This paper deals with minimizing
‖
B
−
(
X
∗
X
−
X
X
∗
)
‖
p
\| B - (X^* X - X X^*) \|_p
, where
B
B
is fixed, self-adjoint and
B
∈
C
p
B \in \mathcal {C}_p
, and where
X
X
varies such that
B
X
=
X
B
BX = XB
and
X
∗
X
−
X
X
∗
∈
C
p
X^* X - X X^* \in \mathcal {C}_p
,
1
≤
p
>
∞
1 \leq p > \infty
. (Here,
C
p
\mathcal {C}_p
,
1
≤
p
>
∞
1 \leq p > \infty
, denotes the von Neumann-Schatten class and
‖
⋅
‖
p
\| \cdot \|_p
its norm.) The upshot of this paper is that
‖
B
−
(
X
∗
X
−
X
X
∗
)
‖
p
\| B - (X^* X - X X^*) \|_p
,
1
≤
p
>
∞
1 \leq p > \infty
, is minimized if, and for
1
>
p
>
∞
1 > p > \infty
only if,
X
∗
X
−
X
X
∗
=
0
X^* X - X X^* = 0
, and that the map
X
→
‖
B
−
(
X
∗
X
−
X
X
∗
)
‖
p
p
X \rightarrow \| B - (X^* X - X X^*) \|_p^p
,
1
>
p
>
∞
1 > p > \infty
, has a critical point at
X
=
V
X = V
if and only if
V
∗
V
−
V
V
∗
=
0
V^* V - V V^* = 0
(with related results for normal
B
B
if
p
=
1
p = 1
or
2
2
).