Von Neumann's inequality for commuting operator-valued multishifts

2018/05/09 by Gupta, Rajeev, Kumar, Surjit, Trivedi, Shailesh · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1805.03547

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, …, 1) is in the algebraic spectrum of B, then the tensor product A ⊗ 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.

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• Weakly \mathcal U(d)-homogeneous commuting tuple of bounded operators
• On similarity to contraction semigroups and tensor products, I
• Unitary equivalence of balanced weighted shifts on rooted directed trees

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