A criterion for a contraction
T
T
on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function
Θ
T
\Theta _{T}
of
T
T
in 2007 (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). To further classify unitary equivalent complex symmetric contractions, we notice a simple condition of when
Θ
T
1
\Theta _{T_{1}}
and
Θ
T
2
\Theta _{T_{2}}
coincide for two complex symmetric contractions
T
1
T_{1}
and
T
2
.
T_{2}.
As an application, surprisingly we solve the problem for any defect index
n
n
, when the defect indexes of contractions are
2
,
2,
this problem was left open by Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]. Furthermore, a construction of
3
×
3
3\times 3
symmetric inner matrices is proposed, which extends some results on
2
×
2
2\times 2
inner matrices (see Stephan Ramon Garcia [J. Operator Theory 54 (2005), pp. 239–250]) and
2
×
2
2\times 2
symmetric inner matrices (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]).