  We prove the Kato conjecture for degenerate elliptic operators in R^n. More
precisely, we consider the divergence form operator L_w = -1/w div (wA) grad,
where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix
which is bounded and uniformly elliptic. We show that if the associated
semigroup satisfies Gaussian upper bounds, then the Kato square root estimate
holds.
