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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 $H:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ and also all weighted dyadic martingale transforms $T_{\sigma }:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.