V$ is a Hilbert space, the above is true with $2$-nuclear embedding of
$V$ into $\mathcal{H}$.
\end{theorem}

For a proof of the above theorem, see \cite[Theorem 230]{Kauffman}.

\begin{theorem}\label{acp2nuc}
Let $\mathcal{H}$ and $W$ be Hilbert spaces such that $W$ is dense in
$\mathcal{H}$ and the embedding of $W$ into $\mathcal{H}$ is $2$-nuclear. Denote
$\mathcal{D}$ as a commuting family of bounded normal operators on
$\mathcal{H}$ such that  the restriction to $W$ takes $W$ into $W$
continuously. Let $\mathcal{C}$ be a commutative $C^*$-algebra
generated by $\mathcal{D}$. Also, let $\mathfrak{C}$ be the maximal ideal
space of $\mathcal{C}$ (which is a locally compact Hausdorff space).
Then $W$ has the almost continuity property with respect to
$\mathcal{S}_e$ and $\mathfrak{C}$.
\end{theorem}

\begin{proof}
Let $\mu_e$ be a positive measure on $\mathfrak{C}$ given by
Riesz  Representation Theorem (RRT) for $e \in \mathcal{H}$. Let
$\epsilon >0$. Then we need to show that there exists a compact
set $\mathfrak{F}$ such that $\mu_e (\mathfrak{C}\backslash
\mathfrak{F})<\epsilon$ and $\hat{G}_e ( P[\mathcal{S}_e] \phi)$ is
continuous on $\mathfrak{F}$ for any $\phi \in W$ (i.e.,
$\mathfrak{F}$ does not depend on $f$). Using Lemma \ref{bob}, we
construct a compact subset $\hat{K}$ such that $\mu_e
(\mathfrak{C} \backslash \hat{K}) < \frac{\epsilon}{2}$ and, for any
$\phi \in W$,
\begin{equation}\label{gelfand}
|\hat{G}_e ( P[\mathcal{S}_e] \phi )(x)| < N \| \phi \| \quad
 \text{for } x \in \hat{K}.
\end{equation}
Also, using Lusin's theorem, we construct another compact subset
$\tilde{K}$ such that $\mu_e (\mathfrak{C} \backslash \tilde{K})
< \frac{\epsilon}{2}$ and $\hat{G}_e ( P[\mathcal{S}_e] \phi_i )$ is
continuous on $\tilde{K}$ for all $\phi_i$. Let $\mathfrak{F} =
\hat{K} \cap \tilde{K}$. Then $\mu_e (\mathfrak{C} \backslash
\mathfrak{F} ) \leq \epsilon$. Now let $\phi \in W$. Then there is
a sequence $\phi_i \in S$ such that $\| \phi_i \| < \| \phi \|$
and $\phi_i \to \phi$. Then, for all $x \in \mathfrak{F}$ in the
complement of a set of measure zero,
\begin{equation}\label{cont}
\hat{G}_e ( P[\mathcal{S}_e] \phi_i )(x) \to \hat{G}_e ( P[\mathcal{S}_e]
\phi)(x)
\end{equation}
On $\mathfrak{F}$, we have
$| \hat{G}_e ( P[\mathcal{S}_e] \phi_i )(x)| < N \| \phi_i \|$ by
(\ref{gelfand}), hence $\{ \hat{G}_e ( P[\mathcal{S}_e] \phi_i )\}$ is uniformly
Cauchy.
Also, recall that $\{ \hat{G}_e ( P[\mathcal{S}_e] \phi_i )\}$ is continuous on
$\mathfrak{F}$. Thus it converges to a continuous function $G(\phi)$ on
$\mathfrak{F}$. However, $G(\phi)$ must agree with
$\hat{G}_e ( P[\mathcal{S}_e] \phi)$ almost
everywhere on $\mathfrak{F}$ by (\ref{cont}). Hence the result follows.
\end{proof}

Here is our main theorem.

\begin{theorem}\label{main}
Let $\mathcal{H}$ and $W$ be given as in Definition \ref{geprojection}
and $\mathcal{C}$ and $\mathfrak{C}$ as in Theorem \ref{acp2nuc}. Let
$\mathcal{A}$ be the smallest von Neumann algebra such that $\mathcal{C}
\subset \mathcal{A}$ (cf. Remark \ref{remark}). Also let $V$ be a Banach
space such that $W \subset V \subset \mathcal{H}$ all dense. Suppose that
the embedding $E_1 : V \to \mathcal{H}$ is $1$-nuclear and $E_2 : W \to
V$ is $2$-nuclear. Suppose also that $W$ has the closed graph
property. Then, for any cyclic vector $e$ for $\mathcal{A}'$ (the
commutant of $\mathcal{A}$) and for any $\epsilon >0$, there exists a
compact set $\mathfrak{F} \subset \mathfrak{C}$ such that $\mu_e
(\mathfrak{C} \backslash \mathfrak{F})< \epsilon$ and there exists a
generalized eigenfamily for $W, \mathcal{C}, \mathfrak{F}$.
\end{theorem}

Before we give the proof, let us note that the embedding condition on $W$
is needed in order to get the almost continuity property
(see Theorem \ref{acp2nuc}). Also notice that the
existence of a generalized eigenfamily is given for the compact set $\mathfrak{F}$ instead
of the entire maximal ideal space $\mathfrak{C}$. This is because the almost continuity
property of $W$ gives a compact set in $\mathfrak{C}$ and because we need to get, for each $x
\in \mathfrak{C}$, the generalized eigenprojection $Q_x$ in $C(W, W')$, i.e., $Q_x (\phi) \in W'$
for all $x \in \mathfrak{C}$. Unfortunately, this is not true in the entire
$\mathfrak{C}$, and hence we use $\mathfrak{F}$ instead of $\mathfrak{C}$.

\begin{proof}
We first note that there exists an isometric isomorphism $H: \mathcal{A}
\to L_{\infty}(\mathfrak{C}, \mu_e)$ such that $H(A) = G(A) \in
C(\mathfrak{C}) \cap L_{\infty} (\mathfrak{C}, \mu_e)$ for $A \in
\mathcal{C}$, where $G$ is the Gelfand transform of $\mathcal{C}$. We
can construct $H$ by observing $\hat{G}_e \mathcal{C} \hat{G}_e^{-1}
= \{ T_f | f \in C(\mathfrak{C})\}$. Also, by a basic property of a
Hilbert space, we know there exists a set of orthonormal vectors
$\{ e_i \}$ in $\mathcal{H}$ such that $\mathcal{H} = \oplus \mathcal{S}_{e_i}$.
Define
$$
e = \sum_i (\frac{1}{n_i} ) e_i
$$
where $(\frac{1}{n_i})e_i$ is in $\ell_2$ for any $i$. Then $e$ is
a cyclic vector for $\mathcal{A}'$ (i.e., $\mathcal{A}' e$ is dense in $\mathcal{H}$
where  $\mathcal{A}'$ be a commutant of $\mathcal{A}$). For this $e$, define
$\mu_e$ on $\mathfrak{C}$ using RRT as before. Since $E_1$ is
$1$-nuclear and $E_2$ is $2$-nuclear, the embedding from $W$ into
$\mathcal{H}$ is $2$-nuclear. Then by Theorem \ref{acp2nuc}, $W$ has
almost continuity property with respect to $\mathcal{S}_{e_i}$ and
$\mathfrak{C}$ for each $i$. That is, for $\epsilon > 0$ and for each
$i$, there exists a compact subset $\mathfrak{F}_i \subset
\mathfrak{C}$ such that $\mu_e (\mathfrak{C} \backslash
\mathfrak{F}_i)< \frac{\epsilon}{2^{i+1}}, $ $\hat{G}_{e_i}
( P[\mathcal{S}_e] (\phi ))$ is continuous on $\mathfrak{F}_i$ for any
$\phi \in W$, and every open set in $\mathfrak{F}_i$ has positive
measure. Let $\hat{\mathfrak{F}} = \cap_i \mathfrak{F}_i$. Then
$\mu_e (\mathfrak{C} \backslash \hat{\mathfrak{F}}) <
\frac{\epsilon}{2}$. Also, for any $i$, $\hat{G}_{e_i}( P[\mathcal{S}_e]
(\phi ))$ is continuous on $\hat{\mathfrak{F}}$ for any $\phi \in
W$.

For each $x \in \hat{\mathfrak{F}}$, define $Q_x$ on $W$ such that
$$
[Q_x (\phi )](\psi ) = \sum_{i=1}^\infty n_i \overline{\hat{G}_{e_i}
(P[\mathcal{S}_{e_i}]\phi )
(x )} \cdot \hat{G}_{e_i} (P[\mathcal{S}_{e_i}]\psi )(x)
$$
for $\phi, \psi \in W$. We would like to have $Q_x (\phi ) \in
W'$, however that is not true for all $x \in \hat{\mathfrak{F}}$.
Hence we shall construct a set in $\mathfrak{C}$ , call it
$\mathfrak{F}$, such that $Q_x (\phi ) \in W'$. For $x\in
\mathfrak{F}$, we must show that, for a fixed $\phi \in W$,
$$
\sum_{i=1}^\infty n_i \overline{\hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \phi )
(x )} \cdot \hat{G}_{e_i} (P[\mathcal{S}_{e_i}]\psi )(x)
$$
is continuous on $W$. Since $W$ has the closed graph property, it
is not difficult to see that $\hat{G}_{e_i} (P[\mathcal{S}_{e_i}]
(\cdot) )$ is continuous on $W$. Hence we only need to show that
the sum converges uniformly. We have
$$
(\psi, \phi) = \int_{\hat{\mathfrak{F}}} \sum_{i=1}^\infty \overline{\hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \phi ) (x)} \cdot \hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \psi )(x)
\cdot n_i \, d\mu_e .
$$
We also observe that
$$
(\psi, \phi) = \sum_i (P[\mathcal{S}_{e_i}] \phi,  P[\mathcal{S}_{e_i}] \psi ) =
\sum_i (P[\mathcal{S}_{e_i}] \phi,  \psi) \leq \| \phi \| \| \psi \|.
$$
Hence the integral on the right hand side of the previous equation is bounded.
Now consider the set
$$
\big\{ x \in \mathfrak{C} : \exists \phi \text{ such that }
\sum_{i=1}^\infty n_i \overline{\hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \phi ) (x)} \cdot \hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \psi )(x) = \infty  \big\}
$$
Then there exists a countable dense set $\{ \theta_r \}$ of all
such $\{ \phi \}$ from he above set. By Theorem \ref{bob}, there
exists a constant $\beta$ such that
\begin{eqnarray}\label{eqn:bob}
\sum_r (P(\Delta_r ) \theta_r,  \phi) \leq \beta \| \phi \|
\end{eqnarray}
for any disjoint family of $\{ \Delta_r\}$ of Borel subsets in $\mathfrak{C}$. Let $M
\in \mathbb{N}$ such that $M \geq 4 \beta \| \phi \| / \epsilon$, and define
$$
\Delta_r = \big\{ x \in \mathfrak{C} : \sum_{i=1}^N
n_i\overline{\hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \phi ) (x)} \cdot
\hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \theta_r )(x) \geq M \text{
for some } N \big\}.
$$
Then by (\ref{eqn:bob}), we get
\begin{align*}
\beta \| \phi \|
&\geq \sum_r (P(\Delta_r ) \theta_r,  \phi) \\
&= \sum_r \int_{\Delta_r} \sum_{i=1}^N \overline{\hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \phi ) (x)} \cdot \hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \psi )(x) \cdot n_i \, d\mu_e \\
&\geq M \mu_e (\cup_r \Delta_r ),
\end{align*}
hence
\begin{equation}\label{measure}
\mu_e (\cup_r \Delta_r ) \leq \frac{\beta \| \phi \|}{M} \leq
\frac{\epsilon}{4}.
\end{equation}
Let $V = \mathfrak{C} \backslash (\cup_r \Delta_r )$. Define $\tilde{\mathfrak{F}}$ as a
compact set contained in $V$ such that $\mu_e ( V \backslash
\tilde{\mathfrak{F}}) < \frac{\epsilon }{4}$. Then, on $\tilde{\mathfrak{F}}$, we
have
$$
\sum_{i=1}^\infty n_i \overline{\hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \phi ) (x)} \cdot
 \hat{G}_{e_i} (P[\mathcal{S}_{e_i}] \theta_r )(x)  < M.
$$
Also, by (\ref{measure}), $\mu_e (\mathfrak{C} \backslash \tilde{\mathfrak{F}}) <
\frac{\epsilon}{2}$. Now let $\mathfrak{F} = \hat{\mathfrak{F}} \cap \tilde{\mathfrak{F}}$.
Then $\mu_e (\mathfrak{C} \backslash \mathfrak{F})< \epsilon$, $\hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \phi )$ is continuous on $\mathfrak{F}$ for any $\phi \in W$,
and the sum is uniformly convergence for $x \in \mathfrak{F}$. Hence $Q_x \in C(W,
W')$.

Define now $g:\mathfrak{F} \to C(W, W')$ such that $g(x) = Q_x$.
We shall show that $g$ is a generalized eigenfamily for $W,
\mathcal{C}, \mathfrak{F}$, i.e., by Definition \ref{gefamily}, we must
show that
\renewcommand{\theenumi}{\alph{enumi}}
\begin{itemize}
\item[(a)] $g$ is continuous;
\item[(b)] for $A \in \mathcal{C}$, $A = \int \overline{H(A)} Q_x \,
d\mu_e $;
\item[(c)] each $Q_x$ is a generalized eigenprojection of $\mathcal{C}$,
corresponding to the eigenvalue $H(A)(x)$.

\end{itemize}
(a) This follows by the similar argument as in Theorem
\ref{multi}.

(b) Let $A \in \mathcal{C}$. Since $H(A)=G(A)$, by simple substitution
we get
$$
(\psi, A(\phi)) = \int \overline{H(A)(x)} [Q_x (\phi)](\psi) \, d\mu_e .
$$

(c) We must show that, for each fixed $x \in \mathfrak{F}$,
\begin{itemize}
\item[(i)] there exists $\{ P_n \}$ of projections in $\mathcal{A}$ and $\{ r_n \}$ of real numbers such
that $r_n P_n \to Q_x$ in $C(W, W')$;
\item[(ii)] for each $\epsilon>0$, $A \in \mathcal{C}$, there
is a spectral projection
$E(G(A)(x), \lambda_{G(A)(x)}, \epsilon)$ such that there exists $N \in \mathbb{N}$ with $E(G(A)(x),
\lambda_{G(A)(x)}, \epsilon)P_n = P_n$ for any $n >N$, where $P_n$ is the projection in i.
\end{itemize}
(i) Let $x \in \mathfrak{F}$ and $\{ \Omega_n \}$ be the sets
containing $x$ such that $\Omega_{n+1} \subset \Omega_n$ for all
$n \in \mathbb{N}$. Define
$$
P_n = \sum_{i} \hat{G}_{e_i}^{-1} T_{\chi_{\Omega_n}}
\hat{G}_{e_i} P[\mathcal{S}_{e_i}].
$$
Since $\hat{G}_e \mathcal{C} \hat{G}_e^{-1} = \{ T_f | f \in C(\mathfrak{C})\}$,
we get $P_n \in
\mathcal{C}$. Let $r_n = \frac{1}{\mu_e (\Omega_n )}$. Then
\begin{align*}
(\psi,  r_n P_n (\phi ))
&= \sum_{i=1}^\infty
\Big(\hat{G}_{e_i}(P[\mathcal{S}_{e_i}] \psi),  r_n
\hat{G}_{e_i}(\hat{G}_{e_i}^{-1} T_{\chi_{\Omega_n}}
\hat{G}_{e_i} P[\mathcal{S}_{e_i}] \phi) \Big) \\
&= \sum_{i=1}^{\infty} \Big(\frac{n_i}{\mu_e (\Omega_n )}
\int_{\Omega_n} \overline{ \hat{G}_{e_i}(P[\mathcal{S}_{e_i}]
\phi)(x) } \cdot \hat{G}_{e_i}(P[\mathcal{S}_{e_i}]
\psi)(x) \, d\mu_e  \\
&\quad  +  \frac{n_i}{\mu_e (\Omega_n )} \int_{\Omega_n}
\Big(\overline{ \hat{G}_{e_i}(P[\mathcal{S}_{e_i}] \phi)(s) }
\hat{G}_{e_i}(P[\mathcal{S}_{e_i}] \psi)(s) \\
&\quad - \overline{
\hat{G}_{e_i}(P[\mathcal{S}_{e_i}] \phi)(x) } \hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \psi)(x)\Big) \, d\mu_e\Big).
\end{align*}
The second integral approaches zero as $n\to \infty$. Hence
$$
(\psi,  r_n P_n (\phi )) \to \sum_{i=1}^{\infty}
\frac{n_i}{\mu_e (\Omega_n )} \int_{\Omega_n} \overline{
\hat{G}_{e_i}(P[\mathcal{S}_{e_i}] \phi)(x) } \hat{G}_{e_i}
(P[\mathcal{S}_{e_i}] \psi)(x) \, d\mu_e = [Q_x (\phi
)](\psi).
$$
Thus $r_n P_n \to Q_x$ as $n \to \infty$.

(ii) Let $A \in \mathcal{C}$. Then there exists $f \in C(\mathfrak{C})$
such that $G(A)(x) = f(x)$ for all $x \in \mathfrak{C}$. Let
$\epsilon >0$. Let $E(A, \lambda_A, \epsilon)$ be the spectral
projection of $A$ to $\{ y : \, |y-f(x)|<\epsilon \}$. For $\phi
\in W$, we have
\begin{align*}
E(A, \lambda_A, \epsilon) P_n (\phi)
&= E(A, \lambda_A, \epsilon)
\sum_{i} \hat{G}_{e_i}^{-1}
T_{\chi_{\Omega_n}} \hat{G}_{e_i} P[\mathcal{S}_{e_i}] (\phi) \\
&= \sum_{i} E(A, \lambda_A, \epsilon) \hat{G}_{e_i}^{-1} \chi_{\Omega_n} \hat{G}_{e_i}
P[\mathcal{S}_{e_i}] (\phi).
\end{align*}
Pick the smallest integer $N$ such that $f(\Omega_n ) \subset \{ y : \, |y-f(x)| <
\epsilon \}$. Then, for $n >N$, we have
\begin{align*}
E(A, \lambda_A, \epsilon) P_n (\phi)
& = \sum_{i} E(A, \lambda_A, \epsilon) \hat{G}_{e_i}^{-1}
\chi_{\Omega_n} \hat{G}_{e_i} P[\mathcal{S}_{e_i}] (\phi) \\
&= \sum_i \hat{G}_{e_i}^{-1} T_{\chi_{\Omega_n}} \hat{G}_{e_i}
P[\mathcal{S}_{e_i}] (\phi)
= P_n (\phi).
\end{align*}
\end{proof}


\subsection{\bf Existence of a Generalized Eigenfamily on the Joint Spectrum}

The expansion in the previous section was done in the maximal ideal space, although
most of the applications have the expansions on the joint spectrum. In this section,
we shall consider the existence of a generalized eigenfamily for a family of bounded
operators with finitely many unbounded operators on the joint spectrum.
Define the following:

 $\{ A_i \}$ is the a family of commuting normal operators in $\mathcal{H}$ which
finitely many of them, say $i=1, \dots, N$, are unbounded

 $\mathcal{A}$ is the smallest von Neumann algebra which contains bounded $\{ A_i \}$
and with which $\{ A_i \}_{i=1}^{N}$ are affiliated.

 $e$  is a cyclic vector (in $\mathcal{H}$) for $\mathcal{A}'$.

 $\mathfrak{A}$ is the maximal ideal space of the above $\mathcal{A}$ with measure $\mu_e$

 $ G_{\mathfrak{A}} : \mathfrak{A} \to C(\mathfrak{A} )$ is the Gelfand transform of
$\mathfrak{A}$. \smallskip

Note that the domain of $ G_{\mathfrak{A}} (A_j)$ for $j=1, \dots, N$ is the
complement of the meagre set $S_j$ and each meagre set has spectral measure
zero with respect to any cyclic vector for $\mathcal{A} '$ (\cite{KR}, \cite{Kauffman}).
Hence we shall use the compliment of these meagre sets instead of the entire
$\mathfrak{A}$, i.e., $\mathfrak{A} := \mathfrak{A} \backslash (\cup_{j=1}^N S_j)$.

\begin{definition}\label{jointspectrum} \rm
The \emph{joint spectrum} $\mathfrak{J}$ of $\mathcal{A}$ is
the closure of $\{ G_{\mathfrak{A}} (A_i ) (x)\}$ in the product space of the spectra of $A_i$.
Notice here that $\mathfrak{J}$ is a locally compact metric space.
\end{definition}

We define a measure on $\mathfrak{J}$ as follows. Let $a_i =
 G_{\mathfrak{A}} (A_i )$. Define $F: \mathfrak{A} \to \mathfrak{J}$ such that
$F(x) = (a_1 (x), a_2 (x), \dots )$ and $h_i : \mathfrak{J} \to \mathbb{C}$
such that $h_i (y_1, y_2, \dots, y_i, \dots ) = y_i$. Then
$\sigma_e (\Delta ) := \mu_e (F^{-1} (\Delta ))$, where $\Delta$
is a Borel set, is the measure on $\mathfrak{J}$.

With the above set ups, we have the main theorem.

\begin{theorem}\label{mainforjointsp} \rm
Let $\mathcal{A}, \mathfrak{A},  G_{\mathfrak{A}}$, and $\mathfrak{J}$ be given as above.
Suppose there exist a locally convex topological vector space $W$
and a Banach space $V$ such that $W \subset V \subset \mathcal{H}$ all
dense, the embedding $E_1 : V \to \mathcal{H}$ is $1$-nuclear, and the
embedding $E_2 : W \to V$ is $2$-nuclear. Assume also that $A \in
\mathcal{A}$ takes $W$ into $W$ continuously. Then, for any cyclic vector
$e$ for $\mathcal{A}'$ and for any $\epsilon > 0$, there exists a compact
set $\mathfrak{G} \subset \mathfrak{J}$ such that $\sigma_e (\mathfrak{J}
\backslash \mathfrak{G} ) < \epsilon$ and there exists a
generalized eigenfamily for $W, \mathcal{A}, \mathfrak{G}$.
\end{theorem}

\begin{proof}
In order to show the existence of a generalized eigenfamily for the joint
spectrum, we must construct an isometric isomorphism from $\mathcal{A}$ to
$L_\infty (\mathfrak{J}, \sigma_e)$. The rest of the proof follows by the
same construction as in Theorem \ref{main}.

Let $K$ be a compact subset in $\mathfrak{J}$ and $\chi_K$ be the
characteristic function on $K$. Then $\chi_K \circ F$ is a
uniquely defined characteristic function of a clopen set of
$\mathfrak{A}$. Hence there exists a projection $P[K]$ in $\mathcal{A}$ such
that $ G_{\mathfrak{A}} (P[K] ) = \chi_K \circ F$. Now define $\hat{G}$
on $\mathcal{A}$ such that $\hat{G} (P[K] A_i) = \chi_K \circ h_i$ for
$A_i \in \mathcal{A}$, where the domain of $\chi_K \circ h_i$ is $\{ F(x)
| x \in \mathfrak{A} \} \subset \mathfrak{J}$. Then $\hat{G}: \mathcal{A} \to
C(\mathfrak{J})$. Let $P(y_1, y_2, \dots, y_n )$ denote any
polynomial in $y_1, \dots, y_n$. Define
$$
\hat{G}_\mathfrak{J} (P[K]P(A_1, \dots, A_n )) = \chi_K \circ P(h_1,
\dots, h_n ).
$$
Then $\hat{G}_\mathfrak{J}$ is an isometry from $P_K \mathcal{A}$ into $L_\infty (K)$. Then, we
can extend $\hat{G}_\mathfrak{J}$ to a unitary operator taking $\mathcal{H}$ onto $L_2
(\mathfrak{J})$. Hence $\mathcal{A}$ is isometric to $L_\infty$.
\end{proof}

The main theorem gives the generalized eigenfunction expansion in $C(W, W')$
space. We now extend the theorem to $C(Z, B)$ where $Z$ is a Hilbert space
and $B$ is a
Banach space, so that we can deal with actual uniform convergence.

Let $Z$ be a Hilbert space such that $W \subset Z \subset
\mathcal{H} = \mathcal{H}' \subset Z' \subset W'$ with each embedding to the next is continuous
and each space is dense in the next. Assume that the embedding of $Z$ into $\mathcal{H}$ is
$2$-nuclear. Let $B$ be a Banach space such that $Z\subset B \subset Z'$
and the embedding of $Z$ into $B$ is $1$-nuclear. Assume that each generalized
eigenprojection $Q_x \in C(W, W')$ extends to a bounded linear transformation
$\hat{Q}_x \in C(Z, Z')$. Suppose that if $\psi \in Z' \subset W'$ and $A^t_i \psi \in
Z'$ for some $i$, where $A^t_i$ is the transpose of $A_i \in \mathcal{A}$, then $\psi \in
B$, and for the same $i$, $Z \subset D(A_i^2) =$ domain of $A_i^2$. Then:

\begin{theorem} \label{BandZ}
With the above conditions on $B$ and $Z$, and with $\mathcal{A}, \mathfrak{A},  G_{\mathfrak{A}}$, and $\mathfrak{J}$
given as above, the statement of Theorem \ref{mainforjointsp} holds for $Z, \mathcal{A},
\mathfrak{G}$. Moreover, the generalized eigenprojections are in $C(Z, B)$ and thus the
expansion converges in $C(Z, B)$.
\end{theorem}

\begin{corollary}\label{maincor}
For every $\epsilon > 0$, there exists a compact subset $\mathfrak{G}$ of the joint
spectrum and a positive constant $\delta$ with the following properties: for every
$\delta$-net $\{ \xi_i \}$ of $\mathfrak{G}$,
\begin{enumerate}
\item there exists a set of generalized eigenfunctions $F_i$ (defined on Definition
\ref{gefunction}) such that
$$
A_n^t F_{\xi_i } = (\xi_i )_n F_{\xi_i}
$$
where $A_n^t$ is a transpose of $A_n \in \mathcal{A}$;
\item there exists a set of complex constants $\{ c_i \}$ such that, for every $\theta$
in the unit ball of $W$,
$$
\big\| \theta - \sum_{i=1}^n c_i F_{\xi_i}(\theta) F_{\xi_i} \big\| < \epsilon.
$$
Note here that $\{c_i \}$ works for every $\theta$ in the unit ball
(by applying our theorem to the identity operator in the algebra) because
our expansion converges in $C(Z, B)$ with the usual operator norm topology
for $Z$ and $B$.
\end{enumerate}
\end{corollary}


\section{Comparison to Other Approaches}\label{sec:comp}


One of the major differences between the previous works on a
generalized eigenfunction expansion and our approach is that we
use the limit of the products of spectral projections and some
real numbers instead of using the elements in the dual space. To
illustrate the difference, let us compare with the approach given
by Maurin in \cite{Maurin}. First, Maurin recall the complete
spectral theorem for a single hermitian operator $A$ in a finite
dimensional Hilbert space $\mathcal{H}$. In this case, he identifies the
spectrum $\Lambda$ of $A$ with the eigenvalues of $A$ (i.e., the
elements in the maximal ideal space generated by $A$ in our
approach) and then defines the set $\hat{H} = \{ \hat{x}:
\hat{x}(\lambda ) = (x, e(\lambda )) \}$ where $x \in \mathcal{H}$,
$\lambda \in \Lambda$, and $\{e(\lambda )\}$ is an orthonormal set
of eigenvectors of $A$ (which spans $\mathcal{H}$). The theorem states
that $\{ e(\lambda )\}$ determines a unitary transformation from
$\mathcal{H}$ onto $\hat{H}$. An extension of the theorem to a
commutative family $\mathcal{C}$ of normal operators in $\mathcal{H}$ is
also given: Let $\Lambda$ be the spectrum of $\mathcal{C}$. Then
there exists a direct integral $\hat{H} = \int_{\Lambda} \hat{H}
(\lambda )\, d\mu (\lambda )$ ($= \{ \hat{x} : \Lambda \to \hat{H}
(\lambda ) : \hat{x} \text{ square integrable vector field }
\}$) and there exists a unitary map $F:\mathcal{H} \to \hat{H}$ such that
$(FA x)_k (\lambda ) = \hat{A} (\lambda ) \hat{x}_k (\lambda )$
for $k=1, 2, \dots, $dim$\hat{H} (\lambda)$ and $A \in
\mathcal{C}$, where $\hat{A} (\lambda )$ is the element in the
maximal ideal space which identified with $\lambda$. Now he gives
the fundamental theorem: Let $W$ be a dense linear subset of
$\mathcal{H}$ such that the embedding is nuclear. Then there exists a
transform $F:\mathcal{H} \to \hat{H}$ such that $F(\phi ) = (\phi, e_k
(\lambda ))$ for $k=1, 2, \dots , $dim$\hat{H} (\lambda )$ and
for $\phi \in W$, where $e_k (\lambda )\in W'$, and $F(\lambda ):
W \to \hat{H} (\lambda )$ is continuous; if $A(\phi ) \in W$, then
$(FA \phi )_k (\lambda ) = \hat{A} (\lambda ) \hat{\phi }_k
(\lambda )$, i.e., $(A\phi , e_k (\lambda )) = ( \phi, \hat{A}
(\lambda ) e_k (\lambda ))$ for almost all $\lambda$. From this
fundamental theorem, he states that the spectral synthesis is
given by
\begin{equation}\label{M}
P\phi = \int_{\lambda } \sum \hat{\phi }_k (\lambda ) e_k (\lambda ) \, d\mu
(\lambda ),
\end{equation}
where $\hat{\phi }_k (\lambda ) = (\phi, e_k (\lambda ))$ and $P:W
\to W'$ is an antilinear map.

Notice that the fundamental theorem given here uses the elements $e_k (\lambda ) \in W'$
(called generalized eigenelements in \cite{Maurin})
which are simultaneous eigenvectors for $\mathcal{C}$; hence he uses the spectral
projections. In our approach, we use $Q_x$, the limit of the products of spectral projections
and real numbers (we called a generalized eigenprojection in Definition 3.2) which is in
$C(Z, B)$. Hence Maurin's convergence of the integral (\ref{M}) is in $W'$ and thus
point-wise convergence whereas our
convergence of the integral in Definition \ref{gefamily} is in $C(Z, B)$, which can not be
easily deduced from Maurin's theorem since it is a different construction. Also,
since we use the limit of the products, our approach can be used even when the
spectral projections tend to be zero. This covers more cases than the previous
approach. Another observation is that Maurin's approach does not contain asymptotics
of the generalized eigenfunctions (eigenelements) thus no convergence information
whereas our approach contains some information on the asymptotics of the generalized
eigenfunctions because they are the elements in the range of the generalized
eigenprojection. This fact (i.e., the generalized eigenfunctions are in the range of
the generalized eigenprojection) is needed in order to prove some asymptotic
behaviour of the eigenfunctions in a number theory application (\ref{sec:remark},
\cite{paper2}). As for a single self-adjoint operator, a systematic apparatus for
calculating asymptotics of the generalized eigenfunctions was given by Poerschke,
Stolz, and Weidmann in \cite{PSW}. As for a family of operators, it does not appear
to have been considered by earlier approaches.


\section{Examples}

\subsection{Fourier Transform}

The most well known example of a generalized eigenfunction expansion is the
inverse Fourier transform. Let us illustrate how our theory works in this
simple situation.

As we mentioned in the introduction, the Fourier transform may be derived from
the spectral decomposition of the operator $i\dfrac{d}{dx}$ in $L_2 (\mathbb{R})$.
Since this operator has multiplicity one, $n\chi(\lambda - \frac{1}{n}, \lambda
+ \frac{1}{n})$ converges to $e^{-i \lambda x}$ for each $\lambda$. In this
section, we consider the Laplacian $\mathcal{L} = \dfrac{d^2}{dx^2}$ on $W= C_0^{\infty}
(\mathbb{R})$ with $\mathcal{H}=L_2 (\mathbb{R})$.  This operator has multiplicity two, and our
theorem gives a generalized eigenprojection with $e^{i\lambda x}$ with two
dimensional range and hence the inverse Fourier transform.

Consider commutative von Neumann algebra generated by $\mathcal{L}$ and
the identity. Recall that $\mathcal{L}$ is unbounded on $\mathcal{H}$, thus the
von Neumann algebra generated by $\mathcal{L}$ is the smallest von
Neumann algebra with which $\mathcal{L}$ is affiliated (see \cite{KR}).
Then $\mathcal{L}(f) = \lambda f$ if $f(x) = e^x$ or $e^{-x}$ with
$\lambda = 1$ and $f(x) = e^{ix}$ or $e^{-ix}$ with $\lambda =
-1$. The Fourier transform $\hat{\phi}$ is given by
$$ \hat{\phi
}(t) = \int_\mathbb{R} \phi (x) e^{-ixt} \, dm(x)
$$
where $dm(x) = \frac{1}{\sqrt{2\pi}} dx )$.
 We know that the Fourier
transform is the Gelfand transform (\cite{RudinF}), hence, if we
let $G (\phi ) = \hat{\phi}$, then $G$ is an isometric
isomorphism. Since
$$
G (\mathcal{L} \phi )(t) = \int_{\mathbb{R}} \mathcal{L} \phi (x) e^{-ixt}\, dm(x)
= -t^2 \int_{\mathbb{R}}
\phi (x) e^{-ixt}\, dm(x) = -t^2 G (\phi )(t)
$$
by integration by parts, let us define $H (\mathcal{L}) = -t^2$. Then $H$
is isometric isomorphism on $\{ \hat{\phi }\}$. In order to
construct the spectral projection, consider $\lambda = 1$. Define
$\omega_n = [1-\frac{1}{n}, 1+\frac{1}{n}]$, Borel subsets on
$\mathbb{R}$. Then $H^{-1} (\omega_n ):= \{ x \in \mathbb{R}^+ | H(x) = -x^2 \in
\omega_n \}$ is empty. Hence consider $\lambda = -1$. Define now
$\omega_n = [-1-\frac{1}{n}, -1+\frac{1}{n}]$. Then $H^{-1}
(\omega_n )$ is not empty this time. Hence define $\Delta_n =
H^{-1}(\omega_n)$. (i.e.,
$\Delta_n = [\sqrt{\frac{n-1}{n}},
\sqrt{\frac{n+1}{n}}] \to 1$ if $n \to \infty$.) We now
define a projection of $\mathcal{L}$ on $\{ \hat{\phi }\}$ as
$\tilde{E}(\Delta_n ) = \chi_{\Delta_n}$ where $\chi_{\Delta_n}$
is a characteristic function on $\Delta_n$. Hence a spectral
projection of $\mathcal{L}$ on $L_2$ can be defined by
\begin{equation}\label{eqn:projection}
E(\Delta_n ) = G^{-1} \tilde{E}(\Delta_n ) G.
\end{equation}
One should notice here that we can not apply the spectral theorem
to $\mathcal{L}$ using $E(\Delta_n )$ because $E(\Delta_n ) \to 0$ as $n
\to \infty$ (i.e., no expansion).

Now we shall construct the generalized eigenprojection using our
theorems. Let $\epsilon>0$. Define $\Delta = [-1-\epsilon,
-1+\epsilon]$. By (\ref{eqn:projection}), $E(\Delta )$ is a
spectral projection of $\mathcal{L}$ on $L_2$. Let $P_n = E(\Delta_n )$.
Clearly, $E(\Delta )P_n = P_n$ for $n >N$ where $N:= \lfloor
\frac{1}{\epsilon} \rfloor$. Also, let $r_n =
1/\sqrt{\mu(\Delta_n)}$ where $\mu (\Delta_n)$ is the measure of
$\Delta_n$. Then $r_n P_n \to P_\lambda$ such that $P_\lambda (x)=
e^{i\lambda x}$. Hence by Definition \ref{gefamily} and Theorem
\ref{mainforjointsp}, we get
$$
\mathcal{L} (\phi)(t) = \int_{\mathbb{R}} H(\mathcal{L}) P_\lambda (\phi) (t) \, d\lambda = \int_{\mathbb{R}} -t^2
\hat{\phi}(\lambda) e^{i\lambda t} \, d\lambda.
$$
If we apply the identity in the von Neumann algebra, we get
$$
\phi (t) = \int_{\mathbb{R}} \hat{\phi }(\lambda ) e^{i\lambda t}\, d\lambda.
$$
Hence we can consider the inverse Fourier transform as a generalized
eigenfunction expansion of the Laplacian.

\subsection{Application in Number Theory}\label{sec:nt}

In this section, we will apply our theorem to a family of the
Hecke operators (defined below) and the Laplacian in the hyperbolic space.
As a corollary, we will also get a
uniform convergence of the expansion, which seems to be new.
One can find details for this section in \cite{Apostol}.

\begin{definition} \label{def5.1} \rm
The set of all M\"{o}bius transform of the form
$$
\tau' = \frac{a\tau +b}{c\tau +d},
$$
where $a, b, c, d$ are integers with $ad-bc=1$, is called the \emph{modular
group} and denoted by $\Gamma$. The group can be represented by $2 \times 2$
integer matrices
$$
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
with $\det A =1$, provided we identify each matrix with its negative,
since $A$ and $-A$ represent the same transformation.
\end{definition}

\begin{definition} \label{def5.2} \rm
Let $G$ be a subgroup of $\Gamma$. Two points $\tau$ and $\tau'$ in the upper
half-plane $H$ are said to be \emph{equivalent} under $G$ if $\tau' = A\tau$
for some $A$ in $G$. An open subset $R_G$ of $H$ is called a
\emph{fundamental domain of $G$} if it has the following properties:
\begin{enumerate}
\item No two distinct points of $R_G$ are equivalent under $G$.
\item If $\tau \in H$, there is a point $\tau'$ in the closure of $R_G$ such
that $\tau'$ is equivalent to $\tau$ under $G$.
\end{enumerate}
\end{definition}

\begin{theorem} \label{thm5.3}
The open set $\mathbb{D} = \{ \tau \in H : |\tau |>1,  |\tau + \overline{\tau}
|<1 \}$ is a fundamental domain of $\Gamma$.
\end{theorem}

\begin{definition} \label{def5.4} \rm
Let $H$ be the upper half-plane and $\Gamma$ be the modular group. We define
the \emph{Hecke operator $T_n$ of order $n$} as
$$
(T_n f)(\tau) = \frac{1}{n} \sum_{d|n} \sum_{b=1}^{d-1}
f(\frac{n\tau +bd}{d^2} )
$$
where $f$ is an automorphic function under $\Gamma$ and $\tau \in H$.
\end{definition}

\begin{theorem} \label{thm5.5}
Any two Hecke operators commute with each other.
\end{theorem}

\begin{definition} \label{def5.6} \rm
Let $\mathbb{H}$ be the hyperbolic space. The \emph{Laplacian} $\mathcal{L}$ on
$C_0^{\infty} (\mathbb{H} )$ is defined by
$$
\mathcal{L} f = -y^2 (\dfrac{\partial^2 f}{\partial x^2} +
\dfrac{\partial^2 f}{\partial y^2} ), \ \ (x, y) \in \mathbb{H}.
$$
\end{definition}

\begin{theorem} \label{thm5.7}
The Hecke operators commute with the Laplacian. Also, there exists a
commutative von Neumann algebra with which $\mathcal{L}$ is affiliated and
in which the Hecke operators are.
\end{theorem}

One interesting fact about the Hecke operators and the Laplacian is as follows:
let $T_p$ be the Hecke operator with a prime number $p$.
Then we can consider $T_p$
as a self adjoint operator from a Hilbert space $\mathcal{H}$ to $\mathcal{H}$
where
$\mathcal{H}$ is the set of all modular forms with the power $2$ and with
$$
(f, g) = \int_{\mathbb{H} / \Gamma^0 (\mathbb{N})} f(z) \overline{g(z)} \, \frac{dx dy}{y^2} .
$$
Then $\mathcal{H}$ has a finite basis $\{ f_1, \dots, f_r \}$, and the basis is
the set of the simultaneous eigenfunctions of $\{ T_p \}$ and the Laplacian.

Before we move on, let us state two theorems on how to construct $2$-nuclear and $1$-nuclear
maps. These theorems are well known, thus we shall state without proofs (\cite{Kauffman},
\cite{diss}).

\begin{theorem}\label{2nuc}
Let $X$ be a locally compact Hausdorff space with a positive
measure $\mu$ and $T_0 : \mathcal{H} \to C(X)$ be a bounded linear
transformation into the set of bounded elements of $C(X)$, given
supremum norm. Suppose $T:\mathcal{H} \to L_2 (X, \mu)$ is given by
$$
T(g) = \theta \cdot T_0 (g)
$$
where $\theta$ is a fixed element of $L_2 (X, \mu)$. Then $T$ is $2$-nuclear.
\end{theorem}

\begin{theorem}\label{1nuc}
Let $D, J, M$ be Hilbert spaces. Suppose $T:D \to J$ is
$2$-nuclear and $S:J \to M$ is also $2$-nuclear. Then $S \circ T :
D \to M$ is $1$-nuclear.
\end{theorem}



\subsubsection{Generalized Eigenfunction Expansion}

We shall now apply our theorem to the family of Hecke operators
and the Friedrichs extension of the Laplacian (instead of the
Laplacian for simplification; abusing the notation, we call the
Friedrichs extension $\mathcal{L}$ from here on) to get a generalized
eigenfamily with the Hilbert space $L_2 (\mathbb{D} )$. In order to do so,
we must construct the spaces $W$ and $V$ with the necessary
properties.

Let $\mathcal{H} = L_2 (\mathbb{D} )$ (then $\mathcal{H} = \mathcal{H} '$). Let us first
construct the $2$-nuclear map from $\mathcal{H}$ to $\mathcal{H}$.

\begin{proposition}
Let $\beta < 1/2$. Define $T:\mathcal{H} \to \mathcal{H}$ such that
$$
T(g) = y^\beta \big(\frac{1}{y} \mathcal{L}^{-1} (g)\big).
$$
Then $T$ is $2$-nuclear.
\end{proposition}

\begin{proof}
We know that $\mathcal{L}$ is positive definite and $(\mathcal{L} (f),\, f) >
\frac{1}{4}(f, f)$. Hence $\mathcal{L}^{-1}$ is bounded by $\frac{1}{4}$.
Then $\frac{1}{y}\mathcal{L}^{-1}$ is also bounded and has $L_\infty$
norm. Since $\beta <\frac{1}{2}$, we have
$$
\int | y^{\beta} |^2 \frac{1}{y^2} \, dy\,dx = \int y^{2\beta-2}\,
dy\,dx < \int y^{-1} \, dy\,dx < \infty,
$$
hence $y^{\beta} \in L_2 (\mathbb{D} )$. By Theorem \ref{2nuc}, $T$ is $2$-nuclear.
\end{proof}

To construct a space $V$ such that the embedding from $V$ into
$\mathcal{H}$ is $1$-nuclear, recall that $S:= T\circ T$ is $1$-nuclear
by Theorem \ref{1nuc}. Notice that
$$
S(g) = y^{\beta-1} \mathcal{L}^{-1} (y^{\beta-1} \mathcal{L}^{-1} (g) ).
$$
Also note that $S(g) = y^{\beta-1} \mathcal{L}^{-1} (y^{\beta-1}
\mathcal{L}^{-1} (g))$ implies $g = \mathcal{L} (y^{1-\beta} \mathcal{L} ( y^{1-\beta}
S(g)))$. Hence define a space $B =  C_0^{\infty} (\mathbb{D} )$ with the norm
$$
\| f | B \| = \| \mathcal{L} ( y^{1-\beta} \mathcal{L} (y^{1-\beta} f)) | L_2
(\mathbb{D} ) \|
$$
where $\| f | B \|$ denotes the norm of $f$ in the space $B$. Let $V_0$
be the completion of $B$. Then the embedding from $V_0$ into $\mathcal{H}$ is
$1$-nuclear. In fact,
\begin{align*}
\| S(f) | V_0 \| &= \|  \mathcal{L} (y^{1-\beta}\mathcal{L} (y^{1-\beta}
S(f))) | L_2 (\mathbb{D} ) \| \\
&= \|  \mathcal{L} (y^{1-\beta}\mathcal{L} (y^{1-\beta} y^{\beta-1} \mathcal{L}^{-1}
(y^{\beta -1} \mathcal{L}^{-1} (f)) ) | L_2 (\mathbb{D} ) \| \\
&= \| f | L_2 (\mathbb{D} ) \|.
\end{align*}
Since $S$ is $1$-nuclear, the embedding is also $1$-nuclear. Although $V_0$
satisfies the required embedding property, let us simplify a little. Define a space
$D =  C_0^{\infty} (\mathbb{D} )$ with the norm
$$
\| f | D \| = \| \mathcal{L}^2 ( y^{2(1-\beta)} f) | L_2 (\mathbb{D} ) \|
$$
Let $V$ be the completion of $D$. Then it is not difficult to see that $V_0$ and $V$
are equivalent (i.e., two norms are equivalent), hence the embedding from $V$ into
$\mathcal{H}$ is $1$-nuclear. Note here that $V$ is also invariant under $\mathcal{L}^{-1}$.

We now construct a space $W$ such that the embedding from $W$ into $V$ is
$2$-nuclear. Let $X =  C_0^{\infty} (\mathbb{D} )$ with the norm
$$
\| f | X \| = \| \mathcal{L} ( y^{1-\beta} f) | V \|
$$
Let $W_0$ be the completion of $X$. Then the embedding from $W_0$ into $V$ is
$2$-nuclear by the same argument as above. As before, define another space $Y =
 C_0^{\infty} (\mathbb{D} )$ with the norm
$$
\| f | Y \| = \| \mathcal{L}^3 ( y^{3(1-\beta)} f) | L_2 (\mathbb{D} ) \|,
$$
and let $W$ be the completion of $Y$. Again, $W_0$ and $W$ are equivalent, hence the
embedding from $W$ into $V$ is $2$-nuclear. $W$ is also invariant under $\mathcal{L}^{-1}$.

We apply Theorem \ref{main} to get the following result.

\begin{corollary}\label{Cormain}
Let $\mathcal{C}$ be the $C^*$-algebra generated by the Hecke
operators, $\mathcal{L}$, and the identity. Also, let $W$, $V$, and
$\mathcal{H}$ be given as above. Then, for any $A \in \mathcal{C}$, the
generalized eigenfunction expansion converges in $C(W, W')$.
\end{corollary}

\subsubsection*{Remarks for Corollary \ref{Cormain}} \label{sec:remark}

We first note that, if $\Phi \in W'$, then by RRT there exists
$\phi \in L_2 (\mathbb{D} )$ such that
$$
\Phi (\theta ) = (\phi,  \mathcal{L}^3 (y^\alpha \theta)) = (y^\alpha
\mathcal{L}^3 (\phi ),  \theta ) \quad  \text{for } \theta \in W
$$
where $\alpha = 3(1-\beta) > \frac{3}{2}$ ($\beta < \frac{1}{2}
 \Rightarrow -\beta > -\frac{1}{2})$. Hence
$\Phi$ is isometrically isomorphic to $y^\alpha \mathcal{L}^3 (\phi)$.
This gives us some Information on the space $C(W, W')$ in which
the expansion converges. If $\Phi \in W'$ also satisfies $\mathcal{L}
(\Phi) = \lambda \cdot \Phi$ for some $\lambda$, one can show that
$W'$ is isometrically isomorphic to $y^\alpha L_2 (\mathbb{D} )$. It is
known that the eigenfunctions of the Laplacian behaves like
$y^{\frac{1}{2} + \epsilon}$ for any $\epsilon >0$ (that is
$y^{\frac{1}{2}+\epsilon} L_\infty (\mathbb{D})$ instead of $y^\alpha L_2
(\mathbb{D})$ in our result). If we use the fact that the multiplicity of
the Hecke operators and the Laplacian is one, then it seems
possible to show the same result using our theory (\cite{paper2}).
One should note that the multiplicity of the Laplacian itself on
the cusp space is not known.

In number theory, we define the cusp space $L_{2, c}$ to be the
set of automorphic functions $f \in L_2 (\mathbb{D} )$ such that $f_0
(y)=0$ where $f_0 (y)$ is the term independent of $x$ in the
Fourier series
$$
f(x, y) = \sum_{n=-\infty}^{\infty} f_n (y) \exp (2\pi inx).
$$
We also define the Eisenstein space $L_{2, E}$ to be the set of
automorphic functions $f \in L_2 (\mathbb{D})$ orthogonal to the cusp
space, i.e., $L_2 = L_{2, c} + L_{2, E}$. Then it is known that,
for $\psi \in  C_0^{\infty} (\mathbb{D})$,
\begin{equation}\label{eqn:numbertheory}
(\cdot,  \psi) = \frac{1}{4\pi}
\int_{t=-\infty,\;
s=1/2+it}^{\infty} (\psi,  E(z , s))(E(z,\, s),  \cdot) \, dt +
\sum_{i=1}^\infty (\psi,  f_i )(f_i,  \cdot)
\end{equation}
where $E(z, s)$ is an Eisenstein series and $f_i$ is an orthogonal
basis (which is also an eigenfunction of $\mathcal{L}$: \cite{Martin}).
Applying our theory (Theorem \ref{main}) to the identity operator
$I$ in $\mathcal{C}$, we get
\begin{equation}\label{eqn:ourtheory}
(\cdot,  I(\psi)) = \int \overline{H(I)(x)}[Q_x (\psi)(\cdot)]\, d\mu_e.
\end{equation}
Since both (\ref{eqn:numbertheory}) and (\ref{eqn:ourtheory}) use the same spectral decomposition,
they are the same. However, there is a major difference for the convergence. In number theory, the
convergence of the integral (\ref{eqn:numbertheory}) is shown for one function at a time,
i.e., one fixes a function $\psi$ and shows the convergence. (Since we consider $\psi$ as a
``point'' in $W$, it is like ``point-wise'' convergence.) In our theory, we are not expanding
a function $\psi$ but expanding the identity operator in the algebra generated by $T_n$ and
$\mathcal{L}$. Hence we know the integral converges in $C(W, W')$ by Corollary
\ref{Cormain}. By using the same set ups as in Theorem \ref{BandZ}, we can also show
the integral converges in $C(Z, B)$; that means
the integral converges for every $\psi$ in the unit ball of $Z$. Hence we get some sort of uniform
convergence. Also, since we do not use anything specific about the fundamental domain of $\Gamma$,
we believe that we can apply our method to so-called
modified Hecke operators (i.e., Hecke operators defined on a set of automorphic functions
under a subgroup of $\Gamma$). Nothing about expansions for those operators seems to
be known.

\subsection*{Acknowledgment}
This paper is a part of my dissertation at the University of
Alabama  at Birmingham, under the direction of Prof. Robert M.
Kauffman.


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