Let
T
T
be a compact operator on a Hilbert space such that the operators
A
=
1
2
(
T
+
T
∗
)
A = \frac {1}{2} (T + T^{*})
and
B
=
1
2
i
(
T
−
T
∗
)
B = \frac {1}{2i}(T-T^{*})
are positive. Let
{
s
j
}
\{ s_{j}\}
be the singular values of
T
T
and
{
α
j
}
,
{
β
j
}
\{ \alpha _{j}\} , \{ \beta _{j}\}
the eigenvalues of
A
,
B
A,B
, all enumerated in decreasing order. We show that the sequence
{
s
j
2
}
\{ s^{2}_{j}\}
is majorised by
{
α
j
2
+
β
j
2
}
\{ \alpha ^{2}_{j} + \beta ^{2}_{j}\}
. An important consequence is that, when
p
≥
2
,
‖
T
‖
p
2
p \ge 2, ~\| T\| ^{2}_{p}
is less than or equal to
‖
A
‖
p
2
+
‖
B
‖
p
2
\| A\| ^{2}_{p} + \| B\| ^{2}_{p}
, and when
1
≤
p
≤
2
,
1\le p \le 2,
this inequality is reversed.