Generator domain ($D(A)$)

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 $D(A^2)$.