Dilation theorems for positive definite operator kernels having majorants

Dilation theorems for Banach space valued stochastic processes and operator valued positive definite kernels are considered. It is shown, e.g., that a Banach space valued stochastic process X can be dilated to another process Y, if and only if the covariance kernel of Y is a majorant of the covariance kernel of X. Positive definite operator kernels having majorants of certain special type are characterized.