On the Kato problem for elliptic operators in non-divergence form

We consider the Kato square root problem for non-divergence second order elliptic operators $L=-a_{ij}{D}_i{D}_j$, and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real coefficients having sufficiently small BMO norm. We assume that the coefficients of the operator are smooth, but our estimates do not depend on the assumption of smoothness.