Von Neumann's inequality for commuting operator-valued multishifts

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 $(1,\dots ,1)$ 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.

Theorem 1.2 revised, accepted in Proceedings of American Mathematical Society