  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.
