Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L^p (Rⁿ ; X) of X -valued functions on Rⁿ. We characterize Kato's square root estimates {norm of matrix} sqrt(L) u {norm of matrix}_p {minus tilde} {norm of matrix} ∇ u {norm of matrix}_p and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L^p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L^p theory of square roots of elliptic operators, as well as an L^p version of Carleson's inequality.