On dilation of local semigroups of contractions and some applications

Abstract

We show that a multiplicative family of contractions on a separable Hilbert space, with parameter on the interval [0, 1) of the dyadic rationals, has a unitary dilation with parameter on the dyadic rationals and values on a larger Hilbert space. This result is used to prove a dilation result for strongly continuous local semigroups of contractions. As an application we give results of extension of positive definite functions on the line, generalizing the Kreĭn extension theorem.