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Let $X$ be a Banach space, $M\ge 1$, and $\omega \in \mathbb{R}$. A linear operator $A:D(A)\subseteq X\to X$ generates a C0-semigroup $S(t)$ with $\Vert S(t)\Vert \le M{e}^{\omega t}$ exactly when $A$ is closed, $D(A)$ is dense, every real $\lambda >\omega$ belongs to the resolvent set, andThe 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.^[ref]

Necessity follows from the Laplace-transform formula for a semigroup resolvent:The Bochner integral is bounded by $M\Vert x\Vert /(\lambda -\omega {)}^n$. 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.

If $A$ generates a C0-semigroup satisfying $\Vert U(t)\Vert \le M{e}^{\omega t}$, then for $\mathrm{Re}z>\omega$ its resolvent operator isRepeated differentiation gives $n!(zI-A{)}^{-(n+1)}x={\int }_0^{\infty }{t}^n{e}^{-zt}U(t)xdt$.

The familyinherits the identity, semigroup property, and strong continuity from $U$, whileIts difference quotient satisfiesfor $u\in D(A)$. Conversely, existence of this limit implies existence of the generator limit for $U$, so $D(\hat{A})=D(A)$ andThis is the exponentially shifted semigroup construction.

The Hille-Yosida theorem in the uniformly bounded case says that a linear operator $B$ on a Banach space generates a C0-semigroup with $\Vert e^{tB}\Vert \le M$ if and only if:

A C0-semigroup on a Banach space $X$ is a family $U(t)\in \mathcal{B}(X)$ such thatfor every $x\in X$. Its infinitesimal generator of a semigroup iswith generator domain

For $M\ge 1$ and $\omega \in \mathbb{R}$, write $A\in \mathcal{G}(M,\omega )$ when $A$ generates a $C_0$-semigroup satisfying $\Vert U(t)\Vert \le M{e}^{\omega t}$. The Hille-Yosida theorem states that this holds exactly when $A$ is closed and densely defined,and, for every real $\lambda >\omega$ and every integer $n\ge 1$,The estimates for every resolvent power, rather than only $n=1$, are essential when $M>1$.

Equip $D(A)$ with its graph norm. The restrictions $U(t){|}_{D(A)}$ form a C0-semigroup on this Banach space; its generator is $A$ restricted to $D(A^2)$.