Unitarily equivalent bilateral weighted shifts with operator weights

We study unitarily equivalent bilateral weighted shifts with operator weights. We establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. We prove that under certain assumptions unitary equivalence of bilateral weighted shifts with operator weights defined on ${\mathbb{C}}^2$ can always be given by a unitary operator with at most two non-zero diagonals. We provide examples of unitarily equivalent shifts with weights defined on ${\mathbb{C}}^k$ such that every unitary operator, which intertwines them has at least $k$ non-zero diagonals.