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Journal of Operator Theory

Volume 72, Issue 1, Summer 2014  pp. 219-239.

A hierarchy of von Neumann inequalities?

Authors:  Quanlei Fang (1) and Jingbo Xia (2)
Author institution: (1) Department of Mathematics and Computer Science, Bronx Community College, CUNY, Bronx, NY 10453, U.S.A.
(2) Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, U.S.A.

Summary: The well-known von Neumann inequality for commuting row contractions can be interpreted as saying that the tuple $(M_{z_1},\dots ,M_{z_n})$ on the Drury-Arveson space $H^2_n$ dominates every other commuting row contraction $(A_1,\dots ,A_n)$. We show that a similar domination relation exists among certain pairs of `lesser' row contractions $(B_1,\dots ,B_n)$ and $(A_1,\dots ,A_n)$. This hints at a possible hierarchical structure among the family of commuting row contractions.

Keywords: von Neumann inequality, row contraction, reproducing-kernel Hilbert space

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