\documentclass[reqno]{amsart} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 95, pp. 1--14.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/95\hfil A new look at boundary perturbations] {A new look at boundary perturbations of generators} \author[G. Nickel\hfil EJDE-2004/95\hfilneg] {Gregor Nickel} \address{Gregor Nickel\hfill\break Mathematisches Institut, Universit\"at T\"ubingen, Auf der Morgenstelle 10, D-72076 T\"ubingen, Germany} \email{grni@fa.uni-tuebingen.de} \date{} \thanks{Submitted July 2, 2004. Published August 6, 2004.} \subjclass[2000]{47D06, 35K05, 93B28, 93C05} \keywords{Boundary value problems; boundary perturbation; \hfill\break\indent strongly continuous semigroups; operator matrices} \begin{abstract} In this paper we show that Greiner's results on boundary perturbation can be obtained {\em systematically} and partially generalised by applying additive perturbation theorems to appropriate operator matrices. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theo}{Theorem}[section] \newtheorem{lem}[theo]{Lemma} \newtheorem{prop}[theo]{Proposition} \newtheorem{cor}[theo]{Corollary} \newtheorem{rem}[theo]{Remark} \newtheorem{exa}[theo]{Example} \newtheorem{defi}[theo]{Definition} \newtheorem{genasu}[theo]{General Assumption} \newtheorem{asu}[theo]{Assumption} \section{Introduction} In a fundamental and much quoted paper \cite{greiner} G\"unther Greiner developed a perturbation theory for generators of strongly continuous semigroups where the perturbation does {\em not} change the {\em mapping} but rather its {\em domain}. His approach is motivated by a semigroup approach to abstract boundary value problems of the form \begin{equation} \label{BPf0} % (BP)_{f_0} \begin{gathered} \dot{f}(t) = A f(t), \quad t\ge 0, \\ Lf(t) = \Psi f(t), \quad t\ge 0, \\ f(0) = f_0 \in X\,. \end{gathered} \end{equation} for a linear operator $(A, D(A))$ defined on a Banach space $X$. This ``maximal'' operator is restricted by a ``boundary condition'' given by operators $L : D(A) \to \partial X$ and $\Psi : X \to \partial X$, where $\partial X$ is another Banach space called ``boundary space''. If we assume the problem to be well posed for zero boundary condition, i.e., for $\Psi = 0$, then the following problem arises. {\em For which perturbation $\Psi$ is the problem $(BP)$ well posed again?} In \cite{greiner} it is shown how delay and other functional differential equations (see \cite{wu:1996}, \cite{hale/lunel:1993}) and difference equations as well as diffusion equations fit into his abstract framework (see \cite{greiner} and the references cited therein). In \cite{greiner/kuhn:19??} this approach has been applied to semilinear problems and unbounded $\Psi$ supposing additional analyticity conditions. A more recent application to age structured population equations may be found in \cite{rhandi2}, \cite{rhandi:1998}. This approach is in contrast to the well established {\em additive} perturbation theory for generators (see \cite[Chap.\ III]{engel/nagel:2000}). In this paper, however, we show that Greiner's results can be obtained systematically and even generalised by applying %a multiplicative perturbation theorem on the one side and an additive perturbation theorems to appropriate operator matrices. % on the other. The paper is organised as follows. In Section \ref{section-2} we define wellposedness for boundary value problems in an abstract setting and characterise it by the generator property of a certain operator. Moreover, we state the general assumptions and a central lemma for the following approach. In Section \ref{generator-speziell} we use non-densely defined operator matrices to obtain wellposedness under specific conditions on $\Psi$. In the final Section \ref{reduction} we show how the boundary value problem can be solved by associating a {\em dynamical} boundary value problem and then using one-sided coupled operator matrices as introduced by Engel \cite{engel:1997}, \cite{engel:1996}, \cite{engel:1994}. \section{Abstract boundary-value problems} \label{section-2} \label{setting} We assume $X$ to be a Banach space, called {\bf (inner) state space}, and $\partial X$ to be another Banach space, called {\bf boundary space}. On $X$ we consider a linear operator \[ A : D(A) \subset X \to X, \] called the {\bf maximal operator}, describing the (internal) dynamics of the system. The connection between the state space and the boundary space is given by a linear operator \[ L : D(A) \subset X \to \partial X, \] the {\bf boundary operator}, relating the state $f \in D(A)$ to its boundary value $x := L f \in \partial X$. It is thus explicitly assumed that all $f \in D(A)$ ``have'' boundary values in $\partial X$. Moreover, we consider a linear {\bf boundary condition operator} \[ \Psi : X \to \partial X, \] which can be interpreted as a ''perturbation of the boundary condition.'' With the exception of Subsection \ref{section-unbound-Psi-bounded-ext}, we will assume $\Psi$ to be bounded. In the abstract perspective we consider the following {\bf boundary-value problem} $(BP)_{f_0}$, \begin{equation*} %\label{BPf0} %(BP)_{f_0} \begin{gathered} \dot{f}(t) = A f(t), \quad t\ge 0, \\ Lf(t) = \Psi f(t), \quad t\ge 0, \\ f(0) = f_0 \in X\,. \end{gathered} \end{equation*} Discussing wellposedness we write just $(BP)$ if the initial value is not fixed. We denote by $(A_0, D(A_0))$ the restriction of $(A, D(A))$ to zero boundary conditions, i.e., \begin{equation} D(A_0) := \{f \in D(A) : Lf = 0\}, \quad A_0f := Af. \end{equation} We now give an example of a difference equation illustrating the abstract concepts. \begin{exa} \label{subsection-paradigm} \rm Let $\partial X$ be a Banach space and $X_1 := L^1([-1,0], \partial X)$ be the state space. On this space we consider the maximal operator \[ Af := f' \] defined on the domain \[ D(A) := W^{1,1}([-1,0], \partial X). \] The boundary operator is given by \[ L \, : D(A) \to \partial X, \quad Lf := f(0). \] The operator $(A_0, D(A_0))$ is then the generator of the (nilpotent) left shift semigroup $(T_0(t))_{t \geq 0}$ on $X$. Finally, the boundary condition operator is $\Psi \in {\mathcal{L}}(X, \partial X)$. So, the boundary condition in $(BP)$ means $f(0) = \Psi(f)$ and $(BP)$ may be called ``difference equation'' (see, e.g., \cite[Sect.\ I.1.]{hale/lunel:1993}). \end{exa} In the sequel we define and characterise wellposedness for $(BP)$ by the generator property of an associated operator on $X$. To that purpose we use the following definition of classical %and mild solutions and wellposedness. \begin{defi} \label{def-wellposed-ABP} \rm A function $f: {\mathbb{R}}_+ \to X$ is called a {\bf classical solution} of \eqref{BPf0} if \begin{itemize} \item[$(i)$] $f(\cdot) \in C^1({\mathbb{R}}_+, X)$, \item[$(ii)$] $f(t) \in D(A)$ %\cap D(\Psi)$ for every $t \ge 0$, and \item[$(iii)$] $f(\cdot)$ satisfies \eqref{BPf0}. \end{itemize} The problem $(BP)$ is called {\bf wellposed} if \begin{itemize} \item[$(i)$] for all $f_0 \in D(A) \cap D(\Psi)$ with $Lf_0 = \Psi f_0$ there exists a unique classical solution $f(\cdot, f_0)$ of \eqref{BPf0}, \item[$(ii)$] the set \[ {\widetilde{D}} := \{f \in D(A) \cap D(\Psi) %\cap D(\Psi) : Lf = \Psi f\} \] of initial values admitting classical solutions is dense in $X$, and \item[$(iii)$] the solutions depend continuously on the initial data, i.e., for every sequence of initial data ${\widetilde{D}} \supset (f_n)_n \to 0$ %with $x_n := Lf_n \to 0$ the corresponding solutions $f(\cdot, f_n)$ fulfill $\lim_{n \to \infty} f(t, f_n) = 0$ %and $\lim_{n \to \infty} Lf(t, f_n,x_n) = 0$ uniformly for $t$ in compact subsets of ${\mathbb{R}}_+$. \end{itemize} \end{defi} % \begin{rem} \label{avoidtrivial}\rm Supposing $(BP)$ to be well posed we immediately infer that --- to avoid a trivial situation --- at least one of the operators $(L,D(A))$ or $(\Psi, D(\Psi))$ has to be unbounded. Otherwise, we would obtain the relation $Lf = \Psi f$ on the dense subset $\{f \in D(A) \cap D(\Psi) : Lf = \Psi f\} \subset X$ implying $L = \Psi$ by the boundedness of both operators. This, however, implies that the boundary condition $Lf = \Psi f$ is trivially fulfilled for all $f \in X$, and problem $(BP)$ is equivalent to an abstract Cauchy problem without boundary condition. \end{rem} We now characterise wellposedness of $(BP)$ by the generator property of a restriction of the maximal operator $(A, D(A))$ to an operator with ``perturbed domain''. \begin{defi} \rm Consider the linear operator $(A_\Psi, D(A_\Psi))$ on $X$ defined by \begin{equation} D(A_\Psi) := \{f \in D(A) \cap D(\Psi) : Lf = \Psi f\}, \quad A_\Psi f := Af, \end{equation} and the associated abstract Cauchy problem %(ACP)_{f_0} \begin{equation} \label{ACPf0} \begin{gathered} \dot{f}(t) = A_\Psi f(t), \quad t\ge 0, \\ f (0) = f_0 \in X\,. \end{gathered} \end{equation} \end{defi} The following result connects between wellposedness of $(BP)$ and the generator property of $A_\Psi$. The proof (using \cite[Thm.\ II.6.7]{engel/nagel:2000}) is straightforward and will be omitted. \begin{prop} \label{well-ABP} The abstract boundary value problem $(BP)$ is wellposed if and only if $(A_\Psi, D(A_\Psi))$ is the generator of a strongly continuous semigroup $(T_\Psi(t))_{t \ge 0}$ on $X$. In that case, $t \mapsto T_\Psi(t) f_0$ gives the classical solutions of \eqref{BPf0} for all $f_0 \in D(A_\Psi)$. %and mild solutions for all $f_0 \in X$. \end{prop} \subsection{Greiner's lemma and the Dirichlet operators} In the spirit of Greiner's approach we assume the following properties of $(A, D(A))$ and $L$ (see also \cite{casarino:2001}). \begin{genasu} \label{genasu-static}\rm In the general setting of Section \ref{setting} we assume that \begin{itemize} \item[$(S1)$] the boundary operator $L : D(A) \subset X \to \partial X$ is surjective and the operator \[ \begin{pmatrix} A\\ L \end{pmatrix} : D(A) \to X \times \partial X, \quad D(A) \ni f \mapsto \begin{pmatrix} Af \\ Lf \end{pmatrix} \] is closed. \item[$(S2)$] The operator $A_0 := A_{|\ker L}$ defined as the restriction of $A$ to the kernel of $L$ generates a strongly continuous semigroup $(T_0(t))_{t \geq 0}$ on the state space $X$. \end{itemize} \end{genasu} The above assumptions imply a decomposition of the domain $D(A)$ which is fundamental for the following approach (see also \cite[Lem.\ 2.2]{casarino:2001}). \begin{lem}[Greiner \mbox{\cite[Lem.\ 1.2]{greiner}}] \label{Greiner} %(1) Assume $(S1)$ and $(S2)$ of the General Assumptions \ref{genasu-static} and take $\lambda \in \rho(A_0)$. Then the restriction of $L$ to $\ker (\lambda - A)$ \[ L_\lambda:= L|_{\ker (\lambda - A)} : \ker (\lambda - A) \to \partial X \] is invertible with bounded inverse $D_{\lambda}$. Moreover, for all $\mu, \lambda \in \rho (A_0)$ we have \begin{eqnarray} R(\mu, A_0) D_\lambda &= R(\lambda, A_0) D_\mu, \label{ResolvDlambda}\\ D_\lambda &= (1- (\lambda - \mu)R(\lambda, A_0))D_\mu. \label{L} %\\ %\|D_{\lambda}\| &\leq& C < \infty \label{NormD} \end{eqnarray} \end{lem} The operators $D_\lambda \in {\mathcal{L}}(\partial X,X)$ play a key role in our approach and correspond to the Dirichlet map in the case of boundary value problems for partial differential equations (see \cite[Section 3]{casarino:2001}). Therefore we use the following terminology. \begin{defi} \rm The operator \[ D_{\lambda} : \partial X \to \ker (\lambda - A) \subset X \] is called {\bf Dirichlet operator} corresponding to the boundary operator $L$, the maximal operator $(A, D(A))$ and the value $\lambda \in {\mathbb{C}}$. \end{defi} \begin{exa} \label{exa-static-Dirichlet-operators} \rm We consider the situation of Example \ref{subsection-paradigm}. The Dirichlet operators corresponding to the boundary operator $L$ and the maximal operator $(A, D(A))$ are given by \[ D_\lambda : \partial X \to L^1([-1,0], \partial X), \quad x \mapsto D_\lambda x := \epsilon_\lambda x, \] where \[ \epsilon_\lambda x (\tau) := e^{(\lambda \tau)} x, \quad \tau \in [-1, 0], \] for all $\lambda \in {\mathbb{C}}$. \end{exa} The characterisation of wellposedness of $(BP)$ given in Proposition \ref{well-ABP} does not contain explicit conditions on the operators $(A, D(A))$, $(L, D(A))$, and $(\Psi, D(\Psi))$. So, the following two sections are devoted to find conditions implying the generator property of $(A_\Psi, D(A_\Psi))$. \section{Wellposedness by non-densely defined operator matrices} \label{generator-speziell} The first question is whether, assuming the General Assumptions \ref{genasu-static}, $(BP)$ is wellposed for {\em any} bounded operator $\Psi \in {\mathcal{L}}(X, \partial X)$. % In fact, Greiner gives an example in \cite{greiner} illustrating that this fails in general. However, assuming %certain special conditions more on $(A, D(A))$ and $L$ this holds true. In the following section we show these results (and a slight generalisation) by using operator matrices and additive perturbation. To do so, we will now enlarge the state space by ``adding'' the boundary values, i.e., we consider the product space \[ {\mathcal{X}} := X \times \partial X, \] and embed $X$ as ${\mathcal{X}}_0 := X \times \{ 0 \}$. The projections on the two factor spaces are denoted by $\Pi_1 : {\mathcal{X}} \to X, \quad \Pi_1 \begin{pmatrix}f \\ x\end{pmatrix} := f$ and $\Pi_2 : {\mathcal{X}} \to \partial X, \quad \Pi_2 \begin{pmatrix}f \\ x\end{pmatrix} := x$, respectively. \begin{defi} \rm Consider on ${\mathcal{X}}$ the operator matrix $({\mathcal{L}}, D({\mathcal{L}}))$ defined by \begin{equation} {\mathcal{L}} := \begin{pmatrix} A & 0 \\ -L & 0 \end{pmatrix} \end{equation} on the domain $D({\mathcal{L}}) := D(A) \times \{ 0 \} \subset {{\mathcal{X}}}$. \end{defi} \begin{rem} \label{triv-part} \rm Clearly, ${\mathcal{L}}$ is not densely defined on the part $({\mathcal{L}}_0, D({\mathcal{L}}_0))$ of $({\mathcal{L}}, D({\mathcal{L}}))$ in ${\mathcal{X}}_0 := X \times \{0\}$, i.e., \begin{gather*} D({\mathcal{L}}_0) := D(A_0) \times \{0\}, \\ {\mathcal{L}}_0 \begin{pmatrix}f \\ x\end{pmatrix} := \begin{pmatrix} A_0 f \\ 0 \end{pmatrix}, \end{gather*} can be identified with $(A_0, D(A_0))$. \end{rem} Consider now the perturbed matrix \[ {\mathcal{M}} := \begin{pmatrix} A & 0 \\ \Psi-L & 0 \end{pmatrix} = {\mathcal{L}} + \begin{pmatrix} 0 & 0 \\ \Psi &0 \end{pmatrix} =: {\mathcal{L}} + {\mathcal{P}} \] still defined on $D({\mathcal{L}}) \subset {\mathcal{X}}$. As before, the part of ${\mathcal{M}}$ in ${\mathcal{X}}_0$ is (isomorphic to) $A_\Psi$. Therefore, if we can show that this part in ${\mathcal{X}}_0$ generates a strongly continuous semigroup, we obtain the semigroup solving $(BP)$. We state this observation explicitly. \begin{lem} Consider the operator matrix $({\mathcal{M}}, D({\mathcal{M}}))$ defined by \[ {\mathcal{M}} := \begin{pmatrix} A & \quad 0 \\ \Psi-L & \quad 0 \end{pmatrix} \] on the domain \[ D({\mathcal{M}}) := D(A) \times \{ 0 \} \] and denote its part in ${\mathcal{X}}_0$ by ${\mathcal{M}}_0$, i.e., \[ D({\mathcal{M}}_0) := \Big\{ \begin{pmatrix}f \\ x\end{pmatrix} \in D(A) \times \{0\} : {\mathcal{M}} \begin{pmatrix}f \\ x\end{pmatrix} \in X \times \{0\} \Big\}. \] Then we have $D({\mathcal{M}}_0) = D(A_{\Psi}) \times \{ 0 \}$ and \[ {\mathcal{M}}_0 = \begin{pmatrix} A_\Psi & 0 \\ 0 & 0 \end{pmatrix}. \] Thus ${\mathcal{M}}_0$ is a generator of a strongly continuous semigroup $({\mathcal{T}}_{\Psi}(t))_{t \ge 0}$ if and only if $A_\Psi$ is and the classical solutions of \eqref{BPf0} are obtained as \[ {\mathbb{R}}_+ \ni t \mapsto \Pi_1 \Big[ {\mathcal{T}}_{\Psi}(t) \begin{pmatrix}f_0\\ 0\end{pmatrix} \Big] \] for every $f_0 \in D(A_\Psi)$.%generates a strongly continuous semigroup if and only if $(A_\Psi, D(A_\Psi))$ generates a strongly continuous semigroup. \end{lem} This simple observation allows the use of %result leads, however, to powerful tools for showing wellposedness of $(BP)$. We only have to show that ${\mathcal{M}}$ satisfies the Hille-Yosida estimates. Then it follows from \cite[Cor. II.3.21]{engel/nagel:2000} that its part in the closure of its domain is a generator. The following two subsections are devoted to follow this path. \subsection[Hille-Yosida operator matrices]{Hille-Yosida operator matrices} \label{section-HY-operatormatrix} In this section we assume $\Psi \in {\mathcal{L}}(X, \partial X)$ to be bounded, show that ${\mathcal{L}}$ is a Hille-Yosida operator, and then apply the bounded perturbation theorem for these operators. The idea of this approach is due to \cite{rhandi2}. In addition to the General Assumptions \ref{genasu-static} we now make an additional boundedness assumption on the operators $D_\lambda$. \begin{asu} \label{Dlambda-HY} \rm ~ \begin{itemize} \item[$(S3)$] Assume that there exists $\omega_3 \in {\mathbb{R}} $ %> \omega_0$ and $C \ge 0$ such that for all $\lambda > \omega_3$, \begin{equation} \label{Dlambda1} \| D_\lambda \|_{{\mathcal{L}} (\partial X, X)} \leq \frac{C}{(\lambda - \omega_3)}\,. \end{equation} \end{itemize} \end{asu} \begin{lem} \label{lemma-resolv-ungestoert} Under the Assumptions %\ref{genasu-static} $(S1), (S2)$, and $(S3)$ the operator $({\mathcal{L}}, D({\mathcal{L}}))$ is a Hille-Yosida operator on the space $X \times \partial X$. Its resolvent is given by the operator matrix %${\mathcal{R}}_\lambda \in {{\mathcal{L}}}({\mathcal{X}})$ \begin{equation} \label{resolv-abstr-HY} {\mathcal{R}}_\lambda := \begin{pmatrix} R(\lambda, A_0) & D_\lambda \\ 0 & 0 \end{pmatrix} \in {{\mathcal{L}}}({\mathcal{X}}) \end{equation} for all $\lambda \in \rho(A_0)$. \end{lem} % \begin{proof} To show that ${\mathcal{R}}_\lambda$ is the resolvent of ${\mathcal{L}}$ for $\lambda \in \rho(A_0)$ we first remark that ${\mathcal{R}}_\lambda$ is a bounded operator on ${\mathcal{X}}$ by Lemma \ref{Greiner}. Second, for all $\begin{pmatrix}f \\ x\end{pmatrix} \in {\mathcal{X}}$ we obtain \[ {\mathcal{R}}_\lambda \begin{pmatrix}f \\ x\end{pmatrix} = \begin{pmatrix} R(\lambda, A_0) f + D_\lambda x \\ 0 \end{pmatrix} \in D({\mathcal{L}}_0) \] and \begin{align*} (\lambda - {\mathcal{L}}){\mathcal{R}}_\lambda \begin{pmatrix}f \\ x\end{pmatrix} &= \begin{pmatrix} \lambda - A & 0 \\ L & \lambda \end{pmatrix} \begin{pmatrix} D_\lambda x + R(\lambda, A_0) f \\ 0 \end{pmatrix} \\ &= \begin{pmatrix} (\lambda - A ) (D_\lambda x + R(\lambda, A_0) f) \\ LD_\lambda x + L R(\lambda, A_0) f \end{pmatrix} \\ &= \begin{pmatrix} (\lambda - A_0) R(\lambda, A_0) f\\ LD_\lambda x \end{pmatrix} = \begin{pmatrix}f \\ x\end{pmatrix} . \end{align*} Moreover, for $\begin{pmatrix}f \\ x\end{pmatrix} \in D({\mathcal{L}})$, i.e., $x = 0$ and $f \in D(A)$, we obtain \begin{align*} {\mathcal{R}}_\lambda (\lambda - {\mathcal{L}}) \begin{pmatrix}f \\ x\end{pmatrix} &= \begin{pmatrix} R(\lambda, A_0) & D_\lambda \\ 0 & 0 \end{pmatrix} \begin{pmatrix} (\lambda - A)f \\ Lf \end{pmatrix} \\ &= \begin{pmatrix} R(\lambda, A_0) (\lambda - A)f + D_\lambda L f \\ 0 \end{pmatrix} = \begin{pmatrix}f \\ x\end{pmatrix} \end{align*} since \begin{align*} R(\lambda, A_0) (\lambda - A)f + D_\lambda L f &= (\lambda, A_0)(\lambda - A) [f - D_\lambda L f] + D_\lambda L f \\ &= R(\lambda, A_0)(\lambda - A_0) [f - D_\lambda L f] + D_\lambda L f \\ &= f - D_\lambda L f + D_\lambda L f = f. \end{align*} % Third, the powers of ${\mathcal{R}}_\lambda$ can be obtained easily as \[ {\mathcal{R}}_\lambda^{n+1} = \begin{pmatrix} R(\lambda, A_0)^{n+1} & R(\lambda, A_0)^{n} D_\lambda \\ 0 & 0 \end{pmatrix} \] for $n \in {\mathbb{N}}$. Thus for $\omega > \tilde{\omega} := \max \{\omega_0, \omega_3\}$ there exists $M \ge 1$ such that \begin{align*} \|{\mathcal{R}}_\lambda^{n+1}\| &\leq \max \{ \|R(\lambda, A_0)^{n+1}\|, \|R(\lambda, A_0)^{n} D_\lambda\|\} \\ &\leq \max \{ \frac{M}{(\lambda - \omega_0)^{n+1}}, \frac{CM}{(\lambda - \omega_3)(\lambda - \omega_0)^{n}}\} \leq \frac{{\widetilde{M}}}{(\lambda - \tilde{\omega})^{n+1}} \end{align*} by $(S3)$ for some ${\widetilde{M}} \ge 1$ and all $\lambda \ge \tilde{\omega}$. \end{proof} The bounded perturbation of a Hille-Yosida operator is again a Hille-Yosida operator (see \cite[Thm.\ III.1.3]{engel/nagel:2000}) and the part of a Hille-Yosida operator is a generator on the closure of its domain (see \cite[Cor. II.3.21]{engel/nagel:2000}). We thus immediately obtain one of Greiner's results. \begin{theo}[\mbox{\cite[Thm. 2.1]{greiner}}] \label{H-Y-bounded} Let the Assumptions %\ref{genasu-static} $(S1),(S2)$, and %\ref{Dlambda-HY} $(S3)$ hold and assume $\Psi \in {\mathcal{L}} (X , \partial X)$. Then the matrix $({\mathcal{M}}, D({\mathcal{L}}))$ defined by \[ {\mathcal{M}} := \begin{pmatrix} A & \quad 0 \\ \Psi-L & \quad 0 \end{pmatrix} =: {\mathcal{L}} + {\mathcal{P}} \] is a Hille-Yosida operator on the space $X \times \partial X$. Thus, its part $({\mathcal{M}}_0, D({\mathcal{M}}_0))$ in ${\mathcal{X}}_0 := X \times \{0\}$ is the generator of a strongly continuous semigroup $({\mathcal{T}}_{\Psi}(t))_{t \ge 0}$. \end{theo} Summing up, we obtain the following wellposedness result for $(BP)$. \begin{cor} Under the conditions of Theorem \ref{H-Y-bounded} the boundary value problem $(BP)$ is wellposed. \end{cor} \begin{exa} \label{exa-static-boundedPsi} \rm We consider the situation of Example \ref{subsection-paradigm}. An estimate of the norm of the Dirichlet operators yields \[ \| D_\lambda \| \leq \int_{-1}^0 e^{(\lambda \tau)} \, d\tau = \frac{1}{\lambda} [1 -e^{-\lambda}] \leq \frac{1}{\lambda} \] for all $\lambda > 0$. Thus the conditions $(S1),(S2)$, and $(S3)$ are fulfilled and for every bounded operator $\Psi : L^1 ([-1,0],\partial X) \to \partial X$ the ``difference equation'' % (\Delta)_{f_0} \begin{equation} \label{Df0} \begin{gathered} \dot{f}(t) = f'(t), \quad t\ge 0, \\ f(t)(0) = \Psi f(t), \quad t\ge 0, \\ f(0) = f_0 \in L^1 ([-1,0],\partial X) \end{gathered} \end{equation} is well posed. \end{exa} \subsection{Unbounded boundary condition $\Psi$ with bounded extension $D_\lambda \Psi$} \label{section-unbound-Psi-bounded-ext} In this subsection we do not assume boundedness of $(\lambda - \omega_3) D_\lambda$ as in $(S3)$ and boundedness of the operator $\Psi$, separately. Instead, we assume %operators $\Psi : D(A) \to \partial X$ and the following smallness condition of $\Psi$ with respect to $D_\lambda$. \begin{asu} \label{asu-Psi-unbound1} \rm ~ \begin{itemize} \item[$(S4)$] Let $\Psi : D(A) \to \partial X$ be a linear operator and assume that each \[ D_\lambda \Psi : D(A) \to D(A) \subset X \] can be extended continuously to bounded operators $D_\lambda \Psi : X \to X$ such that \begin{equation} \label{D-Psi} \| \lambda D_\lambda \Psi \| \leq C < \infty \end{equation} for some $\omega_4 \in {\mathbb{R}}$ and all $\lambda > \omega_4$. \end{itemize} \end{asu} Assuming $(S4)$ we obtain the following generation result. %(see \cite{greiner/kuhn:19??}). \begin{theo}%[\mbox{\cite[Thm. ??]{greiner/kuhn:19??}}] \label{H-Y-unbounded} Let the Assumptions %\ref{genasu-static} $(S1),(S2)$, and $(S4)$ hold and consider the operator matrix $({\mathcal{M}}, D({\mathcal{L}}))$ defined by \[ {\mathcal{M}} := \begin{pmatrix} A & 0 \\ \Psi-L & 0 \end{pmatrix} = \begin{pmatrix} A & 0 \\ -L & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ \Psi & 0 \end{pmatrix} =: {\mathcal{L}} + {\mathcal{P}}. \] Then there exists $c > 0$ such that for all $\lambda > c$ we have $\lambda \in \rho({\mathcal{M}})$ and \[ R(\lambda, {\mathcal{M}}) = \begin{pmatrix} \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n R(\lambda, A_0) & \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n D_\lambda \\ 0 & 0 \end{pmatrix}. \] The part $({\mathcal{M}}_0, D({\mathcal{M}}_0))$ of $({\mathcal{M}}, D({\mathcal{L}}))$ in ${\mathcal{X}}_0 := X \times \{0\}$ is the generator of a strongly continuous semigroup and its resolvent is given by \[ R(\lambda, {\mathcal{M}}_0) = R(\lambda, {\mathcal{M}})_{|{\mathcal{X}}_0} = \begin{pmatrix} \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n R(\lambda, A_0) & 0 \\ 0 & 0 \end{pmatrix} \] for all $\lambda > c$. \end{theo} \begin{proof} By rescaling (see \cite[Ex.\ II.2.2]{engel/nagel:2000}) and renorming the space $X$ (see \cite[Lem.\ II.3.10]{engel/nagel:2000}) we may assume without loss of generality that $(A_0, D(A_0))$ generates a contraction semigroup. Condition (\ref{D-Psi}) still holds with another constant ${\widetilde{C}}$. For every $\lambda > 0$ we thus obtain $\lambda \in \rho(A_0) \cap \rho ({\mathcal{L}})$ and \[ R(\lambda, {\mathcal{L}}) = \begin{pmatrix} R(\lambda, A_0) &D_\lambda \\ 0 &0 \end{pmatrix} \] by the same argument as in Lemma \ref{lemma-resolv-ungestoert}. We now write %with $\RRe \lambda > 0$ \begin{equation} \label{resolv-faktor} (\lambda - {\mathcal{M}}) = (\lambda - {\mathcal{L}} - {\mathcal{P}}) = (\lambda - {\mathcal{L}})[\mbox{Id} - R(\lambda, {\mathcal{L}}) {\mathcal{P}}] \end{equation} which is invertible if and only if $[\mbox{Id} - R(\lambda, {\mathcal{L}}){\mathcal{P}}]$ is. We thus consider \[ R(\lambda, {\mathcal{L}}) {\mathcal{P}} = \begin{pmatrix} D_\lambda \Psi & 0 \\ 0 & 0 \end{pmatrix} \in \mathcal{L} ({\mathcal{X}}) \] and calculate \[ [R(\lambda, {\mathcal{L}}){\mathcal{P}}]^n = \begin{pmatrix} (D_\lambda \Psi)^n & 0 \\ 0 & 0 \end{pmatrix} \quad \mbox{for } n \ge 1. \] For all $\lambda > c := \max\{{\widetilde{C}}, \omega_4\}$ we thus conclude from (\ref{D-Psi}) that \[ \|R(\lambda, {\mathcal{L}}){\mathcal{P}}\| < \frac{\widetilde{C}}{\lambda} < 1 \] and \[ [\mbox{Id} - R(\lambda, {\mathcal{L}}){\mathcal{P}}]^{-1} = \begin{pmatrix} \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n & 0 \\ 0 & \mbox{Id} \end{pmatrix} = \begin{pmatrix} [\mbox{Id} - D_\lambda \Psi]^{-1} & 0 \\ 0 & \mbox{Id} \end{pmatrix} \] with \[ \|[\mbox{Id} - R(\lambda, {\mathcal{L}}){\mathcal{P}}]^{-1}\| \leq \frac{1}{1-{\widetilde{C}}/ \lambda}. \] We therefore obtain the inverse of $(\lambda - {\mathcal{M}})$ as \[ R(\lambda, {\mathcal{M}}) = \begin{pmatrix} \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n R(\lambda, A_0) & \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n D_\lambda \\ 0 & 0 \end{pmatrix}. \] Restricting $R(\lambda, {\mathcal{M}})$ to ${\mathcal{X}}_0 = X \times \{0\}$ the resolvent becomes \[ R(\lambda, {\mathcal{M}}_0) = R(\lambda, {\mathcal{M}})_{|{\mathcal{X}}_0} = \begin{pmatrix} \sum_{n = 0}^{\infty}(D_\lambda \Psi)^n R(\lambda, A_0) & 0 \\ 0 & 0 \end{pmatrix}, \] which is the resolvent of the part $({\mathcal{M}}_0, D({\mathcal{M}}_0))$ of $({\mathcal{M}}, D({\mathcal{L}}))$ in ${\mathcal{X}}_0$. Its norm can now be estimated for all $\lambda > {\widetilde{C}}$ by \[ \|R(\lambda, {\mathcal{M}}_0)\| = \Big\|\sum_{n = 0}^{\infty}(D_\lambda \Psi)^n R(\lambda, A_0)\Big\| \leq \frac{1}{1-{\widetilde{C}}/\lambda} \frac{1}{ \lambda} = \frac{1}{ \lambda - {\widetilde{C}}}. \] The operator $({\mathcal{M}}_0, D({\mathcal{M}}_0))$ is densely defined since $D(A_0)$ is dense in $X$ by assumption and since $\sum_{n = 0}^{\infty}(D_\lambda \Psi)^n$ is invertible. We thus obtain $({\mathcal{M}}_0, D({\mathcal{M}}_0))$ as the generator of a (quasicontractive) strongly continuous semigroup. \end{proof} \begin{rem} \rm It is easy to see that the condition $(S4)$ is weaker than condition $(S3)$. We thus obtained a generalization of Greiner's Theorem 2.1 \cite{greiner} to unbounded operators $(\Psi, D(A))$. %which was obtained in \cite{greiner/kuhn:19??}. \end{rem} By the same argument as in Section \ref{section-HY-operatormatrix} we obtain a wellposedness result for the boundary value problem. \begin{cor} If the conditions of Theorem \ref{H-Y-unbounded} are fulfilled, then $(BP)$ is well posed. \end{cor} \section{Wellposedness by reduction to dynamic boundary-value problems} \label{reduction} In this section we show how wellposedness %the above Theorem \ref{Teo-by-mult} can also be obtained by associating to the boundary value problem a {\em dynamic} boundary value problem and then solve it by operator matrix techniques as developed, e.g., in \cite{engel:1994}, \cite{casarino:2001}, \cite{kramar:2002}. If $\Psi : X \to \partial X$ is a bounded operator and $f(\cdot) \in C^1({\mathbb{R}}_+, X)$, then the function $\Psi f(\cdot)$ is also differentiable. Therefore, if $f(\cdot)$ solves \eqref{BPf0}, then also $Lf(\cdot) = \Psi f(\cdot)$ is differentiable, and $f(\cdot)$ solves the {\em dynamic} boundary value problem % (DBP)_{f_0,\Psi (f_0)} \begin{equation} \label{DBPfPf0} \begin{gathered} \dot{f}(t) = A f(t), \quad t\ge 0, \\ x(t) := Lf(t), \quad t\ge 0, \\ \dot{x}(t) = (\Psi A) f(t), \quad t\ge 0, \\ f(0) = f_0 \in X, \qquad x(0) = \Psi f_0 \in \partial X\,. \end{gathered} \end{equation} This observation leads to the following approach using a characterisation for wellposedness of $(DBP)$ by the generator property of an operator matrix with coupled domain. \begin{defi} \label{Definition-matrix-BP} \rm On ${\mathcal{X}}:=X\times\partial X$ we define the operator matrix \begin{equation} {\mathcal{A}}_\Psi := \begin{pmatrix} A &0\\ \Psi A &0 \end{pmatrix} \end{equation} with domain \begin{equation} D({\mathcal{A}}_\Psi):= \Big\{ \begin{pmatrix}f \\ x\end{pmatrix} \in D(A)\times \partial X : Lf=x \Big\}. \end{equation} Moreover, we consider the corresponding abstract Cauchy problem % (ACP)_{f_0, x_0} \begin{equation} \label{ACPf0x0} \begin{gathered} \dot{U}(t) = {\mathcal{A}}_\Psi U(t), \quad t\ge 0, \\ U (0) = \begin{pmatrix} f_0 \\ x_0\end{pmatrix} \in {\mathcal{X}}\,. \end{gathered} \end{equation} with initial values $f_0 \in X$ and $x_0 \in \partial X$. \end{defi} As suggested by the above observation connecting $(BP)$ and $(DBP)$, the generator property of ${\mathcal{A}}_\Psi$ implies the generator property of $A_\Psi$. \begin{prop} \label{wp-wp} Assume $D(A_\Psi)$ to be dense in $X$. If the matrix $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ generates a strongly continuous semigroup on ${\mathcal{X}}$ %and $D(A_\Psi)$ is dense in $X$, then so does $(A_\Psi, D(A_\Psi))$ on $X$. \end{prop} \begin{proof} Let $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ be a generator and consider $f_0 \in D(A_\Psi)$. Then \[ \begin{pmatrix}f_0 \\ \Psi f_0 \end{pmatrix} = \begin{pmatrix} f_0 \\ L f_0\end{pmatrix} \in D({\mathcal{A}}_\Psi), \] and we consequently obtain a classical solution for \eqref{DBPfPf0}. Denote its first component by $f(\cdot)$. Then $f(0) = f_0$, $Lf(0) = \Psi f(0)$ and the equations \[ \dot{f}(t) = Af(t), \quad t \ge 0, \] and \[ \frac{d}{dt}{Lf}(t) = \Psi A f(t) = \Psi \frac{d}{dt}{f}(t) = \frac{d}{dt}{\Psi f}(t), \quad t \ge 0 \] hold. This in turn implies $Lf(t) = \Psi f(t)$ for all $t \ge 0$ and thus $f(\cdot)$ is a classical solution of \eqref{BPf0}. Continuous dependence of the solutions is obtained easily. Moreover, the closedness of ${\mathcal{A}}_\Psi$ implies the closedness of $A_\Psi$. To see this, consider $D(A_\Psi) \supset f_n \to f_0 \in X$ and $A_\Psi f_n = A f_n \to g \in X$. Then we infer $\Psi A f_n \to \Psi g$ and $Lf_n = \Psi f_n \to \Psi f_0 \in X$ by the boundedness of $\Psi$. Since ${\mathcal{A}}_\Psi$ is closed and ${\mathcal{A}}_\Psi \begin{pmatrix}f_n \\ L f_n\end{pmatrix} \to \begin{pmatrix}g \\ \Psi g\end{pmatrix}$, this implies $\begin{pmatrix}f_0 \\ \Psi f_0\end{pmatrix} \in D({\mathcal{A}}_\Psi)$ and ${\mathcal{A}}_\Psi \begin{pmatrix}f_0 \\ \Psi f_0\end{pmatrix} = \begin{pmatrix}g \\ \Psi g\end{pmatrix}$. Explicitly this means that $f_0 \in D(A)$, $A f_0 = g$, and $Lf_0 = \Psi f_0$. Thus $f_0 \in D(A_\Psi)$ and $A_\Psi f_0 = g$ which means that $A_\Psi$ is closed. By a well known theorem (see \cite[Thm.\ II.6.7]{engel/nagel:2000} we infer that $(A_\Psi, D(A_\Psi))$ is a generator. \end{proof} With respect to wellposedness the two systems are, however, {\em not} equivalent. This is due to the fact that there are mild solutions for $(DBP)_{f_0, x_0}$ for {\em all} $f_0 \in X$ and $x_0 \in \partial X$, while for \eqref{BPf0} the condition $x_0 = \Psi f_0$ must always hold. So there are more mild solutions for $(DBP)$ and wellposedness of $(DBP)$ implies wellposedness of $(BP)$, but not conversely. Here is an example. \begin{exa} \rm Let $(A, D(A))$ be a generator on $X$ and $L = \Psi \in {\mathcal{L}}(X, \partial X)$ be any bounded operator. Then $A_\Psi = A$ and the boundary value problem $(BP)$ is equivalent to the abstract Cauchy problem for the generator $A$, thus wellposed. However, the matrix \begin{equation} {\mathcal{A}}_\Psi := \begin{pmatrix} A &0\\ \Psi A &0 \end{pmatrix} \end{equation} with domain \begin{equation} D({\mathcal{A}}_\Psi):= \Big\{ \begin{pmatrix}f \\ x\end{pmatrix} \in D(A)\times \partial X : Lf=x \Big\} \end{equation} is not even densely defined, thus not a generator, and the coresponding dynamical boundary value problem $(DBP)$ is not wellposed. \end{exa} In view of the preceding Proposition \ref{wp-wp} we have to find conditions implying the matrix $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ to be the generator of a strongly continuous semigroup. This situation has been studied in \cite{kramar:2002} based on the theory of one-sided coupled operator matrices developed by Engel (see \cite{engel:1994}). We will now apply these results, in part \cite[Prop.\ 4.3]{kramar:2002} to our situation. We sketch the proof and refer to \cite{engel:1994} and \cite{kramar:2002} for more details. It turns out that the condition for the generator property of the matrix $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ is exactly the condition obtained by applying multiplicative perturbation theory (see Remark \ref{Teo-by-mult} below). \begin{theo} \label{theo-BP-reduction} Assume the General Assumptions \ref{genasu-static} and let $\Psi : X \to \partial X$ be a bounded operator. %and assume $(C1)$ and $(C2)$ %for $C := \Psi A$. Moreover, assume that $\Psi A_0$ is relatively $(\mbox{Id} - \Psi D_\lambda)A_0$-bounded. \noindent (1) Then the matrix $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ of Definition \ref{Definition-matrix-BP} is the generator of a strongly continuous (analytic) semigroup if and only if the operator $(A_0 - D_\lambda \Psi A_0, D(A_0))$ is the generator of a strongly continuous (analytic) semigroup for some $\lambda \in \rho(A_0)$. \noindent (2) In that case, the operator $(A_\Psi, D(A_\Psi))$ is the generator of a strongly continuous (analytic) semigroup. \end{theo} \begin{proof} (1) For any fixed $\lambda \in \rho(A_0)$ we can factor the matrix ${\mathcal{A}}_\Psi - \lambda$ as \begin{equation} \label{fact} {\mathcal{A}}_\Psi - \lambda = \begin{pmatrix} A_0 - \lambda & 0 \\ \Psi A_0 &\lambda \Psi D_\lambda - \lambda \end{pmatrix} \begin{pmatrix} \mbox{Id}_X & - D_\lambda \\ 0 & \mbox{Id}_{\partial X} \end{pmatrix} =: {\mathcal{A}}_d \mathcal{D}_\lambda \end{equation} with the bounded and invertible operator $\mathcal{D}_\lambda \in {\mathcal{L}} ({\mathcal{X}})$ and an operator matrix $({\mathcal{A}}_d, D({\mathcal{A}}_d))$ with diagonal domain $D({\mathcal{A}}_d) := D(A_0) \times \partial X$. To verify this factorisation we first remark that $\begin{pmatrix}f \\ x\end{pmatrix} \in D({\mathcal{A}}_\Psi)$ is equivalent to $f \in D(A)$, $x \in \partial X$, and $x = Lf$. This in turn is equivalent to $f \in X$, $x \in \partial X$, and $f - D_\lambda x \in D(A_0)$, i.e., $\begin{pmatrix}f \\ x\end{pmatrix} \in D({\mathcal{A}}_d \mathcal{D}_\lambda)$. The equality $(\ref{fact})$ is now obtained by considering $\begin{pmatrix}f \\ x\end{pmatrix} \in D({\mathcal{A}}_\Psi)$ and calculating \begin{align*} {\mathcal{A}}_d \mathcal{D}_\lambda \begin{pmatrix}f \\ x\end{pmatrix} &= {\mathcal{A}}_d \begin{pmatrix} f -D_\lambda x \\ x\end{pmatrix} \\ &= \begin{pmatrix}(A_0 - \lambda) (f -D_\lambda x) \\ \Psi A_0 (f -D_\lambda x) + (\lambda \Psi D_\lambda - \lambda)x \end{pmatrix}\\ &= \begin{pmatrix} (A- \lambda) f \\ \Psi A f - \lambda x \end{pmatrix}\\ &= ({\mathcal{A}}_\Psi - \lambda) \begin{pmatrix}f \\ x\end{pmatrix} \end{align*} since $A_0 (f - D_\lambda f) = Af - \lambda f$. Due to the invertibility of $\mathcal{D}_\lambda$ the matrix ${\mathcal{A}}_\Psi - \lambda$ is similar to \[ {\tilde{\mathcal{A}}} := \mathcal{D}_\lambda {\mathcal{A}}_d = \begin{pmatrix} A_0 - \lambda - D_\lambda \Psi A_0 &\lambda D_\lambda - \lambda D_\lambda \Psi D_\lambda \\ \Psi A_0 & \lambda \Psi D_\lambda - \lambda \end{pmatrix} \] on the diagonal domain $D({\tilde{\mathcal{A}}}) := D(A_0) \times \partial X$. This operator is a bounded perturbation of the operator \[ {\mathcal{G}} := \begin{pmatrix} A_0 - D_\lambda \Psi A_0 & 0 \\ \Psi A_0 & 0 \end{pmatrix} \] on the domain $D({\tilde{\mathcal{A}}})$. Observe further that the lower left entry is, by assumption, relatively bounded with respect to the upper left entry. Hence, by well-known results on matrices with diagonal domain (see, e.g., \cite[Cor.\ 3.2 and Cor.\ 3.3]{nagel:1989}) we finally conclude that ${\mathcal{G}}$ (thus ${\mathcal{A}}_\Psi$) generates a strongly continuous (analytic) semigroup on ${\mathcal{X}}$ if and only if $(A_0 - D_\lambda \Psi A_0, D(A_0))$ does so on $X$. (2) By Proposition \ref{wp-wp} it remains to show that $D(A_\Psi)$ is dense in $X$. Assume without restriction that $0 \in \rho(A_0)$ and suppose the condition in (1). We first remark that $D(A_\Psi)$ can be written as \[ D(A_\Psi) = %{\widetilde{D}} := \{f \in X : (\mbox{Id} - D_\lambda \Psi)f \in D(A_0)\} \] with the bounded operator $P_\lambda := \mbox{Id} - D_\lambda \Psi \in {\mathcal{L}} (X)$. Since $(P_\lambda A_0, D(A_0))$ and $(A_0, D(A_0))$ are generators on $X$, we infer that $D(A_0)$ is a Banach space with respect to the norms $\|\cdot\|_{P_\lambda A_0}$ and $\|\cdot\|_{A_0}$ while $\|\cdot\|_{A_0}$ is finer than $\|\cdot\|_{P_\lambda A_0}$. By the open mapping theorem both norms are equivalent and thus $D((P_\lambda A_0)^2)$ is dense in $(D(A_0), \|\cdot\|_{A_0})$. Take now $f \in X$ and $\epsilon > 0$. Then $A_0^{-1}f \in D(A_0)$ and there exists $g_\epsilon \in D((P_\lambda A_0)^2)$ with $\|A_0^{-1}f - g_\epsilon\|_{A_0} \leq \epsilon$. This implies $P_\lambda A_0 g_\epsilon \in D(A_0)$ and thus $f_\epsilon := A_0 g_\epsilon \in D(A_\Psi)$ and, finally, \[ \|f - f_\epsilon\| = \|f - A_0 g_\epsilon \| = \|A_0 [A_0^{-1}f g_\epsilon] \| \leq \|A_0^{-1} f - g_\epsilon \|_{A_0} \leq \epsilon. \] Thus $D(A_\Psi)$ is dense in $X$. \end{proof} The following results are immediate consequences of this theorem and cover all of Greiner's results not yet contained in the preceding Section \ref{generator-speziell}. Corollary \ref{greiner2.3} follows by the bounded perturbation theorem, Corollary \ref{greiner2.4} by a perturbation result for analytic semigroups, see, e.g. \cite[Cor. 2.17 (ii)]{engel/nagel:2000}. Remark that $\Psi : X \to \partial X$ is automatically compact if the boundary space $\partial X$ is finite dimensional. Corollary \ref{greiner2.1'} follows by the perturbation theorem for analytic semigroups, see, e.g., \cite[Thm. 2.10]{engel/nagel:2000}. It is a slight generalization of \cite[Thm. 2.1']{greiner}. Finally, Corollary \ref{selfadjoint} follows by Rellich's perturbation theorem for selfadjoint operators and Stone's theorem. In all four situations, $(BP)$ is wellposed. \begin{cor}[\mbox{\cite[Thm. 2.3]{greiner}}] \label{greiner2.3} In the situation of Theorem \ref{theo-BP-reduction} assume that \break $(A_0, D(A_0))$ is the generator of a strongly continuous semigroup and assume that $(\Psi A_0, D(A_0))$ has a bounded extension. Then the operator $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ is the generator of a strongly continuous semigroup. If $(A_0, D(A_0))$ generates an analytic (compact) semigroup, $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ generates an analytic (compact) semigroup. \end{cor} \begin{cor}[\mbox{\cite[Thm. 2.1']{greiner}}] \label{greiner2.1'} In the situation of Theorem \ref{theo-BP-reduction} assume that \break $(A_0, D(A_0))$ is the generator of an analytic semigroup. Moreover, assume that \[ \inf_{\lambda \in \rho(A_0)} \|D_\lambda \Psi \| = 0. \] %with $\liminf_{\lambda \in \rho(A_0)} c(\lambda) = 0$. Then also $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ generates an analytic semigroup. \end{cor} \begin{cor}[\mbox{\cite[Thm. 2.4]{greiner}}] \label{greiner2.4} In the situation of Theorem \ref{theo-BP-reduction} assume that \break $(A_0, D(A_0))$ is the generator of an analytic semigroup. Moreover, assume that $\Psi : X \to \partial X$ is a compact operator. Then also $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ generates an analytic semigroup. \end{cor} \begin{cor} \label{selfadjoint} In the situation of Theorem \ref{theo-BP-reduction} assume that $(iA_0, D(A_0))$ is a selfadjoint operator on a Hilbert space $X$. Moreover, assume that $(iD_\lambda \Psi A_0, D(A_0)))$ is symmetric and \[ \|D_\lambda \Psi \| < 1 \] for some $\lambda \in \rho(A_0)$. Then also the operator $(i(A_0 - D_\lambda \Psi A_0), D(A_0))$ is selfadjoint on $X$. Thus $({\mathcal{A}}_\Psi, D({\mathcal{A}}_\Psi))$ generates a strongly continuous (semi)group. %TO BE COMPLETED!!!!!!!!!!! \end{cor} \begin{rem} \label{Teo-by-mult} \rm The content of Theorem \ref{theo-BP-reduction} can also be obtained by using the theory of multiplicative perturbations developed, e.g., in \cite{DLS}, \cite{DH}, \cite{PS}. %, \cite{}. We take any $\lambda \in \rho(A_0)$ and observe that \[ D(A_\Psi) = {\widetilde{D}} := \{f \in X : (\mbox{Id} - D_\lambda \Psi)f \in D(A_0)\} \] and \[ A_\Psi f = A_0(\mbox{Id} - D_\lambda \Psi)f + \lambda D_\lambda \Psi f \] for $f \in {\widetilde{D}}$. Since the operator $\lambda D_\lambda \Psi$ is bounded on $X$, the operator $A_\Psi$ is a generator if and only if $(A_0(\mbox{Id} - D_\lambda \Psi), D(A_\Psi))$ is a generator. Applying a result on multiplicative perturbation \cite[Thm.\ III.3. 20]{engel/nagel:2000} we finally draw the following consequence. If the operator $((\mbox{Id} - D_\lambda \Psi)A_0, D(A_0))$ is the generator of a strongly continuous (analytic) semigroup, the same holds for $A_\Psi$. \end{rem} %\footnotesize \begin{thebibliography}{00} % \bibitem{casarino:2001} \sc V. Casarino, K. Engel, R. Nagel, G. Nickel: {\it A semigroup approach to boundary feedback systems}, \rm Integr.\ Equ.\ Oper.\ Theory {\bf 47} (2003), 289--306. % \bibitem{DLS}{\sc W. Desch, I. 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