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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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von Neumann’s inequality for commuting, diagonalizable contractions. I
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by B. A. Lotto
Proc. Amer. Math. Soc. 120 (1994), 889-895
DOI: https://doi.org/10.1090/S0002-9939-1994-1169881-8

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 $||p(T)|| \leqslant ||p|{|_\infty }$ for any polynomial $p$ in $n$ variables, where $||p|{|_\infty }$ denotes the supremum of $|p|$ 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.
References
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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 120 (1994), 889-895
  • MSC: Primary 47A30; Secondary 15A60, 47B99
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1169881-8
  • MathSciNet review: 1169881