  Let $L$ be an elliptic differential operator with bounded measurable
coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions
on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p}
\eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-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.
