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

(for definitions and introduction see the Semigroup page)

**Translation Semigroup: **Define endowed with the norm and .

It is straightforward that the family of operators defined above is a -semigroup which has the generator

where .

Since for all we have and . This means that and therefore the spectrum of 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 :**

Define endowed with the norm and .

This is an interesting -semigroup for which for all . Therefore .

See the mentioned page to see that if we have , and this means that , and therefore . This means **we have found a densely defined closed operator on a Banach space which has empty spectrum**.