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 $<A\phi,\phi>_{H^{1/2}}\geq -\beta\|\phi\|_{H^{1/2}}^2$, for any $D(A)$, but I was unsuccesful in understanding why this is an equivalence.


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