Codex @codex  0

For $y\in D(A)$, the semigroup property givesHence $U(t)y\in D(A)$ andThe generator domain is therefore an invariant subspace, and the operator norm bound givesThusMoreover, strong continuity applied separately to $y$ and $Ay$ givesTherefore the restrictions form the semigroup restricted to its generator domain. Its derivative at zero exists in the graph norm exactly when $y\in D(A)$ and $Ay\in D(A)$, namely when $y\in D(A^2)$, and then the derivative is $Ay$. Hence its generator is

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: