\magnification = \magstep1
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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1993/01\hfil Lie Generators for Semigroups\hfil\folio}
\def\leftheadline{\folio\hfil J. R. Dorroh and J. W. Neuberger\hfil EJDE--1993/01}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations\hfil\break
Vol. 1993(1993), No. 01, pp. 1-7. Published: August 27, 1993.\hfil\break
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break
ftp (login: ftp) 147.26.103.110 or 129.120.3.113 }
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 AMS {\eighti Subject Classification:} Primary 47H20, 47D20.\hfil\break
{\eighti Key words and phrases:} semigroups of operators,
infinitesimal generator.\hfil\break
\copyright 1993 Southwest Texas State University and
University of North Texas.\hfil\break
Submitted: May 8, 1993.} }
\bigskip\bigskip
\relpenalty=10000
\binoppenalty=10000
\def\qed{\vrule height8pt width4pt depth1pt}
\centerline{\bf LIE GENERATORS FOR SEMIGROUPS OF}
\centerline{\bf TRANSFORMATIONS ON A POLISH SPACE}
\smallskip
\centerline{
J. R. Dorroh and J. W. Neuberger}
\medskip
\midinsert\narrower
{\bf Abstract.}
Let $X$ be a separable complete metric space.
We characterize completely the infinitesimal generators of semigroups
of linear transformations in $C_b(X)$, the bounded real-valued
continuous functions on $X$, that are induced by
strongly continuous semigroups of continuous transformations in $X$.
In order to do this, $C_b(X)$ is equipped with a locally convex topology
known as the {\it strict topology}.
\endinsert
%\def\vf{\varphi}
\def\ve{\varepsilon}
\bigskip
{\bf Introduction.}
A strongly continuous semigroup of transformations on a topological space
$X$ is a function $T$ from $[0,\infty)$ into the collection of
continuous transformations from $X$ into $X$ such that
\item{(1)}
$T(0)=I$, the identity transformation on $X$,
\item{(2)}
$T(t)\circ T(s)=T(t+s)$ for all $t,s\ge0$, and
\item{(3)}
if $x\in X$, then the function $T(\cdot)x$ is continuous from
$[0,\infty)$ into $X$.\par\noindent
We will follow the standard practice writing the semigroup as a
collection
and denoting such a semigroup as $\{T(t):t\ge0\}$
or just $\{T(t)\}$.
Since at least the time of Sophus Lie, mathematicians have been
investigating generators, often called {\it infinitesimal generators},
of such semigroups. The case in which $X$ is a Banach space
and $T(t)$ is a bounded linear operator on $X$ for each $t\ge0$
has become particularly well understood.
A series of results, now called Hille-Yosida theorems, or
Hille-Yosida-Phillips theorems, forms the heart of this understanding.
The
{\it infinitesimal generator} of a strongly continuous semigroup of
(bounded) linear operators on $X$ is defined by
$$
Ax=\lim\limits_{t\to0}{1\over t}\left[T(t)x-x\right],\leqno(4)
$$
with domain ${\cal D}(A)$ consisting of all $x$ for which this limit
exists.
The simplest Hille-Yosida theorem characterizes infinitesimal
generators of strongly continuous semigroups of nonexpansive linear
transformations. If $A$ is the infinitesimal
generator of a strongly continuous nonexpansive semigroup $\{T(t)\}$,
then
${\cal D}(A)$ is dense in $X$, and
\smallskip\noindent
\line{(5)\hfil
$I-\ve A$ has a nonexpansive inverse defined on all of $X$
for each $\ve>0$.\hfil}
\smallskip\noindent
Furthermore, the semigroup is constructed from its
infinitesimal generator by
$$
T(t)x=\lim\limits_{n\to\infty}\left(I-(t/n)A\right)^{-n}x
\quad\hbox{ for }\quad t>0,\;x\in X.
\leqno(6)
$$
Conversely, any
densely defined operator $A$ in $X$ that satisfies
(5) is the infinitesimal generator of a strongly continuous
nonexpansive semigroup, which is given by (6).
\smallskip
All Hille-Yosida
theorems deal with a collection ${\cal G}$ of generators
and a collection ${\cal S}$ of semigroups. Each $A\in{\cal G}$
is obtained from a $T\in{\cal S}$ by means of a formula like (4),
and each $T\in{\cal S}$ is obtained from an $A\in{\cal G}$ by a formula
like (6). The main motivation for this activity is to solve
initial value problems for abstract ordinary differential equations,
which, by allowing $A$ to be discontinuous, include partial differential
equations (with boundary
conditions incorporated into the domain of $A$).
\smallskip
Considerable effort was made in the late 1960's
and early 1970's to find analogous results for semigroups of nonlinear
transformations. The only complete success was in the case of
strongly continuous semigroups of nonexpansive transformations on convex
sets in a Hilbert space; see [Br].
The infinitesimal generator of such a semigroup was characterized
as a single-valued
selection (the element of minimum norm)
of a possibly multivalued
maximal dissipative operator.
There similar results for Banach spaces with smooth norm ([Ba],[R]).
Another very important result was the Crandall-Liggett theorem, [CL],
which gave
sufficient conditions on a possibly multivalued operator in a general
Banach space $X$ in order that the formula (6) give a strongly
continuous semigroup of nonexpansive transformations on a convex subset
of $X$, but there is no proof that every strongly continuous
nonexpansive semigroup arises in this way.
Progress toward a fairly complete
nonlinear Hille-Yosida theory in general Banach
spaces (even for nonexpansive semigroups) has been minimal.
\smallskip
A second approach to the semigroup-generator problem goes back to Sophus
Lie himself in his search for a theory of ordinary differential
equations in terms of integrating factors; see [In].
This second approach shares common ground with Koopman and
von~Neumann on representation
by means of unitary groups of groups generated by Hamiltonian systems;
see [Ko], [Nm].
For a general strongly continuous semigroup $\{T(t)\}$
of continuous transformations
in a metric space
$X$, one may consider the induced linear semigroup $\{U(t)\}$ in the space
$C_b(X)$ of bounded real-valued continuous functions on $X$ given by
$U(t)f=f\circ T(t)$. It was essentially shown in [Nb] that the semigroup
$\{U(t)\}$,
while not generally strongly continuous, has a generator
$A$
that is dense
in the topology of pointwise convergence (the limit in the definition of
generator is also taken pointwise), and that $I-\ve A$ has a
nonexpansive inverse defined on all of $C_b(X)$ for each $\ve>0$.
Furthermore, $\{U(t)\}$, and hence $\{T(t)\}$, may be recovered from $A$
by means of
$$
U(t)f=\lim\limits_{n\to\infty}\left(I-(t/n)A\right)^{-n}f
\quad\hbox{ for }\quad t>0,f\in C_b(X),
$$
where the limit is taken pointwise on $X$. This is at best half
the Hille-Yosida result, since a characterization of such Lie generators
was not found.
\smallskip
It is the purpose of this paper to give a complete characterization of
the Lie generators of strongly continuous semigroups of continuous
transformations on a separable complete metric space. We hope that
this characterization will be a useful substitute for a still
nonexistent
Hille-Yosida characterization of such semigroups in terms of ordinary
generators.
Let $X$ denote a separable complete metric space, and let
$\{T(t): t\ge0\}$ denote a strongly continuous semigroup of
continuous transformations on $X$. That is, each $T(t)$ is a
continuous transformation from
$X$ into $X$, and for each $x\in X$, the mapping
$t\to T(t)x$ is continuous from $[0,\infty)$ into $X$.
It follows that
$\{T(t)\}$ is {\it jointly continuous}, that is,
the mapping $(t,x)\to T(t)x$ is continuous from $[0,\infty)\times X$
into $X$.
Now let $\{U(t)\}$ denote the semigroup of linear transformations in
$C_b(X)$, the linear space of bounded real-valued continuous functions
on $X$, given by $U(t)f=f\circ T(t)$. It is clear that each
$U(t)$ is a multiplicative homomorphism.
We describe a topology $\beta$ on $C_b(X)$ such that $\{U(t)\}$
is a strongly continuous semigroup of continuous transformations in
$(C_b(X),\beta)$, such that $\{U(t):0\le t\le b\}$ is
$\beta$-equicontinuous for each $b>0$, and such that if $\alpha>0$, then
the semigroup $\{e^{-\alpha t}U(t)\}$ is $\beta$-equicontinuous.
There is a version of the Hille-Yosida-Phillips Theorem for such
semigroups of linear transformations in topological vector spaces.
The semigroup consists of homomorphisms if and only if the generator is
a derivation.
We also show that every
$\beta$-strongly
continuous semigroup of
$\beta$-continuous homomorphisms in $C_b(X)$ is induced by
a strongly continuous semigroup of continuous transformations in $X$.
This gives a correspondence between the
strongly continuous semigroups of continuous transformations in $X$
and a well-described
class of linear derivations in $C_b(X)$.
Probably the most interesting case is that in
which $X$ is a separable $G_\delta$ set in a Banach space,
equipped with a complete metric that induces the topology inherited from
the Banach space. A similar characterization was
obtained by the first author [D1], [D2]
for strongly continuous semigroups of
transformations in a locally compact Hausdorff space.
\medskip
{\bf
Section 1. Preliminaries.}
In this section we give some facts about topological vector spaces that
will be needed in order to establish our main results. We begin by
stating a
simple and straight-forward generalization of the usual
Hille-Yosida-Phillips Theorem to the setting of topological vector
spaces. Many extensions have been given, and many complications can
occur. However, the following theorem,
which is essentially given in [Y, Chapter IX, Section 7],
will suit our purposes.
\smallskip
\noindent
\proclaim
1.1 Theorem. Let $E$ be a sequentially complete locally convex
topological vector space, and let $A$ be a linear operator in $E$.
Then the following two statements are equivalent:
\item{i)}
$A$ is the infinitesimal generator of a strongly continuous
equicontinuous semigroup of transformations in $E$.
\item{ii)}
The domain of $A$ is dense in $E$ and
$$
\left\{\left(I-n^{-1}A\right)^{-m}: m,n=1,2,\ldots\right\}
$$
is an equicontinuous collection of linear operators on $E$.
\par
\smallskip
\noindent
{\bf Remark.}
Formula (6) of the introduction
for obtaining the semigroup from its generator
is not established
for this setting in Yosida's book;
rather, the related formula
$$
U(t)x=\lim\limits_{n\to\infty}
\exp\left(tn\left[(I-n^{-1}A)^{-1}-I\right]\right)x
\quad\hbox{ for }t>0,x\in E
$$
is proven, where $\{U(t)\}$ is the semigroup generated by $A$.
\par
\medskip
The following proposition
characterizes the infinitesimal generators of strongly continuous
semigroups of continuous multiplicative homomorphisms in a topological
vector space that is also an algebra.
The proof is completely routine and is
omitted.
\medskip\noindent
\proclaim
1.2 Proposition.
Let
$\{U(t)\}$
be a strongly continuous semigroup of continuous linear transformations
in the sequentially complete
locally convex topological vector space
$E$, and let $A$ be its
infinitesimal generator. Suppose $E$ is also an algebra and that
multiplication is jointly continuous. Then
$\{U(t)\}$
is a semigroup of homomorphisms if and only if $A$ is a derivation;
that is, $f,g\in{\cal D}(A)$ implies that $fg\in{\cal D}(A)$ and
$A(fg)=f(Ag)+(Af)g$.
\par
\medskip
Now
let $E=C_b(X)$, the linear space of all bounded real-valued continuous
functions on
the separable complete metric space $X$.
For $r>0$, let $B_r$ denote the closed ball
$\{f\in E: \|f\|\le r\}$, where
$\|\cdot\|$ denotes the supremum norm.
Let $\kappa$ denote the compact-open topology on
$E$, and let
$\beta$ denote the strongest locally convex topology on $E$
that agrees with $\kappa$ on each set $B_r$.
We call $\beta$ the strict topology on $E$.
Throughout this paper,
``norm'' will mean the supremum norm, and $E^\ast$ will
denote the space of $\beta$-continuous linear functionals on $E$.
The rest of this section is devoted
to establishing the properties of the topology $\beta$ that are
needed for our main results.
Mainly, the properties we need are documented in
the paper [S], and what we do here is to provide a guide for finding the
documentation.
Sentilles defines three ``strict'' topologies
[S, p~315] in the case that $X$ is a completely regular Hausdorff space,
and then establishes that the three topologies coincide if $X$
is a complete separable metric space [S, Theorem~9.1, p~332].
\medskip\noindent
\proclaim
1.3 Proposition. $\varphi\in E^\ast$ if and only if there is a bounded
Borel measure $\mu$ on $X$ such that $\varphi(f)=\int_X f\,d\mu$ for all
$f\in E$.
\par
\smallskip
\noindent
{\bf Proof.}
This is (c) of [S, Theorem~9.1, p~332], together with the fact that
every
bounded Borel measure on a complete separable metric space is compact
regular; see for example [P, Thm.~3.2, p~29],
or [K, Cor. to Thm.~3.3, p~147] (applied to the total variation of a
signed measure). \qed
\medskip\noindent
\proclaim
1.4 Proposition.
Let $V$ be an absolutely convex absorbent set in $E$
having the property ($\cal P$):
for each $r>0$, there is a $\beta$-neighborhood $V_r$ of 0
such that $V\supset V _r\cap B_r$. Then $V$ is a $\beta$-neighborhood
of 0.
\par
\smallskip\noindent
{\bf Proof.}
Since $\beta$ is the strongest locally convex topology
on $E$ agreeing
with $\kappa$ on
each set $B_r$,
then $\beta$ is the strongest
locally convex topology
on $E$ agreeing with itself on each set $B_r$.
The collection of all
absolutely convex absorbent sets $V$
having property $(\cal P)$
is a base for a locally convex topology
$\gamma$ on $E$ by [RR, Thm.~2, p~10].
Clearly, $\gamma$ is stronger than $\beta$, and
the restriction of $\beta$ to any set $B_r$
coincides with the restriction of $\gamma$ to $B_r$.
Therefore, $\beta=\gamma$. \qed
\medskip\noindent
\proclaim
1.5 Proposition.
The strict topology has a base of 0-neighborhoods of the form
$$
W\{(K_n,a_n)\}=\{f\in E: \|f\|_{K_n} \le a_n \hbox { for all } n\},
$$
where $K_n$ is a compact set in $X$ for each $n$, $\{a_n\}$
is a sequence of positive numbers converging to $\infty$,
and $\|f\|_K=\sup_{x\in K}|f(x)|$ for $f\in E$ and $K\subset X$.
\par
\smallskip\noindent
{\bf Proof.}
See [S, Thm.~2.4(a), p~316]. \qed
\medskip\noindent
\proclaim
1.6 Proposition.
If $m$ is a nonzero $\beta$-continuous multiplicative linear functional
on $E$, then there is a point $\hat m \in E$ such that
$m(f)=f(\hat m)$ for all $f\in E$.
\par
\smallskip\noindent
{\bf Proof.}
By Proposition 1.3,
$m$ is given by $m(f)=\int_X f\,d\mu$
for some bounded Borel measure $\mu$ on $X$.
The fact that $m$ is
a norm continuous multiplicative linear functional implies that there
is a point $z$ of the Stone-\v Cech compactification
$\widehat X$ of $X$ such that
$m(f)=\bar f(z)$ for all $f\in E$,
where $\bar f$ denotes the continuous extension of $f$ to $\widehat X$.
Therefore,
$\int_X f\,d\mu=\bar f(z)$ for all $f\in E$. By [K, Thm.~2.1, p 142],
$z\in X$. \qed
\medskip
{\bf Section 2. The Induced Linear Semigroup.}
Let $\{T(t)\}$ denote a strongly continuous semigroup of continuous
transformations
from $X$ into $X$. That is, we assume that the mapping
$(t,x)\to T(t) x$ is separately continuous. It follows that this
mapping is jointly continuous; see [CM, Thm.~4]. Now let $\{U(t)\}$
denote the semigroup of linear transformations in $C_b(X)$ defined by
$U(t)f=f\circ T(t)$. $\{U(t)\}$ is clearly a semigroup of
multiplicative homomorphisms.
\medskip\noindent
\proclaim
2.1 Theorem.
If $b>0$, then the collection $\{U(t): 0\le t\le b\}$ is
$\beta$-equicontinuous.
\par
\smallskip\noindent
{\bf Proof.}
Let $V=W\{(K_n,a_n)\}$ (see Proposition 1.5), and define the
transformation
$\Phi:[0,\infty)\times X\to X$ by $\Phi(t,x)=T(t)x$.
Let $K_n^\prime=\Phi([0,b]\times K_n)$ for each $n$, and let
$V^\prime=W\{(K_n^\prime,a_n)\}$. Then
$U(t)V^\prime\subset V$ for $0\le t\le b$. \qed
\medskip\noindent
\proclaim
2.2 Theorem.
$\{U(t)\}$ is $\beta$-strongly continuous; that is, for each $f\in E$,
$U(\cdot)f$ is $\beta$-continuous from $[0,\infty)$ to $E$.
\par
\smallskip
\noindent
{\bf Proof.}
We prove strong continuity from the right at $0$. Two-sided strong
continuity everywhere follow from this and Proposition 1.1. Let
$f\in E$ and $V=W\{(K_n,a_n)\}$. Choose $N$ so that $a_n>2\|f\|$
for $n>N$, for $n=1,\dots,N$, choose $\delta_n>0$ so that
$|f(T(t)x)-f(x)|0$ then $\{e^{-\alpha t}U(t): t\ge 0\}$ is
$\beta$-equicontinuous.
\par
\smallskip\noindent
{\bf Proof.}
Let $V=W\{(K_n,a_n)\}$, $a=\min a_n$. For each $r>0$, choose $b_r>0$
so that $re^{-\alpha b_r}0$ and is this a $\beta$-neighborhood of $0$ by
Proposition 1.4. \qed
\medskip
Now let $A$ denote the infinitesimal generator of $U(t)$. Then $A$
is a derivation by Proposition 1.2, and for each $\alpha>0$, the
collection
$$
\left\{\left[({n+\alpha\over\
n})\left(I-(n+\alpha)^{-1}A\right)\right]^{-m}:
m,n=1,2,\dots\right\}
$$
is a $\beta$-equicontinuous collection, by Theorem 1.1.
\medskip
{\bf
Section 3. The Induced Nonlinear Semigroup.}
Now let $\{U(t)\}$ be an arbitrary $\beta$-strongly continuous
semigroup of linear $\beta$-continuous multiplicative homomorphisms in
$E$. We want to prove that there is a strongly continuous semigroup
$\{T(t)\}$ of continuous mappings in $X$ such that $U(t)f=f\circ T(t)$
for all $f$ and $t$. Incidentally, this will prove that
$\{e^{-\alpha t}U(t)\}$ is $\beta$-equicontinuous for all
$\alpha>0$, by Theorem 2.3.
If $t\ge 0$, and $x\in X$, then $m\in E^\ast$ defined by
$m(f)=\left[U(t)f\right](x)$ is a nonzero multiplicative linear
functional, and therefore, by Proposition 1.6, there is a point
$y\in X$ such that $m(f)=f(y)$ for all $f\in E$. Since $C_b(X)$
separates points, this point $y$ is unique. Thus, for each $t\ge0$,
we can define the transformation $T(t)$ from $X$ into $X$ by
$$
f\left(T(t)x\right)=\left[U(t)f\right](x)
$$
for $f\in E$ and $x\in X$. From the fact that $C_b(X)$ separates
points, it also follows that $\{T(t)\}$ is a semigroup of
transformations. We need to prove that each transformation
$T(t)$ is continuous and that the semigroup
$\{T(t)\}$ is strongly continuous. Let $\rho$ denote the metric
on $X$.
\medskip\noindent
\proclaim
3.1 Lemma.
If $t\ge 0$, then $T(t)$ is continuous.
\par
\smallskip\noindent
{\bf Proof.}
Suppose not, and choose $x\in X$, $\{x_n\}\subset X$
converging to $x$, and $r>0$ so that
$\rho\left(T(t)x_n,T(t)x\right)\ge r$ for all $n$.
Now choose $f\in E$ so that $f(x)=1$ and
$f(y)=0$ for $\rho(y,x)\ge r$. Then
$\left[U(t)f\right](x_n)=0$ for all $n$, which contradicts the
continuity
of $U(t)f$. \qed
\medskip\noindent
\proclaim
3.2 Lemma.
If $x\in X$, $\{t_n\}\subset(0,\infty)$, $\{x_n\}\subset X$,
$x_n\to x$, and $t_n\to 0$, then $T(t_n)x_n\to x$.
\par
\smallskip\noindent
{\bf Proof.}
Suppose not, and choose subsequences $\{y_n\}$ of $\{x_n\}$
and $\{u_n\}$ of $\{t_n\}$ such that for some $r>0$,
$\rho\left(T(u_n)y_n,x\right)\ge r$ for all $n$. Now choose
$f\in E$ such that $f(x)=1$ and $f(y)=0$ for $\rho(y,x)\ge r$.
The sequence $\{U(u_n)f\}$ converges to $f$ in the topology $\beta$,
and therefore, it converges uniformly on the compact set
$\{x,y_1,y_2,\ldots\}$. This yields a contradiction.
\qed
\medskip\noindent
\proclaim
3.3 Theorem.
The transformation $(t,x)\to T(t)x$ is separately continuous
from $[0,\infty)\times X$ into $X$. \par
\smallskip
\noindent
{\bf Proof.}
Define $\Phi$ on $[0,\infty)\times X$ by $\Phi(t,x)=T(t)x$.
We want to show that $\Phi$ is separately continuous.
By Lemma 1.2, $\Phi$ is separately continuous at $(0,x)$ for each
$x\in X$. Now, let $t>0$ and $x\in X$. To show that $\Phi$ is
separately continuous at $(t,x)$ it is sufficient to prove that
if $\{h_n\}\subset(0,t)$, $\{x_n\}\subset X$, $h_n\to0$, and
$x_n\to x$, then $\Phi(t+h_n,x_n)\to\Phi(t,x)$ and
$\Phi(t-h_n,x_n)\to\Phi(t,x)$.
The first conclusion follows from Lemmas 3.1 and 3.2, since
$\Phi(t+h_n,x_n)=T(t)\left(T(h_n)x_n\right)$.
To prove that
$\Phi(t-h_n,x_n)\to\Phi(t,x)$,
we use the fact that
if $f\in E$, then
$\{U(t-h_n)f\}$ converges to $U(t)f$ uniformly on the
compact set
$\{x,x_1,x_2,\dots,T(h_1)x,T(h_2)x,\dots\}$ and that
$$
f\left(\Phi(t-h_n,x_n)\right)-f\left(\Phi(t,x)\right)
=\left[U(t-h_n)f\right](x_n)-\left[U(t-h_n)f\right]
\left(T(h_n)x_n\right).\quad\qed
$$
\bigskip
\centerline{
\bf References}
\medskip\noindent
\item{[Ba]}
J.-B. Baillon, G\'en\'erateurs et semi-groupes dans les espaces de
Banach uniform\'ent lisses, J. Funct. Anal. {\bf 29} (1978), 199 - 213.
\item{[Br]}
H. Br\'ezis, {\it Operateurs Maximaux Monotones et Semigroupes de
Contractions dans les Espaces de Hilbert}, North Holland, 1973.
\item{[CL]}
M. G. Crandall and T. M. Liggett,
Generation of semigroups of nonlinear transformations on general Banach
spaces, American J. Math. {\bf 93} (1971), 265 - 298.
\item{[CM]}
P. Chernoff and J. E. Marsden, On continuity and smoothness of group
actions, Bull. American Math. Society {\bf 76} (1970), 1044 - 1049.
\item{[D1]}
J. R. Dorroh, Semigroups of maps in a locally compact space,
Canadian J. Math. {\bf 19} (1967), 688 - 696.
\item{[D2]}
$\underline{\hskip 1truein}$, The localization of the strict topology via
bounded sets, Proc. American Math. Soc. {\bf 20} (1969), 413 - 414.
\item{[K]}
J. D. Knowles,
Measures on topological spaces, Proc. London Math. Society {\bf 17}
(1967), 139 - 156.
\item{[Ko]}
B. O. Koopman, Hamiltonian systems and transformations in Hilbert space,
Proc. Nat. Acad. Sci. {\bf 17} (1931), 315-318.
\item{[Nb]}
J. W. Neuberger,
Lie generators for one parameter semigroups of transformations,
J. reine angew. Math. {\bf 258} (1973), 315-318.
\item{[Nm]}
J. von Neumann, Dynamical systems of continuous spectra,
Proc. Nat. Acad. Sci. {\bf 18} (1932), 278-286.
\item{[P]}
K. R. Parthasarathy,
{\it Probability Measures on Metric Spaces}, Academic Press, 1967.
\item{[R]}
Simeon Reich,
A nonlinear Hille-Yosida Theorem in Banach spaces, J. Math. Anal. App.
{\bf 84} (1981), 1 - 5.
\item{[RR]}
A. P. Robertson and W. J. Robertson,
{\it Topological Vector Spaces}, Cambridge University Press, 1964.
\item{[S]}
F. Dennis Sentilles, Bounded continuous functions on a completely
regular space, Trans. American Math. Society {\bf 168} (1972),
311 - 336.
\item{[Y]}
K. Yosida, {\it Functional Analysis}, Springer Verlag,
vol 123, Grundlehren der Math. Wiss., 1977(5$^{\rm th}$ed).
\medskip
Department of Mathematics, Louisiana State University, Baton Rouge,
LA 70803\hfil\break\noindent
{\it E-mail address}: mmdorr@lsuvax.sncc.lsu.edu
\medskip
Department of Mathematics, University of North Texas, Denton, TX
76203 \hfil\break\noindent
{\it E-mail address}: jwn@vaxb.acs.unt.edu
\bigskip
\centerline{\bf ADDENDUM}
\medskip
{\bf November 17, 1994.}
We have stated that a strongly continuous semigroup $T$ of continuous
transformations on a Polish space $X$ is necessarily {\it jointly
continuous}; that is, the mapping $(t,x)\to T(t)x$ is jointly continuous
from $[0,\infty)\times X$ into $X$. This fact was attributed to
[CM, Thm. 4], but in fact
[CM, Thm. 4] asserts joint continuity only on $(0,\infty)\times X$, and in
fact, an example is given in [C] to show that joint continuity at $t=0$
does not follow in this general a setting. Therefore, in Section 2,
we must assume that the semigroup $T$ is jointly continuous in order to
establish the necessary properties of the induced linear semigroup $U$
on $C_b(X)$. Fortunately, the argument for Theorem 3.3 proves that the
semigroup $T$ on $X$ induced by
be an arbitrary $\beta$-strongly continuous
semigroup $U$ of linear $\beta$-continuous multiplicative homomorphisms in
$C_b(X)$ is jointly continuous on $[0,\infty)\times X$.
\smallskip
Thus, in the statement of Theorem~3.3, the word ``separately'' should be
changed to ``jointly'', and our paper characterizes
Lie generators of jointly continuous semigroups of maps on a Polish
space.
\smallskip
\item{[C]}
P. Chernoff, Note on continuity of semigroups of maps, Proc. American
Math. Society {\bf 53} (1975), 318-320.
\bye