### A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse ${W}^{-1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H:L{\xb2}_{W}(\mathbb{R},)\to L{\xb2}_{W}(\mathbb{R},)$ and also all weighted dyadic martingale transforms ${T}_{\sigma}:L{\xb2}_{W}(\mathbb{R},)\to L{\xb2}_{W}(\mathbb{R},)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.