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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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Self-commutator approximants
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by P. J. Maher
Proc. Amer. Math. Soc. 134 (2006), 157-165
DOI: https://doi.org/10.1090/S0002-9939-05-07871-8
Published electronically: August 15, 2005

Abstract:

This paper deals with minimizing $\| B - (X^* X - X X^*) \|_p$, where $B$ is fixed, self-adjoint and $B \in \mathcal {C}_p$, and where $X$ varies such that $BX = XB$ and $X^* X - X X^* \in \mathcal {C}_p$, $1 \leq p < \infty$. (Here, $\mathcal {C}_p$, $1 \leq p < \infty$, denotes the von Neumann-Schatten class and $\| \cdot \|_p$ its norm.) The upshot of this paper is that $\| B - (X^* X - X X^*) \|_p$, $1 \leq p < \infty$, is minimized if, and for $1 < p < \infty$ only if, $X^* X - X X^* = 0$, and that the map $X \rightarrow \| B - (X^* X - X X^*) \|_p^p$, $1 < p < \infty$, has a critical point at $X = V$ if and only if $V^* V - V V^* = 0$ (with related results for normal $B$ if $p = 1$ or $2$).
References
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Bibliographic Information
  • P. J. Maher
  • Affiliation: Department of Mathematics, Middlesex University, The Burroughs, London NW4 4BT, United Kingdom
  • Email: p.maher@mdx.ac.uk
  • Received by editor(s): March 5, 2003
  • Received by editor(s) in revised form: March 25, 2004
  • Published electronically: August 15, 2005
  • Communicated by: Joseph A. Ball
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 157-165
  • MSC (2000): Primary 47B20, 47A30; Secondary 47B10
  • DOI: https://doi.org/10.1090/S0002-9939-05-07871-8
  • MathSciNet review: 2170555