Structure and Entropy for Positive-Definite Toeplitz Kernels on Free Semigroups
✍ Scribed by Gelu Popescu
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 198 KB
- Volume
- 254
- Category
- Article
- ISSN
- 0022-247X
No coin nor oath required. For personal study only.
✦ Synopsis
In this article, we study the structure of positive-definite Toeplitz kernels on free Ž . semigroups called 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, Szego 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 Ž . Schur parameters resp., maximal outer factor associated with the covariance kernel of the process, and a connection with a Szego type infimum problem is ëstablished.
We solve the Caratheodory interpolation problem for positive-definite ḿulti-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.