Application of the Weyl–Hörmander calculus to generators of Feller semigroups

In this article we apply the S ( M , g )–calculus of L. Hörmander and, in particular, results concerning the spectral invariance of the algebra of operators of order zero in ℒ( L 2 (ℝ n )) to study generators of Feller semigroups. The core of the article is the proof of the invertibility of λ Id + P for a strongly elliptic operator P in Ψ( M , g ) and suitable weight functions M and metrics g . The proof depends highly on precise estimates of the remainder term in asymptotic expansions of the product symbol in Weyl and Kohn–Nirenberg quantization. Due to the Hille–Yosida–Ray theorem and a theorem of Courrège, the result concerning the invertibility of λ Id + P is applicable to obtain sufficient conditions for an operator to extend to a generator of a Feller semigroup. Moreover, we discuss Sobolev spaces adapted to our weight functions and apply our results to study the generator of a subordinate Feller semigroup. As for the Feller semigroups and the subordination in the sense of Bochner, we build on results of N. Jacob and R. L. Schilling.