Unitary equivalence of complex symmetric contractions with finite defect

Abstract:

A criterion for a contraction $T$ on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function $\Theta _{T}$ 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 $\Theta _{T_{1}}$ and $\Theta _{T_{2}}$ coincide for two complex symmetric contractions $T_{1}$ and $T_{2}.$ 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 $3\times 3$ symmetric inner matrices is proposed, which extends some results on $2\times 2$ inner matrices (see Stephan Ramon Garcia [J. Operator Theory 54 (2005), pp. 239–250]) and $2\times 2$ symmetric inner matrices (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]).