A version of Burkholder's theorem for operator-weighted spaces

Let $W$ be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space $\mathcal{H}$. We prove that if the dyadic martingale transforms are uniformly bounded on $L^2_{\mathbb{R} }(W)$ for each dyadic grid in $\mathbb{R}$, then the Hilbert transform is bounded on $L^2_{\mathbb{R} }(W)$ as well, thus providing an analogue of Burkholder's theorem for operator-weighted $L^2$-spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts''.