Schur parameters, Toeplitz matrices, and Kre\u{\i}n shorted operators

We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kre\u\i n shorted operators. By means of Schur parameters or shorted operators for defect operators of Toeplitz matrices necessary and sufficient conditions for a simple conservative discrete-time system to be controllable/observable and for a completely non-unitary contraction to be completely non-isometric/completely non-co-isometric are obtained. For the Schur problem a characterization of central solution and uniqueness criteria to the solution are given in terms of shorted operators for defect operators of contractive Toeplitz matrices, corresponding to data.