 Hille–Yosida theorem

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.

The Hille–Yosida theorem is a fundamental result in functional analysis that characterizes the generators of strongly continuous semigroups of linear operators on Banach spaces. It provides a set of conditions under which a certain type of linear operator can be considered the generator of a strongly continuous semigroup. This theorem is particularly important in the study of evolution equations and the analysis of time-dependent systems.