Integral Representations for Markov Transition Probabilities

The following results (and one or two others like them) are proved in [3]; the continuous-parameter analogues will appear in [4] and some applications will be discussed in [5]. The proofs are based on Sz.-Nagy's group of theorems [9] which show how a discreteor continuous-parameter semigroup of contraction operators on Hilbert space can be "dilated" to a similar semigroup (actually a group) of unitary operators acting on a larger Hilbert space. The WintnerStone theorems on the spectral representation of discrete/continuousparameter groups of unitary operators then lead easily to integral representations for the Markov transition probabilities which appear in the matrix elements of the operator to be dilated.