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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.

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