Generalized Bochner theorem and Levy-Khinchine representation for completely positive definite generalized Toeplitz kernels

Abstract

The so-called generalized Bochner theorem, stated by Cotlar and Sadosky, provides an integral representation of the positive definite generalized Toeplitz kernels. In this paper we derive noncommutative analogues of lifting theorems and the generalized Bochner theorem for completely positive definite generalized Toeplitz kernels, also considered by Cotlar and Sadosky, through the theory of unitary extensions of isometric operators. Moreover, in the case where the kernel is defined inℤ×ℤ, we can associate to each unitary extension an interpolation colligation providing thus a wide class of liftings. Also a Levy-Khinchine type formula for this kind of kernels is obtained.