## 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
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## 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