  We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier
theorem to operator-valued multipliers on Banach function spaces. Our results
involve a new boundedness condition on sets of operators which we call
$\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness in
many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type
estimates in Banach function spaces which were recently obtained by the
authors.
