Naimark Dilations, State-Space Generators and Transmission Lines,

With any positive Toeplitz form we can associate, via an inverse scattering algorithm, a discrete transmission-line model, parametrized by a so called “choice sequence” of local reflection coefficients. The infinite-dimensional state-space representation of this model then readily yields the structure of the Naimark dilation and of the state-space moment generator for the unit circle measure that corresponds to the Toeplitz form. This point of view motivates and considerably simplifies many operator-theoretic manipulations aimed at analyzing the structure of Naimark dilations and also connects this theory to some recent results on state-space generators for moment matrices associated with positive measures on the unit circle. The new insights readily yield several interesting matrix indentities and also suggest the extension of Naimark dilation results and state-space generator theory to the continuous case.