Von Neumann's inequality for commuting operator-valued multishifts

TL;DR

This paper investigates the validity of von Neumann's inequality for commuting operator-valued multishifts, showing limitations of previous results and identifying specific cases where the inequality holds.

Contribution

It demonstrates that von Neumann's inequality does not generally extend to operator-valued multishifts with invertible weights and characterizes conditions for its validity.

Findings

Von Neumann's inequality fails for certain operator-valued multishifts.

Tensor products satisfy von Neumann's inequality iff the component does.

Identifies families of multishifts where the inequality always holds.

Abstract

Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In particular, we show that if $A$ and $B$ are commuting contractive $d$-tuples of operators such that $B$ satisfies the matrix-version of von Neumann's inequality and $(1, \ldots, 1)$ is in the algebraic spectrum of $B$, then the tensor product $A \otimes B$ satisfies the von Neumann's inequality if and only if $A$ satisfies the von Neumann's inequality. We also exhibit several families of operator-valued multishifts for which the von Neumann's inequality always holds.