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## Semigroup Examples (ex. empty/non-compact spectrum)

(for definitions and introduction see the Semigroup page)

Translation Semigroup: Define $X=\{f:\Bbb{R}_+ \to \Bbb{R} : f \text{ is uniform continuous and bounded on }\Bbb{R}_+\}$ endowed with the norm $||f||=\displaystyle \sup_{t \geq 0} |f(t)|$ and $T(t): X \to X,\ (T(t)f)(s)=f(t+s)$.

It is straightforward that the family of operators defined above is a $\mathcal{C}_0$-semigroup which has the generator

$A: D(A) \subset X \to X,\ Af=f^\prime$ where $D(A)=\{f \in X : f\text{ is differentiable on }\Bbb{R}_+,\ f^\prime \in X\}$.

Since for all $\lambda \in (-\infty,0)$ we have $e^{\lambda t} \in X$ and $A(e^{\lambda (\cdot)})=\lambda e^{\lambda (\cdot)}$. This means that $(-\infty,0) \subset \sigma(A)$ and therefore the spectrum of $A$ is not compact. Thus we found an example of densely defined closed operator on a Banach space which has a non-compact spectrum.

Example of a Semigroup with $\omega_0(T)=-\infty$ :

Define $X=\{f:\Bbb{R}_+ \to \Bbb{R} : f \text{ is continuous and }f(1)=0\}$ endowed with the norm $||f||=\displaystyle \sup_{t \geq 0} |f(t)|$ and $T(t): X \to X,\ (T(t)f)(s)=\begin{cases} f(t+s) & 0\leq t+s\leq 1 \\ 0 & t+s>1 \end{cases}$.

This is an interesting $\mathcal{C}_0$-semigroup for which $\|T(t)\|=0$ for all $t >1$. Therefore $\displaystyle \omega_0(T)=\lim_{t \to \infty} \frac{\ln \|T(t)\|}{t}=-\infty$.

See the mentioned page to see that if $Re\lambda> \omega_0(T)=-\infty$ we have $\lambda \in \rho(A)$, and this means that $\Bbb{C} \subset \rho(A)$, and therefore $\sigma(A)=\emptyset$. This means we have found a densely defined closed operator on a Banach space which has empty spectrum.