# Determining domain of differential operators in a concrete semigroup problem

Let s>3/2, $$H^s=H^s(\mathbb{R})$$ be the Sobolev space of order $$s$$, $$B$$ be the set of bounded operators from $$H^{1/2}$$ to itself, $$u\in H^s$$ and $$A(u):=u\partial_x$$ an operator. How can I determine the domain $$D(A)$$ such that:

1.) $$U_s:=e^{-sA(u)}$$ is an operator from $$H^{1/2}$$ to $$H^{1/2}$$;

2.) $$\|U_s\|_B\leq e^{\beta s}$$, for a certain constant $$\beta$$? Here the norm is the usual norm on the space $$B$$.

As far as I discovered, this would be equivalent to prove that $$_{H^{1/2}}\geq -\beta\|\phi\|_{H^{1/2}}^2$$, for any $$D(A)$$, but I was unsuccesful in understanding why this is an equivalence.