If
T
T
is a complex symmetric operator on a separable complex Hilbert space
H
\mathcal H
, then the spectrum
σ
(
|
T
|
)
\sigma (|T|)
of
T
∗
T
\sqrt {T^*T}
can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator
T
:
H
→
H
T:\mathcal H\rightarrow \mathcal H
), in terms of the spectra of the selfadjoint operators
Re
T
\operatorname {Re} T
and
Im
T
\operatorname {Im} T
, restricting the possible location of elements of
σ
(
|
T
|
)
\sigma (|T|)
. A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.