Lototsky-Schnabl operators on compact convex sets and their associated limit semigroups

Abstract

Given a metrizable compact convex setX of a locally convex Hausdorff space, a positive projectionT:C(X, ℝ)→C(X, ℝ) and a continuous function λ:X→[0, 1], it is shown that under suitable assumptions there exists a positive contraction semigroup onC(X, ℝ) that can be represented in terms of the Lototsky-Schnabl operators associated withT and λ. Several properties of this semigroup are investigated. In particular, its infinitesimal generator is determined in a core of its domain. WhenX⊂ℝ p for somep≥1, then the generator is shown to be a degenerate elliptic second order differential operator.