Say, we have a Hilbert space $H$ with a semibounded self-adjoint operator $A:D(A)\to H$ generating a strongly continuous semigroup $T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of $A$ in such a way that the restriction is again a strongly continuous semigroup, say $T_1(t)$?

How does the generator of $T(t)$ relate to the generator of $T_1(t)$? Is the latter self-adjoint w.r.t. a product $(x,y)_1:=(x,Ay)+(x,y)$?

Is there any relation of $Q(A)$ to the fractional Favard space $F_{1/2}$?