Can a single point uniquely determine the whole semi-group?

Question

We have two strongly continuous semi-group $\{T(t)\}_{t\geq0}$ and $\{S(t)\}_{t\geq0}$ on a Banach space, and we know for some $t_0>0$, the two operators coincide, i.e., $T(t_0) = S(t_0)$. Can we conclude that $T(t) = S(t)$ for all $t \geq0 $? What if $\{T(t)\}_{t\geq0}$ and $\{S(t)\}_{t\geq0}$ are strongly continuous unitary groups on a Hilbert space? Any hint will be appreciated.

Answer 1

$$ e^{0\cdot t} = e^{2\pi i \cdot t} \mbox{ at } t=1. $$