Let $E$ be a $\mathbb R$-Banach space and $A$ be the generator of a uniformly continuous contraction semigroup $(T(t))_{t\ge0}$ on $E$. Are we able to derive some bound on $\left\|A\right\|_{\mathfrak L(E)}$? Is $A$ even a contraction (i.e. has operator norm at most $1$)?
By assumption, $T(t)=e^{tA}$ and $\left\|T(t)\right\|_{\mathfrak L(E)}\le1$ for all $t\ge0$.