Domain contingency of infinitesimal generators

Question

I'm beginner in Semigroup theory(study alone) and i'm trying "to solve":(maybe through hille yosida's theorem) Let A and B be infinitesimal generators of two semigroups in space X. Show an example where if D(A) ⊂ D(B), then not necessarily we have that D(A) = D(B).

Answer 1

For $L^2(1,\infty)$ let $A$ and $B$ act by multiplying by $-x^2$ and $-x$ respectively with domains $$D(A)=\{ f\in L^2\,:\, x^2f\in L^2\}, \quad D(B)=\{ f\in L^2\,:\, xf\in L^2\}$$ Then $D(A)\subsetneq D(B).$ These operators are self-adjoint and nonpositive. They generate semigroups of contractions $S_t$ and $T_t$ respectively, where $$S_tf=e^{-tx^2}f,\qquad T_tf=e^{-tx}f$$