Von Neumann's inequality for commuting weighted shifts

Abstract: We show that every multivariable contractive weighted shift dilates to a tuple of commuting unitaries, and hence satisfies von Neumann's inequality. This answers a question of Lubin and Shields. We also exhibit a closely related $3$-tuple of commuting contractions, similar to Parrott's example, which does not dilate to a $3$-tuple of commuting unitaries.