The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds

, which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations (see [16]). We remark that (1.3) is equivalent to the oppositeinequality for the square root of the adjoint operator L:In [13, 14], Kato conjectured that an abstract version of (1.3) might hold,for \regularly accretive operators" (see [14, 18] for the details). A counter-example to this abstract conjecture was obtained by McIntosh [17], whothen reformulated the conjecture in the following form, bearing in mind thatKato’s interest in the problem had been motivated by the special case ofelliptic di erential operators.Conjecture 1.4. The estimate (1.3) holds, for Lde ned as in (1.2), for anyL