Resolvents, Transition Functions, and Strongly Markovian Processes

Abstract

Let X be a compact Hausdorff space and A a complex linear algebra of continuous complex-valued functions defined on X. Suppose A is normal on X, i.e., for every pair of disjoint closed sets Ko, Ki in X, there exists a function /G-4 such that /(2£ 0) = 0 and f(Ki) = 1. Does it follow that every continuous complex-valued function on X can be uniformly approximated by functions in A? With the additional assumption that A is closed under complex conjugation, it follows by the Stone-Weierstrass theorem. (Trivially, if A is normal then A separates points.) The same theorem implies that the analogous question in the case of real-valued functions has an affirmative answer. However, in the complex-valued case it need not be so. An example will be given which demonstrates this. In this example, the space X is a suitably chosen compact set in the complex plane. The algebra is R(X), the algebra of all functions which can be uniformly approximated on X by rational functions whose poles lie outside X. It will be shown that R(X) is normal on X and is a proper sub-algebra of C(X) t the algebra of all continuous complex-valued functions on X. Since R(X) is closed under uniform limits, this will be sufficient. Two lemmas are needed to accomplish this. One is a modification of an observation of Mergelyan [l]. The second represents a slight extension of a result due to Beurling [2].

Représentation de temps terminaux et applications aux fonctionnelles additives et aux systèmes de Lévy

“…Nous désignerons alors, d'après un théorème de Ray [11], par (Pi),_, l'unique semi-groupe sur E admettant (V^>o pour résolvante et tel que pour tout xçE l'application t^->Pi{x, .) soit étroitement continue à droite sur R+.…”

Familles d'opérateurs et frontière en théorie du potentiel

“…Ray s'est préoccupé de savoir sous quelles hypothèses on peut conclure que chaque opérateur V^ est l'intégrale d'un semi-groupe sous-markovien (voir [30]). Le théorème de Hille-Yosida (voir chapitre v) permet de ramener cette question à la suivante :…”

“…Dans la lecture d'un article de D. Ray [30], nous avons puisé l'idée d'une hypothèse de séparation moins restrictive que l'hypothèse de densité de Hunt, et permettant d'associer un semi-groupe à la famille résolvante (V^).…”

Excessive measures and the existence of right semigroups and processes

“…Extensions of these classical results to more general settings have been formulated by Engelbert [7,8]. The most general method of construction of processes so far is the Ray (-Knight) compactification originally introduced by Ray [19]; it provides a Ray semigroup and a Ray process on an enlarged compact state space (see, e.g., [4,11,21]). In order to obtain an answer to the initial question via that construction one has to make sure that the Ray semigroup (resp.…”