SOLUTION OF THE MPC PROBLEM IN TERMS OF THE SZ.-NAGY–FOIAŞ DILATION THEORY

Abstract

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space [Formula: see text], find a Hilbert space [Formula: see text], a contraction semigroup {Wt} on [Formula: see text] and an injective operator [Formula: see text] with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt= UtΛ). More precisely, they studied the case where [Formula: see text] is an L2-space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L2-spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.