Let $E$ be a $\mathbb R$-Banach space and $T_t:E'\to\mathbb C$ for $t\ge0$ with $$T_{s+t}(\varphi)=T_s(\varphi)T_t(\varphi)\;\;\;\text{for all }\varphi\in E'\text{ and }s,t\ge0\tag1$$ and $$T_0(\varphi)=1\;\;\;\text{for all }\varphi\in E'\tag2.$$
Are we able to conclude that $$T_t(\varphi)=e^{-tf(\varphi)}\;\;\;\text{for all }\varphi\in E'\text{ and }t\ge0\tag3$$ for some $f:E'\to\mathbb C$ with $f(0)=0$? Moreover, assuming $[0,\infty)\ni t\mapsto T_t(\varphi)$ is continuous for all $\varphi\in E'$, are we able to conclude that $f$ is continuous?
I guess the pure existence of $f$ is somehow an easy consequence of the functional equation solved by the exponential function.
If it simplifies the matter, I'm also interested in the case where $E$ is a $\mathbb R$-Hilbert space and/or separable.