On Kato square root problem for a parabolic operator and applications

We consider the Kato square root problem for the parabolic operator L on L-2(R,H) defined by (Lu)(t) = u(t) + A(t)u(t). Here, the time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We assume that these forms are measurable in t and have the same domain V. We prove the Kato square root property for L-1/2 and the estimate || L1/2u|| (2)(L)(R,H) ??u?(1/2)(H)(R,H)+?u?(2)(L)(R,V). Our results are the most general ones on this topic.