Structure and entropy for Toeplitz kernels

Abstract

In this paper we study the structure of positive-definite Toeplitz kernels on free semigroups and its implications in noncommutative dilation theory, harmonic analysis on Fock spaces, prediction and interpolation theory for stationary stochastic processes. A parametrization of positive-definite Toeplitz kernels in terms of generalized Schur sequences of contractions leads to a series of applications: explicit minimal Naimark dilations, Cholesky factorizations, Szegö type limit theorems, entropy, maximal outer factors, and factorizations for positive-definite Toeplitz kernels. The Kolmogorov-Wiener prediction-error operator associated to any stochastic process having as covariance kernel a positive-definite Toeplitz kernel is calculated in terms of its Schur sequence (resp. maximal outer factor), and a connection with a Szegö type infimum problem is established. We solve the Carathéodory interpolation problem for positive-definite Toeplitz kernels, obtain a parametrization of all solutions in terms of Schur sequences, and find the maximal entropy solution. The results of this paper can be used to develop a theory of stochastic n-linear systems.