Self-commutator approximants

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 $2$).