# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2012dec12.1998

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

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