Self-commutator approximants

Maher, P.

doi:10.1090/S0002-9939-05-07871-8

Self-commutator approximants
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by P. J. Maher PDF

Proc. Amer. Math. Soc. 134 (2006), 157-165 Request permission

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 $2$).

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