Von Neumann's inequality for commuting weighted shifts

TL;DR

This paper proves that multivariable contractive weighted shifts satisfy von Neumann's inequality by dilation to commuting unitaries, but also provides a counterexample showing not all commuting contractions dilate similarly.

Contribution

It establishes dilation results for multivariable weighted shifts and presents a counterexample for certain commuting contractions, advancing understanding of operator dilation theory.

Findings

Weighted shifts dilate to commuting unitaries, satisfying von Neumann's inequality.

Counterexample of a 3-tuple of commuting contractions that does not dilate to unitaries.

Answers a longstanding question in multivariable operator theory.

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.