# Extension of domain of contactive semigroups, and why contractivity is important?

I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this space. Do you think that am I right?

Also, I have another question that, why contactivity of semigroup is important. Why if we have $$\| S(t) \| \leq 2$$ the semigroup $$S(t)$$ does not enjoy the properties of the contractive semigroups?

• I am not too familiar with semigroups but here what I think about your last question: I would think that contractivity is not the important part but boundedness. If I remember correctly It can be shown that for a bounded $C_0$-semigroup that it is possible to find an equivalent norm on the Banach/Hilbertspace such that the semigroup is contractive. – Jonas Lenz Feb 17 at 8:43

Suppose $$\|S(t)\|\le M$$ for some constant and $$\lim_{t\downarrow 0}S(t)x=x$$ for all $$x$$. Then you can renorm by defining $$\|x\|_S=\sup_{t \ge 0}\|S(t)x\|$$. The two norms are equivalent because $$\|x\| \le \|x\|_S \le M\|x\|.$$ $$S$$ becomes a contractive semigroup in the new norm: $$\|S(t)x\|_{S} = \sup_{t' \ge 0 }\|S(t')S(t)x\|=\sup_{t' \ge 0}\|S(t'+t)x\| \le \|x\|_S.$$