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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

von Neumann's inequality for commuting, diagonalizable contractions. I


Author: B. A. Lotto
Journal: Proc. Amer. Math. Soc. 120 (1994), 889-895
MSC: Primary 47A30; Secondary 15A60, 47B99
MathSciNet review: 1169881
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Abstract: We obtain a sufficient condition for an $ n$-tuple $ T$ of commuting, diagonalizable contractions on a finite-dimensional space to satisfy von Neumann's inequality $ \vert\vert p(T)\vert\vert \leqslant \vert\vert p\vert{\vert _\infty }$ for any polynomial $ p$ in $ n$ variables, where $ \vert\vert p\vert{\vert _\infty }$ denotes the supremum of $ \vert p\vert$ over the unit polydisk in $ {{\mathbf{C}}^n}$. We apply this condition to the case where $ T$ acts on a two- or three-dimensional space. In addition, we prove that von Neumann's inequality for commuting, diagonalizable contractions on a three-dimensional space implies von Neumann's inequality for arbitrary commuting contractions on a three-dimensional space.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1169881-8
PII: S 0002-9939(1994)1169881-8
Article copyright: © Copyright 1994 American Mathematical Society