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

 

Self-commutator approximants


Author: P. J. Maher
Journal: Proc. Amer. Math. Soc. 134 (2006), 157-165
MSC (2000): Primary 47B20, 47A30; Secondary 47B10
Published electronically: August 15, 2005
MathSciNet review: 2170555
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Abstract: This paper deals with minimizing $\Vert B - (X^* X - X X^*) \Vert _p$, 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 $\Vert \cdot \Vert _p$ its norm.) The upshot of this paper is that $\Vert B - (X^* X - X X^*) \Vert _p$, $1 \leq p < \infty$, is minimized if, and for $1 < p < \infty$ only if, $X^* X - X X^* = 0$, and that the map $X \rightarrow \Vert B - (X^* X - X X^*) \Vert _p^p$, $1 < p < \infty$, 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 $2$).


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Additional Information

P. J. Maher
Affiliation: Department of Mathematics, Middlesex University, The Burroughs, London NW4 4BT, United Kingdom
Email: p.maher@mdx.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07871-8
PII: S 0002-9939(05)07871-8
Keywords: Self-commutator, von Neumann-Schatten class
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
Article copyright: © Copyright 2005 American Mathematical Society