= Solution
A <C0-semigroup> on a <Banach space> $X$ is a family $U(t)\in\mathcal B(X)$ such that
$$
U(0)=I,
\qquad
U(t+s)=U(t)U(s),
\qquad
\lim_{t\downarrow0}U(t)x=x
$$
for every $x\in X$. Its <infinitesimal generator of a semigroup> is
$$
Ax=\lim_{t\downarrow0}\frac{U(t)x-x}{t},
$$
with <generator domain>
$$
D(A)=\left\{x\in X:
\lim_{t\downarrow0}\frac{U(t)x-x}{t}
\text{ exists in }X\right\}.
$$
For $M\geq1$ and $\omega\in\mathbb R$, write $A\in\mathcal G(M,\omega)$ when $A$ generates a $C_0$-semigroup satisfying $\|U(t)\|\leq Me^{\omega t}$. The <Hille-Yosida theorem> states that this holds exactly when $A$ is closed and densely defined,
$$
(\omega,\infty)\subset\rho(A),
$$
and, for every real $\lambda>\omega$ and every integer $n\geq1$,
$$
\boxed{
\|R(\lambda,A)^n\|
\leq\frac{M}{(\lambda-\omega)^n}},
\qquad
R(\lambda,A)=(\lambda I-A)^{-1}.
$$
The estimates for every resolvent power, rather than only $n=1$, are essential when $M>1$.
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