Unitary equivalence of complex symmetric contractions with finite defect.

A criterion for a contraction T on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function Θ_T of 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_T1 and Θ_T2 coincide for two complex symmetric contractions T₁ and T₂. As an application, surprisingly we solve the problem for any defect index n, when the defect indexes of contractions are 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 symmetric inner matrices is proposed, which extends some results on 2 × 2 inner matrices (see Stephan Ramon Garcia [J. Operator Theory 54 (2005), pp. 239–250]) and 2 × 2 symmetric inner matrices (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). [ABSTRACT FROM AUTHOR]