Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on R N , N ≥ 1. It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C 0 -semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on R N . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.

A b s t r a c t . Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L p spaces on R N , N ≥ 1. Estimates of the rate of convergence are also obtained.Our mean tool is a Korovkin-type theorem which we establish in the context of L p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X . Several examples are explicitly indicated as well.

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss-Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu := u + u + u.Shape-preserving and regularity properties are also investigated.

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