Unitary equivalence of complex symmetric contractions with finite defect

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]).