On Residualities in the Set of Markov Continuous Semigroups on C1

Abstract

Abstract. We show that the set of all stochastic strongly continuous semigroups on Ci such that lim t^o c |||T(i) -Qx, III = 0, where Qx. is one-dimensional projection for some state X*, is norm open and dense. Moreover this set forms a norm dense Gs if a state X» is strictly positive. IntroductionMarkov operators P : L 1 (//) -> (P > 0,P*1 = 1) arise as a generalization of the concept of transition probabilities (or stochastic matrices). Their iterates P n are intensively studied as they describe the evolution of classical (commutative) stochastic dynamical systems. Those systems which have a (unique) stationary density /* G L 1 (fi) and such that they are stable (i.e. after perturbations they return asymptotically to their equilibrium) are called mixing. Using mathematical language mixing simply means limn-Kx, P n (/) = /*, independently of the initial distribution /. The reader is referred to the monograph [8] in this regard. In a quantum mechanics the role of densities play states on a Hilbert space H. A natural model of a quantum system, in this context, is a linear operator P which transforms a state into another state. Again one can ask about stability of such systems.In [5] it was shown that the set of those noncommutative Markov operators on C\ (the class of trace operators on 7i) such that lim^oo |||P n -Q||| = 0, where Q is one-dimensional projection, is norm open and dense. Moreover if we request that the limit projections must be on strictly positive states then such operators P form a norm dense Gs-This kinds of questions, about geometric properties of sets of operators, are very important from

On typical properties of Hilbert space operators

“…Typical properties of various classes of operators have been studied previously: see e.g. [5] for typical properties of measure preserving transformations; [3] and [21] for typical mixing properties of Markov semigroups; [9], [12] and [13] for typical stability properties; [19], [27] and [30] for typical spectral properties in various very special families of operators. Our research is different from these works in several respects.…”