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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 the tensor product $A \otimes B$ satisfies the von Neumann's inequality if and only if $A$ satisfies the von Neumann's inequality, where $A $ is a contractive $d$-tuple of $n \times n$ matrices and $B$ is a contractive $d$-tuple of operators satisfying the matrix-version of von Neumann's inequality with the Taylor spectrum containing the unit $d$-torus. We also exhibit several families of operator-valued multishifts for which the von Neumann's inequality always holds.

Von Neumann’s inequality for commuting operator-valued multishifts