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Self-commutator approximants

Authors:	P J Maher

Institution:	Department of Mathematics, Middlesex University, The Burroughs, London NW4 4BT, United Kingdom

Abstract:	This paper deals with minimizing $\Vert B - (X^* X - X X^) \Vert _p$ , where is fixed, self-adjoint and $B \in \mathcal{C}_p$ , and where varies such that 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, , and that the map $X \rightarrow \Vert B - (X^ X - X X^*) \Vert _p^p$ , $1 < p < \infty$ , has a critical point at if and only if (with related results for normal if or ).

Keywords:	Self-commutator von Neumann-Schatten class

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