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

✍ Scribed by Frank Baldus

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 313 KB
• Volume: 252
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310034

✦ Synopsis

Abstract

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 ℒ︁(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.