Abstract :
In this article, we study the structure of positive-definite Toeplitz kernels on free
semigroupsŽcalled also multi-Toeplitz.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
multi-Toeplitz kernels in terms of generalized Schur sequences of contractions is
obtained. This leads to explicit minimal Naimark dilations, Cholesky factorizations,
Szeg¨o type limit theorems, and entropy for positive-definite multi-Toeplitz kernels.
Maximal outer factors are associated with positive-definite multi-Toeplitz kernels
and are used to obtain inner outer factorization for operators on Fock spaces with
coefficients in Hilbert spaces. The Kolmogorov Wiener prediction problem for
stochastic processes having as covariance kernels positive-definite multi-Toeplitz
kernels is considered. The predication-error operator is calculated in terms of
Schur parametersŽresp., maximal outer factor.associated with the covariance
kernel of the process, and a connection with a Szeg¨o type infimum problem is
established. We solve the Carath´eodory interpolation problem for positive-definite
multi-Toeplitz kernels, we obtain a parametrization of all solutions in terms of
Schur sequences, and we find the maximal entropy solution. The results of this
article can be used to develop a theory of stochastic n-linear systems
