%Bericht 04/96 des SFB 003 "Optimierung und Kontrolle"
%Typeset with Latex 2e
\documentclass[12pt,psamsfonts,reqno]{amsart}
\usepackage{amssymb,a4}
\usepackage{latexsym}
\numberwithin{equation}{section}
%%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{assumption}{Assumption}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{prop}{Proposition}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{example}{Example}[section]
\newtheorem{problem}{Problem}[section]
\theoremstyle{remark}
\newtheorem{rem}[prop]{Remark}
\newtheorem{note}{Note} \renewcommand{\thenote}{}
\newtheorem{summ}{Summary} \renewcommand{\thesumm}{}
\newtheorem{ack}{Acknowledgment} \renewcommand{\theack}{}
\newtheorem{notation}{Notation} \renewcommand{\thenotation}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\fH}{{\mathfrak H}}
\newcommand{\fK}{{\mathfrak K}}
\newcommand{\fP}{{\mathcal P}}
\newcommand{\fD}{{\mathfrak D}}
\newcommand{\fM}{{\mathcal M}}
\newcommand{\fA}{{\mathcal A}}
\newcommand{\Dom}{{\mathfrak D}}
\newcommand{\rmi}{{\mbox{i}}}
\newcommand{\Ker}{\mbox{Ker}\,}
\newcommand{\Ran}{\mbox{Ran}\,}
\newcommand{\DA}{{\mathfrak D}_A}
\newcommand{\id}{{\bf 1}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\sgn}{\mbox{sgn}\,}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\title[]{\Large Approximation of Degenerate Cauchy Problems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[]{\large S. Thaller ${}^*$ and B. Thaller}
\address{Inst. f. Mathematik, Universit\"at Graz, Heinrichstra\ss e 36,\\A-8010 Graz, Austria}
\email{bernd.thaller@kfunigraz.ac.at}
\email{sigrid.thaller@kfunigraz.ac.at}
\maketitle
\bigskip\bigskip
\begin{center}
{\sc Institut f\"ur Mathematik, Universit\"at Graz, Heinrichstra\ss e 36, A-8010 Graz, Austria}\\
\end{center}
\bigskip\bigskip
\begin{abstract}
We investigate the approximation of a linear homogenuous degenerate Cauchy problem
$\frac{d}{dt}Mz(t) = Az(t), z(0) = z_0$, where $M$ and $A$ are closed, densely defined
operators in a Hilbert space $\fH$ and $M$ has a nontrivial kernel. Analoguously to the
Trotter-Kato Theory for nondegenerate equations we obtain conditions when the convergence
of the pseudo resolvents of $M$ and $A$ is equivalent to the convergence of the solutions
of the given Cauchy problem. We study the connection between these approximations and
approximations of the factorized nondegenerate Cauchy problem connected with the given
problem and we give an example where factorization and approximation can be interchanged.
\end{abstract}
\vfill
${}^*$ Supported by SFB F003 ``Optimierung und Kontrolle''
%%%%%%%%%%%%%%%%%%%%%%%%%-------------> Erzwungener Seitenwechsel
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-----Introduction----%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\setcounter{equation}{0}
\setcounter{prop}{0}
\bigskip
%
In this paper we investigate the approximation of a homogeneous
degenerate abstract Cauchy problem given in a Hilbert space $\fH$,
%
\begin{equation}\label{IDEG}
\frac{d}{dt}\,M\,z(t) = A\,z(t),\qquad z(0) = z_0,
\end{equation}
%
by degenerate Cauchy problems
%
\begin{equation}\label{IDEGn}
\frac{d}{dt}\,M^n\,z_n(t) = A^n\,z_n(t),\qquad z_n(0) = z_{n0}
\end{equation}
%
in some subspaces $\fH^n$. The operators $M$, $M^n$, $A$ and $A^n$ are closed and
densely defined operators in $\fH$ resp. $\fH^n$. We give conditions on these operators
such that the strict solutions of equations (\ref{IDEGn}) converge to the strict solution
of (\ref{IDEG}). In Section \ref{sec2} we recapitulate results of \cite{TT1} and \cite{TT2} where
we have shown under which conditions a Cauchy problem (\ref{IDEG}) can be factorized
into a nondegenerate Cauchy problem
%
\begin {equation}\label{IFAK}
\frac{d}{dt}x(t) = A_1 x(t)
\end{equation}
%
on the factor space $\fH/\Ker M$, when strict solutions of (\ref{IDEG})
exist, and when the solutions are described by a semigroup. In Section \ref{sec3} a result
for approximating (\ref{IDEG}) by (\ref{IDEGn}) similarily to the Trotter-Kato Theorem
for nondegenerate Cauchy problems (\cite{KATO}, \cite{TROTTER})
is obtained. We show that if we impose certain conditions
on $M$, $M^n$, $A$ and $A^n$ the convergence of the pseudo resolvents of $A^n$ and $M^n$
to the pseudoresolvent of $A$ and $M$ in the Hilbert space norm is equivalent to the
convergence of the solutions of (\ref{IDEGn}) to the solutions of (\ref{IDEG}) in the
Hilbert space norm uniformly on compact t-intervals.
As the connection between solutions of (\ref{IDEG}) and (\ref{IFAK}) is described by
an unbounded linear operator $Z_A$, it is in general not possible to obtain an
approximation of the degenerate equation (\ref{IDEG}) by approximating the factorized
nondegenerate problem (\ref{IFAK}). Even if this operator $Z_A$ is bounded it is in general
not possible to interchange
factorization and approximation, i.e., factorizing the approximating equations (\ref{IDEGn})
does not lead to the same result as approximating the factorized Cauchy problem (\ref{IFAK}).
In Section \ref{sec4} we describe a special situation where this is possible. The
subspaces $\fH^n$ are finite dimensional and the operators $A^n$ and $M^n$ are
defined by $A^n = \tilde\Pi A \Pi$ and $M^n = \tilde\Pi M \Pi$, where $\Pi$
is a projection on the subspaces $\fH^n$ and $\tilde\Pi$ is a projection on the range
of $M\Pi$.
%
This construction of $A^n$ and $M^n$ leads to Galerkin approximation of the
factorized equation (\ref{IFAK}) and there is a continuous dependence between the
approximating solutions of (\ref{IFAK}) and (\ref{IDEG}).
Approximations of degenerate Cauchy problems have been investigated by Lamm-Rosen
\cite{LR}, Rosen-Raghu \cite{RR}, and Mao-Reich-Rosen \cite{MRR}.
They all used the setting of Carroll-Showalter \cite{CSH}, where the norm of the
Hilbert space is defined by the operator $M$ which has to be positive semidefinite.
Therefore, the approximating equations are situated in Hilbert spaces with different
norms.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------Preliminaries----%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{sec2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\setcounter{equation}{0}
\setcounter{prop}{0}
\bigskip
%
In this section we give a short summary of results in \cite{TT1} and \cite{TT2}
which we will need for this paper. We consider the homogeneous degenerate abstract
Cauchy problem
%
\begin{equation}\label{DEG}
\frac{d}{dt} Mz(t) = Az(t),\qquad z(0) = z_0
\end{equation}
%
where $M$ and $A$ are closed, densely defined, linear operators mapping a Hilbert space
$\fH$ into itself.
For convenience, we assume that the operator $M$ is bounded and has a closed range.
%%
%%%%%%%%%%%%%%%%%%%% NOTATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{notation}
Let $T$ be a linear operator.
By $\Dom(T)$, $\Ker T$, and $\Ran T$ we denote its domain, kernel,
and range, respectively.
The restriction of $T$ to
a set $\Dom$ smaller than the domain of $T$ will be denoted by $T \restriction \Dom$,
the closure of $T$ will be denoted by $T^c$ and the adjoint operator will be written
as $T^*$. The degenerate Cauchy problem (\ref{DEG}) will be denoted by $(M,A)$.
\end{notation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%% Definition 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{DEF1}
A strict solution of the degenerate Cauchy problem Eq.~(\ref{DEG}) is a continuous function
$z:[0,\infty)\longrightarrow \fH$ such that $z(t)\in\Dom(A) \cap \Dom(M)$ for all $t\ge 0$,
$Mz$ is continuously differentiable, and Eq.~(\ref{DEG}) holds.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Let $P$ denote the orthogonal projection onto $\Ker M$, $Q$ the projection onto
$\Ker M^*$ and let $P^\bot = \id - P$, $Q^\bot = \id - Q$.
Our assumptions on $M$ imply that $M^\bot = M\restriction{P^\bot\fH}$
is bounded and defined on all of $P^\bot\fH$.
Moreover, $M^\bot$ is invertible and $(M^\bot)^{-1}$ is bounded on $Q^\bot\fH$.
\begin{rem}
Our restrictive assumptions on the operator $M$ can be weakened,
but they are convenient, because they
simplify certain domain conditions and enable us to apply all results of \cite{TT1} and \cite{TT2}.
In a forthcoming paper \cite{TT3} the theory of degenerate Cauchy problems
will be treated under more relaxed conditions on $M$.
\end{rem}
%
Any strict solution of (\ref{DEG}) must be in the set
%
\begin{align}
\DA &= \{z\in\Dom(A) \mid Az\in (\Ran M)^c\}\\
&= \{z \in \Dom(A) \mid QAz =0 \}
\end{align}
%%
The following assumption is necessary in order to obtain uniqueness of strict solutions.
%%%%%%%% ASSUMPTION 1 %%%%%%%%%%%%%%%
\begin{assumption}
\label{ASS1}
For every $x\in P^\bot \DA$ the set $(P^\bot)^{-1}\{x\} \cap \DA$ consists of
precisely one element $z\in\DA$.
\end{assumption}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
An equivalent way of stating this assumption would be to require
%
\begin{equation*}
\Ker M\cap\DA = \{0\}.
\end{equation*}
%%
%%
%%%%%%%% ASSUMPTION 2 %%%%%%%%%%%%%%%
\begin{assumption}
\label{ASS2}
There is a real constant $\omega$ such that for all $\lambda$ with $\lambda > \omega$
the operator ${(A-\lambda M)\restriction\DA}$ has a bounded inverse which is defined on all of
$\Ran M$. Moreover, there exists a constant $0 < K \le 1$ such that
%
\begin{equation}
\label{rescond}
\|P^\bot (A-\lambda M)^{-1}M\| \le \frac{K}{\lambda - \omega}
\end{equation}
%
for all $\lambda > \omega$.
\end{assumption}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Remark 1 %%%%%%%%
\begin{rem}
By Assumption~\ref{ASS1} we can define a linear operator $Z_A$ on $P^\bot \DA$ by $z = Z_A x$,
where $z$ is the unique element in the set $(P^\bot)^{-1}\{x\}\cap\DA$. This operator
$Z_A$ is the inverse of the projection $P^\bot$ on $\DA$, i.e.,
\begin{equation}
Z_A P^\bot = \id \quad\text{on $\DA$},\qquad P^\bot Z_A = \id\quad \text{on $P^\bot \DA$}
\end{equation}
\end{rem}
%
%%%%%%%%%%%%%%%%%%%%%%%%
We define the operator $A_0$, $\Dom(A_0) = P^\bot (\DA)$, by
\begin{equation}\label{A0}
A_0 = A Z_A
\end{equation}
and we assume that $\Dom(A_0) = P^\bot (\DA)$ is dense in $P^\bot \fH$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Remark 2 %%%%%%%%%%%%
\begin{rem}
In \cite{TT1} and \cite{TT2} it was shown that $(A-\lambda M)Z_A = A_0 - \lambda M^\bot$
is closed, $A_0$ is closed on $P^\bot \DA$, and the operator $A_1 = (M^\bot)^{-1} A_0$
is closed on $\Dom(A_1) = P^\bot \DA$. Furthermore,
%
\begin{equation}\label{RESA1}
(A_1 - \lambda)^{-1} = P^\bot (A-\lambda M)^{-1} M\qquad \text{on $P^\bot \fH$}
\end{equation}
%%
and therefore Eq. (\ref{rescond}) implies that $A_1$ generates a $c_0$-semigroup on
$P^\bot \fH$.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%
%
The following theorem was proved in \cite{TT1}:
%
%%%%%%%%%%%%%%%%% THEOREM 1 %%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{TH2.1}
Let Assumptions \ref{ASS1} and \ref{ASS2} be fulfilled. Then the degenerate Cauchy
problem (\ref{DEG}) has a unique strict solution $z(t)$ for each initial value
$z_0 \in \DA$ and $z(t) = Z_A x(t)$, where $x(t) = e^{A_1 t} P^\bot z_0$ is the solution
of the nondegenerate Cauchy problem
\begin{equation}\label{FAK}
\frac{d}{dt} x(t) = A_1 x(t),\qquad x(0)=x_0=P^\bot z_0.
\end{equation}
\end{theorem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% REMARK 3 %%%%%%%%%%%
\begin{rem}
In general, the operator $Z_A$ is not closable. A necessary and sufficient condition for
the closability of $Z_A$ is that $\Ker M \cap \DA^c = \{0\}$.
\end{rem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% REMARK 4 %%%%%%%%%%%
\begin{rem}\label{REMARK4}
If $Z_A$ is closable we denote by $\fH_0$ the domain of the closure ${Z_A}^c$. By Lemma 2.2
in \cite{TT2} $\fH_0$ is a dense subspace of $(P^\bot \DA)^c$ and, equipped with the graph norm
\begin{equation}
\|x\|^2 _G = \|Z_A ^c x\|^2,
\end{equation}
$\fH_0$ is a Hilbert space which is isometrically
isomorphic to the closure of $\DA$ with respect to the norm in $\fH$. The isomorphism
from $\fH_0$ onto $\DA$ is given by $Z_A$, its inverse by $P^\bot$.
If $\fH_0$ is an invariant set for the semigroup $e^{A_1 t}$, i.e., $e^{A_1 t}\fH_0
\subset \fH_0$, then the operators $T(t) = Z_A ^c e^{A_1 t} P^\bot$ are a semigroup
on the Hilbert space $\DA^c$. In this case the homogeneous degenerate Cauchy problem
(\ref{DEG}) is completely equivalent to a nondegenerate Cauchy problem on the Hilbert
space $(\fH_0,\|.\|_G)$.
It can be shown \cite{TT3} that Assumption~\ref{ASS2} together with $Z_A$ closable and $\fH_0$ invariant
under $e^{A_1 t}$ are implied by the requirement, that for some $K<1$
\begin{equation}
\|(A-\lambda M)^{-1}M\| \le \frac{K}{\lambda - \omega}
\end{equation}
%
for all $\lambda$ with $\lambda > \omega$.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------Approximation----%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Approximation}\label{sec3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\setcounter{equation}{0}
\setcounter{prop}{0}
\bigskip
%
In this section we will approximate a degenerate Cauchy problem (\ref{DEG}) in the
sense of Trotter-Kato. We will use the well-known Trotter-Kato Theorem for nondegenerate
Cauchy problems in the version given in \cite{Kappel-Ito}:
%
%
%%%%%%%%%%%%%%% TROTTER-KATO %%%%%%%%%%%%%%%%%%%%
%
\begin{theorem}\label{TKKI}
Let $Z$ and $X_n$ be Banach spaces with norm $\|.\|$, $\|.\|_n$, $n=1, 2, ...,$ respectively,
and $X$ be a closed linear subspace of $Z$. For every $n=1,2,...$ there exist bounded
linear operators $P_n :X\longrightarrow X_n$ and $E_n:X_n\longrightarrow Z$ satisfying
\begin{align*}
&\text{(A1) $\|P_n\|\le M_1, \|E_n\|\le M_2 $ where $ M_1,M_2$ are independent of $n$,}\\
&\text{(A2) $P_nE_n = \id _n$, where $\id _n$ is the identity operator on $X_n$.}
\end{align*}
Let $A, A_n \in G(M,\omega)$ and let $T(t)$ and $T_n(t)$ be the semigroups generated
by $A$ and $A_n$ on $X$ and $X_n$ respectively. Then the following statements are
equivalent:
\begin{align*}
&\text{(a) There exists a $\lambda _0\in\rho (A)\cap\bigcap_{n=1} ^{\infty} \rho (A)$ such that,
for all $x\in X$,}\\&\text{$\|E_n(A_n-\lambda _0)^{-1}P_n x - (A-\lambda _0)^{-1}x\|\longrightarrow 0$
as $n\longrightarrow\infty$.}\\
&\text{(b) For every $x\in X$ and $t \ge 0$,
$\|E_n T_n (t) P_n x - T(t)x\|\longrightarrow 0$ as $n\longrightarrow\infty$}\\
&\text{uniformly on bounded $t-$intervals.}
\end{align*}
If $(a)$ or $(b)$ is true, then $(a)$ holds for all $\lambda$ with $\lambda >\omega$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
We now consider the degenerate Cauchy problem (\ref{DEG}) in the case where the solution
is described by a $c_0$-semigroup: Let $A$ and $M$ fulfill the assumptions made in Section
\ref{sec2} for factorizing the degenerate Cauchy problem (\ref{DEG}) into the nondegenerate
problem (\ref{FAK})
\begin{equation*}
\frac{d}{dt}x(t) = A_1 x(t)
\end{equation*}
with $A_1 = (M^\bot)^{-1} A_0 = (M^\bot)^{-1} AZ_A$. Furthermore, assume that $Z_A$ is
closable and $e^{A_1 t}(P^\bot \DA)^c \subset (P^\bot \DA)^c$, i.e., $e^{A_1 t}$ is a
$c_0$-semigroup $T(t)=Z_A ^c e^{A_1 t}P^\bot$ in $\DA^c$ where $z(t) = T(t)z_0$,
$z_0\in \DA$,is a strict solution of (\ref{DEG}), and its generator $\tilde A =
Z_A ^c A_1 P^\bot$ is densely defined in $\DA$.
For this situation we now apply the Trotter-Kato Theorem:
%
%
%%%%%%%%%%%%%%%%% PROPOSITION 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{prop}\label{PROP1}
Let $(\DA ^c)^N$ be subspaces of $\DA ^c$ and let $\fP ^N :\DA ^c \longrightarrow
(\DA ^c) ^N$ be orthogonal projections with $\fP ^N z\longrightarrow z$ for all
$z \in \DA ^c$. Furthermore, let $\tilde A ^N, \tilde A$ be infinitesimal generators
of $c_0$-semigroups $T^N (t), T(t)$ on $(\DA ^c) ^N$ and $(\DA ^c)$, respectively,
satisfying $\tilde A ^N \in G(K,\omega), \tilde A \in G(K,\omega)$, and assume that
there exists a $\lambda \in \rho (\tilde A) \cap \bigcap_{N=1} ^\infty \rho (\tilde A ^N)$
such that for all $z \in \DA ^c$
\begin{equation*}
\|(\tilde A ^N - \lambda )^{-1} \fP ^N z - (\tilde A - \lambda )^{-1} z\| \longrightarrow 0
\qquad\text{ as $N \longrightarrow \infty.$}
\end{equation*}
Then for all $z \in \DA ^c$
\begin{equation*}
\|T^N (t) \fP ^N z - T(t) z \| \longrightarrow 0\qquad \text{ as $N \longrightarrow \infty$}
\end{equation*}
uniformly in $t$ on any compact interval.
\end{prop}
%
%
\begin{proof}
Apply Theorem \ref{TKKI} with $Z=\DA ^c$, $X_n = (\DA ^c)^N$, $P_n = \fP^N$ and
$E_n = \iota ^N$, where $\iota ^N$ is the canonical injection, $\iota ^N : (\DA ^c)^N
\longrightarrow \DA ^c$.
\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Remark %%%%%%%%%%%%%
%
\begin{rem}
Instead of $\rho (\tilde A)$, the resolvent set of the generator $\tilde A$, we could
also take the resolvent set of the pseudoresolvent $(A-\lambda M)^{-1} M$.
\end{rem}
%
\begin{rem}
Proposition \ref{PROP1} does not say whether the operators $\tilde A ^N$ belong to an
approximating degenerate Cauchy problem or not.
\end{rem}
%
%%%%%%%%%%%%%%%%%%%%
In the next Theorem we want to give conditions for degenerate equations approximating
a given equation (\ref{DEG}).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% THEOREM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%
\begin{theorem}\label{TH}
Consider the degenerate Cauchy problems
\begin{equation*}
\frac{d}{dt} M z(t) = A z(t)
\end{equation*}
and
\begin{equation}\label{DEGn}
\frac{d}{dt} M^n z_n (t) = A^n z_n (t)\qquad \text{for all $n\ge 1$.}
\end{equation}
Let Assumption~\ref{ASS1} be fulfilled,
%let furthermore
%\begin{equation}\label{HG}
%e^{A_1 t} P^\bot \DA ^c \subset P^\bot \DA ^c
%\end{equation}
%and
%\begin{equation}\label{HGn}
%e^{(A^n)_1 t} P_{M^n} ^\bot \fD_{A^n} ^c \subset P_{M^n} ^\bot \fD_{A^n} ^c
%\end{equation}
and assume that there exist $K<1$ and $\omega$ such that
\begin{equation}\label{RES}
\|(A-\lambda M)^{-1} M\|\le\frac{K}{\lambda -\omega}
\end{equation}
and
\begin{equation}\label{RESn}
\|(A^n - \lambda M^ n)^{-1} M^n \|\le\frac{K}{\lambda - \omega}
\end{equation}
for all $n\ge 0$ and $\lambda >\omega$. Let $\fP ^n$ denote the
orthogonal projection of $\fH$ onto $\fD _{A^n} ^c$.
Then the following statements are equivalent:
a) There is a $\displaystyle{\lambda \in \rho (M,A) \cap \bigcap_{n=1} ^\infty \rho (M^n, A^n)}$ such
that for all $z \in \DA ^c$
\begin{equation*}
\|(A^n - \lambda M^n) ^{-1} M^n \fP ^n z - (A- \lambda M )^{-1} M z \| \longrightarrow 0
\quad\text{as $n \longrightarrow \infty$.}
\end{equation*}
b) For all $z \in \DA ^c$ and for all $t_0 \ge 0$
\begin{equation*}
\| T_n (t) \fP^n z - T(t) z \| \longrightarrow 0\quad\text{ as $n\longrightarrow \infty$}
\end{equation*}
uniformly on bounded $t$-intervals, where $T(t)$ and $T_n (t)$ are $c_0$-semigroups
describing the solutions of (\ref{DEG}) and (\ref{DEGn}), respectively.
\end{theorem}
%
%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%
%%%% PROOF OF THEOREM TH %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{proof}
By condition (\ref{RES}) and Remark~\ref{REMARK4} we find that
%
\begin{equation*}
\tilde A = Z_A ^c A_1 P^\bot
\end{equation*}
%%
is the generator of
a $c_0$-semigroup $T(t)$ and the solution of (\ref{DEG}) is given by $z(t) = T(t) z_0$
(cf. \cite{TT2},Theorem 2.2).
In the same way we get by (\ref{RESn})
for all $n\ge 0$ generators
\begin{equation*}
\tilde A ^n = Z_{A^n} (A^n)_1 P_{M^n} ^\bot
\end{equation*}
of $c_0$-semigroups $T_n(t)$ which describe
the solutions of (\ref{DEGn}) by $z_n (t) = T_n (t) z_{n0}$. Now we can apply Theorem
(\ref{TKKI}), setting $\fH = Z$, $\DA ^c = (X,\|.\|_{\fH})$, $\fD_{A^n} ^c = (X_n, \|.\|_{\fH})$
and $\fP ^n = P_n$, $\id = E_n$. Hence conditions $(A 1)$ and $(A 2)$ in Theorem (\ref{TKKI})
are fulfilled. Equations (\ref{RES}) and (\ref{RESn}) imply that $\tilde A$ and
$\tilde A ^n \in G(K,\omega)$
for all $n\ge 0$. For the resolvent sets $\rho (\tilde A)$ and $\rho (\tilde A^n)$
we get
\begin{equation*}
\rho (\tilde A) =\rho (A_1) = \rho (M,A),
\end{equation*}
where $\rho (M,A)$ isthe resolvent set of the pseudoresolvent $(A- \lambda M)^{-1} M$,
and
\begin{equation*}
\rho (\tilde A ^n)=\rho (M^n, A^n)\qquad \text{for all $n\ge 0$.}
\end{equation*}
Clearly, the resolvents of $\tilde A$ and $\tilde A ^n$
are given by
\begin{equation*}
(\tilde A - \lambda )^{-1} = (A- \lambda M)^{-1} M
\end{equation*}
and
\begin{equation*}
(\tilde A ^n -\lambda )^{-1} = (A^n - \lambda M^n) ^{-1} M^n \text{for all $n \ge 0$}
\end{equation*}
(c.f.\cite{TT1}).
\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{example}
Let $\fH = L^2 (\R)\oplus L^2 (\R)$ and consider the one-dimensional
Dirac equation
%
\begin{equation}\label{DIRn}
\rmi \frac{d}{dt}
\left(\begin{matrix} 1 & 0\\0& \frac{1}{c^2}\end{matrix}\right)
\left(\begin{matrix} f\\g\end{matrix}\right) =
\left(\begin{matrix} V & -\rmi \frac{d}{dx}\\
-\rmi \frac{d}{dx} & -1+ \frac{V}{c^2} \end{matrix}\right)
\left(\begin{matrix}f\\g\end{matrix}\right),
\end{equation}
%%
where $c$ denotes the speed of light, $V$ is a symmetric operator on
$\Dom(V) \supset W^{2,2} (\R)$ describing the potential.
If we take the nonrelativistic
limit $c\longrightarrow \infty$ (c.f. \cite{TT1}, \cite{TT2}, \cite{TB}), we obtain a
degenerate Cauchy problem on $\fH$,
%
\begin{equation}\label{DIR}
\frac{d}{dt}\left(\begin{matrix}1&0\\0&0\end{matrix}\right)\left(\begin{matrix}f\\g
\end{matrix}\right) =
\left(\begin{matrix}-\rmi V&-\frac{d}{dx}\\-\frac{d}{dx}&\rmi
\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)
\end{equation}
%%
with matrix-operators
%
\begin{equation}\label{MA}
M=\left(\begin{matrix}1&0\\0&0\end{matrix}\right)\qquad
\text{ and } A=\left(\begin{matrix}-\rmi V&-\frac{d}{dx}\\-\frac{d}{dx}&\rmi
\end{matrix}\right).
\end{equation}
%%
We have $M=P^\bot=Q^\bot$ and $\rmi A$ is symmetric on
$\Dom(A)= (\Dom(V)\cap W^{1,2}(\R))\oplus W^{1,2}(\R)$ (c.f. \cite{TT2}).
Furthermore, it was proved in \cite{TT2} that $P^\bot \DA= W^{2,2}(\R)$,
$Z_A f=\left(\begin{matrix}f\\ \rmi f'\end{matrix}\right)$ on $P^\bot \DA$,
$\rmi A_0 = \rmi A Z_A = -d^2/dx^2 + V$ is self-adjoint on $\Dom(A_0) = W^{2,2} (\R)$,
$T(t) = Z_A ^c e^{A_0 t} P^\bot$, $t\in \R$, is a strongly continuous group on
$\DA ^c \subset \fH$ and the operators $T(t)$ are bounded uniformly in $t$.
Hence, (\ref{DIR}) is factorizable into a nondegenerate problem and (\ref{RES})
is fulfilled, because
%
\begin{equation*}
(A-\lambda M)^{-1}M = \left(\begin{matrix}
\left(-\rmi(-\frac{d^2}{dx^2} + V) - \lambda\right)^{-1} & 0 \\
\rmi\frac{d}{dx} \left(-\rmi(-\frac{d^2}{dx^2} + V) - \lambda\right)^{-1} & 0
\end{matrix}
\right).
\end{equation*}
%%
Setting $c^2 = n$, $n= 1,2,...$, the equations (\ref{DIRn})
can be written in the form of Eq.~(\ref{DEGn}) with
%
\begin{equation*}
M^n= \left(
\begin{matrix} 1& 0 \\ 0& \frac{1}{n}
\end{matrix} \right),
\qquad
A^n = \left(
\begin{matrix}
-\rmi V & -\frac{d}{dx}\\
-\frac{d}{dx}&\rmi -\rmi\frac{V}{n}
\end{matrix}\right).
\end{equation*}
%%
These operators define a sequence of non-degenerate Cauchy problems, where the operator $\rmi (M^n)^{-1}A_n$ is
the self-adjoint Dirac operator.
%are already
%nondegenerate problems in $X^n = \fH$ for all $n\ge 0$ and therefore clearly
%fulfill (\ref{HG}). As the solutions are described by
%unitary groups, also condition (\ref{RES}) is satisfied for all $n\ge 0$.
The calculations in \cite{TB} imply that the condition $a)$ of Theorem~\ref{TH}
is fulfilled with $\fP^n = \id$.
To this purpose we note that the operator $B(c)(H(c)-mc^2-\lambda)^{-1}B(c)^{-1}$ which occurs in Lemma~2.1 in \cite{TB}
is just another way of writing $(A^n-\lambda M^n)^{-1}M^n$.
Lemma~2.1 in \cite{TB} now implies that $(A^n-\lambda M^n)^{-1}M^n$ converges in operator norm to
$(A-\lambda M)^{-1}M$.Therefore, the solutions of (\ref{DIRn})
converge to the solution of (\ref{DIR}) uniformly on bounded $t$-intervals as
$n \longrightarrow \infty$.
\end{example}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%% REMARK %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{rem}
Assume that the Hilbert space can be written as an orthogonal direct sum
\begin{equation}
\fH = \Ran M \oplus \Ker M.
\end{equation}
The operator $M$ has the matrix representation $M=\left(\begin{matrix} M^\bot&0\\0&0\end{matrix}\right)$
with respect to this decomposition of the Hilbert space. We can regularize the degenerate
Cauchy problem (\ref{DEG}) by replacing $M$ by
%
\begin{equation*}
M^n = \left(\begin{matrix}M^\bot&0\\0&\frac{1}{n}\end{matrix}\right).
\end{equation*}
%
\end{rem}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% LEMMA %%%%%%%%%%%%%%%
%
%
\begin{lemma}
Let Assumption~\ref{ASS1} be fulfilled and let $M$ be as described above.
Assume for some $K<1$ that $\|(A-\lambda M)^{-1}M\|\le K/\lambda$, for all $\lambda>0$. Then for $\lambda > 0$ and for
$n$ large enough,
there exist constants $K'<1$ and $C>0$ such that $\|(A-\lambda M^n)^{-1}M^n\|\le K'/\lambda$ and
\begin{equation}
\|(A-\lambda M^n)^{-1}M^n - (A-\lambda M)^{-1}M\|\le \frac{C}{\lambda n}\to 0,\qquad\text{as $n\to\infty$.}
\end{equation}
Hence Theorem~\ref{TH} implies that the strict solutions of the regularized system (with $M$ replaced by $M^n$)
approximate the strict solutions of (\ref{DEG}).
\end{lemma}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% PROOF %%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{proof}
Let $\lambda > 0$. Since $M^\bot$ is bounded and bounded invertible, we obtain
%
\begin{equation*}
\|(A-\lambda M)^{-1}\| \le \frac{K\|(M^\bot)^{-1}\|}{\lambda} = \frac{\hat K}{\lambda}.
\end{equation*}
%%
From $\|M-M^n\|=1/n$ we find
\begin{align*}
\|(A-&\lambda M^n)^{-1}M^n\| = \\
=&\|(1 + \lambda(A-\lambda M)^{-1}(M-M^n))^{-1} (A-\lambda M)^{-1}(M^n-M+M)\|\\
\le &\|(1 + \lambda(A-\lambda M)^{-1}(M-M^n))^{-1}\|\,\|(A-\lambda M)^{-1}\|\,\|M^n-M\| +\\
+ &\|(1 + \lambda(A-\lambda M)^{-1}(M-M^n))^{-1}\|\,\|(A-\lambda M)^{-1}M\|\\
\le &\frac{1}{1-\hat K/n}\,
\left( \frac{\hat K}{n} + K\right)\,\frac{1}{\lambda}
\le \frac{K'}{\lambda},
\end{align*}
where $K'<1$ provided $K<1$ and $n$ is large enough.
Similarly, we obtain
\begin{align*}
\|(A-&\lambda M^n)^{-1}\| \le
\frac{1}{1- \hat K/n}\,
\frac{\hat K}{\lambda}\le \tilde K/\lambda,\\
\|(A-&\lambda M^n)^{-1}-(A-\lambda M)^{-1}\| =\\
=& \|(A-\lambda M^n)^{-1}\lambda(M^n-M)(A-\lambda M)^{-1}\|\\
\le&
\frac{\tilde K}{\lambda}\frac{\lambda }{n}\frac{\hat K}{\lambda}\to 0\quad\text{as $n\to\infty$},
\end{align*}
and finally
\begin{align*}
\|(A-&\lambda M^n)^{-1}M^n-(A-\lambda M)^{-1}M\| \\
\le & \|\bigl((A-\lambda M^n)^{-1} - (A-\lambda M)^{-1}\bigr)\,M^n\| +
\|(A-\lambda M)^{-1}\|\,\|M^n-M\|\\
\le & \left(2\|M\|\frac{\tilde K \hat K}{\lambda} + \frac{\hat K}{\lambda}\right)\frac{1}{n}
\to 0\quad\text{as $n\to\infty$}.
\end{align*}
\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% 4.GALERKIN APPROXIMATION %%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\section{Galerkin approximation}\label{sec4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\setcounter{equation}{0}
%\setcounter{prop}{0}
%\bigskip
%
%
Let us now consider the case where the given Hilbert space $\fH$ is approximated by
finite dimensional subspaces and the approximating operators are the restrictions to
these subspaces. We will see that if the operators in the factorized system are
Galerkin approximations with some projection $\fP ^n$, the operators in the
degenerate system are restricted by two different projections and in this case we
can interchange factorization and approximation.
Denote by $(M,A)$ a degenerate Cauchy problem in the Hilbert space $\fH$, assume
that it is factorizable and that $Z_A$ is bounded.
Let $U$ be a finite dimensional subspace of $\DA$ and let
\begin{equation*}
\Pi : \DA ^c\longrightarrow U
\end{equation*}
be a projection onto this subspace $U$. Define a
projection $\fP$ from $P^\bot \DA ^c$ in $P^\bot U = P^\bot \Pi (\DA ^c)$ by
\begin{equation}\label{fP}
\fP :x\mapsto P^\bot \Pi z
\end{equation}
where $x=P^\bot z$ and $z = Z_A x$. $\Pi$ can always be chosen such that $\fP$ is
an orthogonal Projection. (For the right choice of $\Pi$ start with an orthogonal
projection $\fP : P^\bot \DA ^c \longrightarrow P^\bot U$ and define $\Pi$ by
$\Pi z = Z_A \fP P^\bot z$).
Hence we have
\begin{equation}\label{PIPE}
\fP P^\bot z = P^\bot \Pi z \quad \text{for all $z \in \DA ^c$}
\end{equation}
and
\begin{equation}\label{PEPI}
\fP x = P^\bot \Pi Z_A x \quad \text{for all $x \in P^\bot \DA ^c$}.
\end{equation}
%
%
Define the operator $\fM : \DA ^c \longrightarrow \fH$ by $\fM = M\Pi$. As the subspace
$U$ is assumed to be finite dimensional the range of $\fM$ is also finite dimensional
and we can define $\tilde\Pi$ as the projection of $Q^\bot \fH$ on $\Ran \fM$. Hence,
\begin{equation}\label{M}
\fM = \tilde\Pi \fM = \tilde\Pi M \Pi \quad\text{ on $\DA ^c$}.
\end{equation}
%
In a similar way we define a restriction $\fA$ of the operator $A$ as an operator
mapping $U \subset \DA ^c$ into $\fH$ by
\begin{equation}\label{A}
\fA = \tilde\Pi A \Pi
\end{equation}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% LEMMA 1 %%%%%%%%%%%%%%%
%
%
\begin{lemma}\label{L1}
Let $(M,A)$ be a degenerate Cauchyproblem and assume that it is factorizable to a
nondegenerate Cauchy problem $\frac{d}{dt} x(t) = A_1 x(t)$, where $A_1$ is defined
as in section (\ref{sec2}). Consider furthermore the degenerate Cauchy problem
$(\fM,\fA)$ with operators $\fM$ and $\fA$ defined by (\ref{A}) and (\ref{M}) and
denote its factorized equation by $\frac{d}{dt} x(t) = \fA_1 x(t)$ with $\fA_1=
(\fM^\bot)^{-1} \fA Z_{\fA}$. Then we have:
\begin{enumerate}
\item $\Pi\DA = \fD _{\fA}$
\item $P^\bot = P_{\fM} ^\bot$ \qquad on $\fD_{\fA}$, where $P_{\fM} ^\bot$ denotes the
orthogonal projection from $\fH$ to $(\Ker \fM)^\bot$
\item $Z_A = Z_{\fA}$ \qquad on $\fP P^\bot \DA$
\end{enumerate}
\end{lemma}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% PROOF %%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{proof}
1. We have $\DA = \{z\in \Dom(A) \mid Az \in (\Ran M)^c\}$ and similarily for the
equation $(\fM,\fA)$ we get $\fD_{\fA} = \{z\in \Dom(\fA) \mid \fA z \in (\Ran \fM)^c\}$.
By definition of $\fM$ and $\tilde \Pi$ we have
\begin{align*}
\Ran \fM &= \Ran \tilde \Pi M \Pi\\
&= \tilde \Pi \Ran M \Pi\\&= \Ran M \Pi\\&= \tilde \Pi \Ran M.
\end{align*}
For all $z\in \DA$ we have
$\Pi z \in U \subset \DA$ and hence $A\Pi z \in (\Ran M)^c$. This implies that
$\tilde \Pi A \Pi z \in \tilde \Pi \Ran M = \Ran \fM$, $z \in \Dom(\fA)$ and $\fA z
\in \Ran \fM$, hence $z \in \fD_{\fA}$ and therefore $\Pi \DA \subset \fD_{\fA}$.
Conversely, by definition of $\fA$, $\fD(\fA) \subset U$, hence, $\fD_{\fA}\subset
\Pi \DA$.
2. By definition of $\fM$, $M\restriction {\Pi \DA} = \fM$. This implies
$P_{\fM} ^\bot \restriction {\Pi\DA} = P^\bot \restriction {\Pi\DA}$.
3. $\fP P^\bot \DA = \text{( by (\ref{PIPE})) } P^\bot \Pi \DA =\text{ (by 1.) }
P^\bot \fD_{\fA} =\text{ (by 2.) } P_{\fM} ^\bot \fD_{\fA}$.
Therefore, we get $Z_A P^\bot \Pi \DA = \Pi \DA = \fD_{\fA}
= Z_{\fA} P_{\fM} ^\bot \fD_{\fA}$.
\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% LEMMA 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{lemma}\label{L2}
Consider the degenerate Cauchy problems $(M,A)$ and $(\fM,\fA)$ with the same assumptions
as in Lemma \ref{L1}. Then we have
\begin{equation}\label{A1A1}
\fP A_1 \fP = \fA_1 \qquad\text{on $P^\bot \fD_{\fA} = \fP P^\bot \DA$}
\end{equation}
\end{lemma}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% PROOF %%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{proof}
By definition, $\fA_1$ is defined on $P^\bot \fD_{\fA}$. The operator
$\fP A_1 \fP$ is defined on $\fP P^\bot \DA$ which is by (\ref{PIPE}) and Lemma
\ref{L1} equal to $P^\bot \fD_{\fA}$. By definition of $A_1$ we get for all
$x \in \fP P^\bot \DA$:
\begin{equation}\label{PAP}
\fP A_1 \fP x = \fP (M^\bot)^{-1} A Z_A \fP x
\end{equation}
$\fP x = P^\bot \Pi Z_A x$ for all $x \in P^\bot \DA$. Therefore we have
\begin{equation}\label{ZA}
Z_A \fP x = \Pi Z_A x \qquad \text{ for all $x \in \fP P^\bot \DA$}.
\end{equation}
By construction of $\fM$ we get
$\fM^\bot \fP = \tilde \Pi \fM^\bot$ and this implies
\begin{equation}\label{MP}
\fP (M^\bot)^{-1} = (\fM ^\bot)^{-1} \tilde \Pi
\end{equation}
Inserting (\ref{ZA}) and (\ref{MP}) into (\ref{PAP}) yields
\begin{align*}
\fP A_1 \fP x &= (\fM^\bot)^{-1} \tilde\Pi A \Pi Z_A x\\
&=(\fM^\bot)^{-1} \fA Z_{\fA} x\\
&=\fA_1 x \qquad\text{ for all $x\in \fP P^\bot \DA = P^\bot \fD_\fA$}.
\end{align*}
\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% THEOREM 4.1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{theorem}\label{THGA}
Let $(M,A)$ be a degenerate Cauchy problem, assume that it is factorizable to a
nondegenerate Cauchy problem and that the operator $Z_A$ is bounded. For every $n= 1,2,...$
let $U^n$ be a
finite dimensional subspace of $\DA^c$ and define for each $U^n$ the projection
$\Pi ^n$, $\tilde \Pi ^n$ and $\fP ^n$ as before. Define the Operators $M^n$ and $A^n$
by
\begin{equation}\label{MnAn}
M^n = \tilde \Pi ^n M \Pi ^ n,\qquad A^n = \tilde\Pi ^n A \Pi ^n \qquad\text{on $\Pi ^n\DA$}
\end{equation}
Furthermore, let
\begin{enumerate}
\item $P^\bot \Pi^n Z_A x \longrightarrow x$ for all $x\in P^\bot \DA^c$
\item there exist constsants $K$ and $\omega$ such that
\begin{equation}\label{KO}
\|P^\bot (A-\lambda M)^{-1} M\|\le \frac{K}{\lambda - \omega}
\end{equation}
and for all $n = 1,2,...$
\begin{equation}\label{KOn}
\|P_{M^n} ^\bot A^n - \lambda M^n)^{-1} M^n\|\ \le \frac{K}{\lambda - \omega}
\end{equation}
\item there is a $\lambda \in \rho(M,A)\cap \bigcap_{n=1} ^\infty \rho (M^n,A^n)$ such that
for all $x\in P^\bot\DA ^c$
\begin{equation}\label{ETWAS}
\|P^\bot(A^n - \lambda M^n)^{-1} M^n \fP^n x - P^\bot(A-\lambda M)^{-1} M x \|
\longrightarrow 0 \quad\text{as $n \longrightarrow \infty$}.
\end{equation}
\end{enumerate}
For all $n\ge 1$ let $z_n(t)$ be the strict solution of the degenerate Cauchy
problem $(M^n,A^n)$. Then
\begin{equation*}
z_n (t) \longrightarrow z(t) \text{ uniformly on compact $t$-intervals}.
\end{equation*}
\end{theorem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Proof %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{proof}
Condition $1.$ implies $\fP^n x \longrightarrow x$ for all $x \in P^\bot \DA^c$. By condition
(\ref{KO}) the operators $A_1$ and $(A^n)_1$ generate semigroups of the class $(M,\omega)$.
Condition (\ref{ETWAS}) implies that the resolvents of $(A^n)_1$ converge. Using Lemma \ref{L2} we
get $(A^n)_1 = (A_1)^n$. So all the conditions for the Trotter-Kato Theorem for
the nondegenerate Cauchy problems
\begin{equation}\label{CPA}
\frac{d}{dt}x(t)= A_1 x(t)
\end{equation}
and
\begin{equation}\label{CPAn}
\frac{d}{dt} x_n (t) = (A_1)^n x_n(t)
\end{equation}
are fulfilled and therefore
\begin{equation*}
x_n(t)\longrightarrow x(t)
\end{equation*}
uniformly on bounded $t$-intervals. By Lemma \ref{L1} we have
$z_n (t) = Z_{A^n} x_n (t) = Z_A x_n (t)$ on $\fP ^n \DA ^c$. As $Z_A$ is assumed to be
bounded we get
\begin{equation*}
z_n (t)\longrightarrow z(t) \qquad \text{uniformly on bounded $t$-intervals}
\end{equation*}
\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% REMARKS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\begin{rem}\label{ENDREM1}
The situation in Theorem \ref{THGA} is an example when we can interchange factorization
and approximation. In this case it doesn't matter wether we factorize the degenerate
equation first and approximate the nondegenerate system by finite dimensional equations
or if we approximate the degenerate equation first. In general, we cannot expect the
approximating solutions of the factorized equation to approximate the degenerate problem,
because $Z_A$ need not be bounded and, in addition, we get for each equation a
different $Z_{A^n}$. Furthermore, approximating subspaces of $P^\bot \DA$ need not
lead to subspaces of $\DA$.
\end{rem}
%
%
%
\begin{rem}\label{ENDREM2}
If we want the projections $\fP^n$ to be orthogonal, we can first define the projections
$\fP^n$ and then the projections $\Pi ^n$ in order to guarantee orthogonality. In the case
where we consider the factorized problem in the graph norm of $Z_A$ both $\fP^n$ and
$\Pi^n$ are orthogonal at the same time because then $Z_A$ is an isometric isomorphism
between $\DA^c$ and $P^\bot \DA ^c$.
\end{rem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{XXXX}
\bibitem{CSH} Carroll, R. W. and Showalter, R. E.: {\em Singular and Degenerate Cauchy Problems},
Mathematics in Science and Engineering {\bf 127}, Academic Press, New York, San Francisco, London
1976.
\bibitem{TB} Gesztesy, F., Grosse, H., and Thaller, B.: {\em A rigorous approach to relativistic corrections of bound state energies for spin-1/2 particles},
Ann. Inst. H. Poincar\'e {\bf 40}, 159--174 (1984).
\bibitem{Kappel-Ito} Ito, K. and Kappel, F.: {\em On variational formulations of the
Trotter-Kato theorem}, Bericht No.39 des SFB "Optimierung und Kontrolle", University of Graz,
1995
\bibitem{KATO} Kato, T.: {\em Perturbation Theory for Linear Operators}, Springer
Verlag, New York 1976.
\bibitem{LR} Lamm, P.K., and Rosen, I.G.: {\em Approximation theory for the estimation
of parameters in degenerate Cauchy problems}, J. Math. Anal. Appl. {\bf 162}, 13--48 (1991)
\bibitem{MRR} Mao, C., Reich, S., and Rosen, I. G.: {\em Approximation in the
identification of nonlinear degenerate distributed parameter systems}, Nonl. Analysis, TMA,
{\bf 22}, 91-120 (1994)
\bibitem{RR} Rosen, I. G., and Raghu, P.: {\em Approximation in the identification of
degenerate distributed parameter systems: Generalized Galerkin schemes and nonconforming
elements}, International Series of Numerical Mathematics {\bf 100}, 273-293, Birkhauser
Basel 1991.
\bibitem{TT1} Thaller, B. and Thaller, S.: {\em Factorization of degenerate Cauchy problems: The linear
case}, Bericht No.18 des SFB "Optimierung und Kontrolle", University of Graz, 1995
\bibitem{TT2} Thaller, B. and Thaller, S.: {\em Some considerations on degenerate control
systems: the LQR problem }, Bericht No. 45 des SFB "Optimierung und Kontrolle", University of Graz, 1995
\bibitem{TT3} Thaller, B. and Thaller, S.: {\em Semigroup theory of degenerate linear Cauchy problems}, in preparation.
\bibitem{TROTTER} Trotter, H. F.: {\em Approximation of semigroups of operators} Pacific J.
Math. {\bf 8}, 887-919 (1958).
\end{thebibliography}
\end{document}