Infinitesimal generator of a semigroup ($A$)

The infinitesimal generator is the generally unbounded operatoron the vectors for which the norm limit exists.

For a strongly continuous semigroup, apply the bounded operator $T_t$ to the difference quotient defining its semigroup generator. The limit proves domain invariance and commutation on the generator domain. This is not an assertion that the unbounded generator acts on every vector.

// Target: analysis.bigb

The Yosida averaging of a semigroup gives $x_h=h^{-1}{\int }_0^{h}S(t)xdt\in D(A)$ with $x_h\to x$, proving that the generator domain is dense. For $x\in D(A)$, the Bochner integral identity $S(t)x-x={\int }_0^{t}S(s)Axds$ holds. If $x_n\to x$ and $A{x}_n\to y$, pass to the limit in this identity and divide by $t$. The strongly continuous semigroup property makes the resulting limit $y$, so $x\in D(A)$ and $Ax=y$. This proves that the infinitesimal generator of a semigroup is a closed operator.

The generator domain consists of the vectors whose semigroup orbit is right-differentiable at zero. It is a dense linear subspace, and the generator is closed.

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 $D(A^2)$.