I am trying to understand some properties of semigroups and linear operators for generators of markov processes. My question is about strong continuity and $C_0$ semigroups.
Lets consider a semigroup of linear contractions $(P_t)_{t \geq 0}$. Then I have the definition of strong continuity, when $\lim_{h \downarrow0}P_h f = f$ holds for $f \in X$.
Now my professor introduced also the $C_0$ semigroups with the definition that if for all $f \in C_0(E)$ we have $P_t f \in C_0(E)$ than we call $(P_t)_t$ a $C_0$ semigroup. And we say $f \in C_0(E)$ if it vanishes in infinity and is continuous.
Maybe it is obvious but are both definition equivalent? Because on Wikipedia (https://en.wikipedia.org/wiki/C0-semigroup) they say our semigroup is in $C_0$ when the first definition holds.
Thank you for your help!