Are every analytic semigroups norm continuous? Are there counterexamples otherwise? What would make analytic semigroup norm continuous as well? (My apology, frankly, I am not sure if norm continuous is equivalent of saying uniform continuous for the semigroups.)
I know the difference between $C_0$ semigroup and norm continuous semigroup is that for norm continuous semigroup we have $$\lim_{t\to 0}\|S(t)-Id\|=0$$ in contrast to $$\lim_{t\to 0}S(t)x=x$$ for $C_0$ semigroup.
I saw from some text and have a feeling that every analytic semigroup is also norm continuous semigroup. (Background: This has a huge impact inspecting on different notions of solutions. For a norm continuous semigroup, it is known that weak, mild and strong solution are equivalent.)
From the Theorem below, you are wrong, since the generator of analytic semigroup is not necessarily bounded.
Note that $T(t)$ is called uniformly continuous if and only if $\lim_{t\to 0^+}\|T(t)-I\|=0$.