In this paper, we prove that if A and B are normal operators on a Hilbert space H, then, for every operator S satisfying
A
S
B
=
S
,
‖
A
X
B
−
X
+
S
‖
≥
‖
A
‖
−
1
‖
B
‖
−
1
‖
S
‖
ASB = S, \left \| {AXB - X + S} \right \| \geq {\left \| A \right \|^{ - 1}}{\left \| B \right \|^{ - 1}}\left \| S \right \|
for all operators
X
∈
B
(
H
)
X \in B(H)
, and that if A and B are contractions, then, for every operator S satisfying
A
S
B
=
S
ASB = S
and
A
∗
S
B
∗
=
S
,
‖
A
X
B
−
X
+
S
‖
≥
‖
S
‖
{A^ \ast }S{B^ \ast } = S,\left \| {AXB - X + S} \right \| \geq \left \| S \right \|
for all operators
X
∈
B
(
H
)
X \in B(H)
, where
B
(
H
)
B(H)
denotes the set of all bounded linear operators on H.