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For a matrix A_2 weight W on R^p, we introduce a new notion ofW-Calder\'on-Zygmund matrix kernels, following earlier work in by Isralowitz.We state and prove a T1 theorem for such operators and give a representationtheorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit ofHyt\"onen's Representation Theorem. Finally, by means of a Bellman functionargument, we give sharp bounds for such operators in terms of bounds forweighted matrix martingale transforms and paraproducts.

Sharp bounds and T1 theorem for Calder\'on-Zygmund operators with matrix kernel on matrix weighted spaces