= Hille-Yosida theorem
{c}
{wiki=Hille–Yosida_theorem}
Let $X$ be a <Banach space>, $M\geq1$, and $\omega\in\mathbb R$. A <linear operator> $A:D(A)\subseteq X\to X$ generates a <C0-semigroup> $S(t)$ with $\|S(t)\|\leq Me^{\omega t}$ exactly when $A$ is closed, $D(A)$ is dense, every real $\lambda>\omega$ belongs to the <resolvent set>, and
$$
\|R(\lambda,A)^n\|\leq\frac{M}{(\lambda-\omega)^n}
\qquad(n\geq1),
\quad R(\lambda,A)=(\lambda I-A)^{-1}.
$$
The same constant $M$ must work for every power. When $M=1$, the first-power bound implies all the power bounds by submultiplicativity. For general $M$, multiplying first-power bounds gives $M^n$, which is insufficient.https://deleo.website/FunctionalAnalysis/section-57.html[Theorem9.7.1; that source uses the opposite sign convention for the resolvent]{ref}
Necessity follows from the <Laplace-transform formula for a semigroup resolvent>:
$$
R(\lambda,A)^nx=\frac1{(n-1)!}\int_0^\infty t^{n-1}e^{-\lambda t}S(t)x\,dt.
$$
The <Bochner integral> is bounded by $M\|x\|/(\lambda-\omega)^n$. Sufficiency constructs the <semigroup> from bounded <Yosida approximations>, using all the resolvent-power estimates for a common growth bound. For $M=1$ and $\omega=0$, this characterizes generators of <contraction semigroups>, providing a route from an unbounded differential <linear operator> to a well-posed time evolution.
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