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?
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: $$ \|S(t)x\|_{S} = \sup_{t' \ge 0 }\|S(t')S(t)x\|=\sup_{t' \ge 0}\|S(t'+t)x\| \le \|x\|_S. $$