We prove that if
A
A
and
B
∗
{B^ * }
are subnormal operators acting on a Hubert space, then for every bounded linear operator
X
X
, the Hilbert-Schmidt norm of
A
X
−
X
B
AX - XB
is greater than or equal to the Hilbert-Schmidt norm of
A
∗
X
−
X
B
∗
{A^ * }X - X{B^ * }
. In particular,
A
X
=
X
B
AX = XB
implies
A
∗
X
=
X
B
∗
{A^ * }X = X{B^ * }
. In addition, if we assume
X
X
is a Hilbert-Schmidt operator, we can relax the subnormality conditions to hyponormality and still retain the inequality.