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Section 9.8 The Lumer-Phillips Theorem

A special case arises when the constant \(M_\gamma\) =1. In this case we only need the resolvent estimate for \(n=1\text{:}\)
\begin{equation} \|R_\lambda(A)\|\leq \frac{1}{\lambda-\omega}\tag{9.8.1} \end{equation}
and the rest follows through iteration
\begin{equation} \|R_\lambda(A)^n\|\leq \frac{1}{(\lambda-\omega)^n}.\tag{9.8.2} \end{equation}
Here condition 2. is much easier to check than condition 2. in the Hille-Yosida Theorem. We give this case a special name.

Definition 9.8.2.
The semigroup \(\{T(t)\}\) is called a quasi contraction if \(\|T(t) \|\leq e^{\omega t}\text{,}\) and it is called a contraction if \(\|T(t)\|\leq 1\) (i.e. \(\omega=0\)).
In addition to the Hille-Yosida Theorem for semigroups, there is an alternative for contractive semigroups on Hilbert spaces, the Lumer-Phillips theorem.
For \(\lambda>\omega\) we estimate
\begin{equation*} \|(A-\lambda I) x \| \|x\| \geq \mbox{Re} (x, (\lambda I - A) x) \geq (\lambda - \omega) (x,x), \end{equation*}
which implies
\begin{equation*} \|(A-\lambda I) x\|\geq (\lambda-\omega) \|x\|. \end{equation*}
Hence \(A-\lambda I\) is bounded below, away from \(0\text{,}\) and by Corollary Corollary 8.2.9 the resolvent \(R_\lambda(A)\) exists and
\begin{equation*} \|R_\lambda(A)\|\leq \frac{1}{\lambda-\omega}. \end{equation*}
Hence for any \(\lambda_0>\omega\) the map \(A-\lambda_0 I\) is onto. Then \(R_{\lambda_0}(A)\) has range \(X\) and is continuous. This implies that \(A-\lambda_0 I\) is closed, which means that \(A\) is closed. By the Corollary Corollary 9.8.1 to the Hille-Yosida Theorem, \(A\) generates a strongly continuous semigroup \(T(t)\) with
\begin{equation*} \|T(t) \|\leq e^{\omega t}. \end{equation*}
There is an immediate corollary, which is often called the Lumer-Phillips theorem \cite{Rhandi}

Assume \(A\) is self-adjoint on a Hilbert space \(H\) with dense domain \(D(A)\text{.}\) Let \(\{\lambda_i\}_i\) denote the (real) eigenvalues and assume
\begin{equation} \omega >\max\{\mbox{Re}(\lambda), \lambda\in \sigma(A)\}.\label{spbd}\tag{9.8.3} \end{equation}
Let \(\{\phi_i\}_i\) denote an orthonormal basis of eigenvectors. Then
\begin{equation*} (x, Ax) = \sum_{i=1}^\infty \lambda_i (x,\phi_i) (x,\phi_i) \leq \omega \sum_{i=1}^\infty (x,\phi_i)(x,\phi_i) = \omega \|x\|^2. \end{equation*}
Then for \(\lambda_0>\omega\) the resolvent \(R_{\lambda_0}(A)\) exists and \(A-\lambda_0 I\) is onto. By the Lumer-Phillips theorem \(A\) generates a quasi contraction semigroup. Note here that the spectral bound ((9.8.3)) is essential.
xml:id="RDEexample" Let \(A=\frac{d}{dx}\) on \(L^2(0,1)\) with \(u(1) =0\text{.}\) As usual, we put the boundary condition into the domain
\begin{equation*} D(A) = \{u\in H^1(0,1), u(1) =0\}. \end{equation*}
\(D(A)\) is dense in \(L^2(0,1)\text{,}\) since \(H^1\) is dense in \(L^2\) and functions in \(L^2\) are only defined up to a set of measure zero. Hence the condition on one of the boundary points is not relevant.

We compute the spectrum of \(A\text{.}\) For \(\varphi\in L^2(0,1)\) we like to solve the resolvent equation \((A-\lambda I) u = \varphi\text{.}\) This is written as an ODE
\begin{equation*} u' = \lambda u +\varphi, \qquad u(1) =0, \end{equation*}
which is a linear ODE that can be solved for any \(\lambda\in \CC\text{.}\) Hence \((A-\lambda I)^{-1}\) exists for all \(\lambda\in \CC\text{,}\) and we can chose a spectral bound of \(\omega=0\text{.}\) To apply Lumer-Phillips we need one more estimate
\begin{equation*} (u,Au) = \int_0^1 u(x) u'(x) dx = - \frac{1}{2} (u(0))^2 \leq 0 . \end{equation*}
Hence \(A\) generates a contraction semigroup \(T(t)\) with
\begin{equation*} \|T(t) \|\leq 1. \end{equation*}