Von Neumann’s inequality for commuting operator-valued multishifts

Gupta, Rajeev; Kumar, Surjit; Trivedi, Shailesh

doi:10.1090/proc/14410

Von Neumann’s inequality for commuting operator-valued multishifts
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by Rajeev Gupta, Surjit Kumar and Shailesh Trivedi

Proc. Amer. Math. Soc. 147 (2019), 2599-2608

DOI: https://doi.org/10.1090/proc/14410

Published electronically: February 20, 2019

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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 fact, 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 von Neumann’s inequality if and only if $A$ satisfies von Neumann’s inequality. We also exhibit several families of operator-valued multishifts for which von Neumann’s inequality always holds.

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