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 quantitative estimates do not depend on the assumption of smoothness.