Pseudodifferential Operators via Harmonic Analysis on Noncompact Symmetric Spaces

Objectives To create a framework for harmonic analysis of pseudodifferential operators on noncompact using spherical functions and the spherical transform, Riemannian symmetric spaces can be connected to spatially dependent Lévy-type processes and sub-Feller semigroups. Material and Methods Gangolli operators are represented as pseudodifferential operators using spherical functions and the spherical transform. The Hille-Yosida-Ray theorem is extended to noncompact symbols and negative-definite symbols are the foundation of the analysis. Symmetric Riemannian spaces. Results We show that Gangolli operators admit pseudodifferential representations with computable negative definite symbols. We also prove that these operators generate strongly continuous sub-Feller semigroups on C 0 (G/K) . These results extend the theory of Markov generating pseudodifferential operators to noncompact symmetric spaces in Euclidean terms. Conclusion The proposed framework suggests a unified harmonic-analytic approach to the study of Lévy-type processes, pseudodifferential operators and semigroup theory on noncompact symmetric spaces. It increases the bond between differential geometry, harmonic analysis and probability theory.