  We consider the Kato square root problem for non-divergence second order
elliptic operators $L =- a_{ij} D_iD_j$, 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.
