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We provide a general framework for the realization of powers or functions ofsuitable operators on Dirichlet spaces. The first contribution is to unify theavailable results dealing with specific geometries; a second one is to providenew results on rather general metric measured spaces that were not consideredbefore and fall naturally in the theory of Dirichlet spaces. The main tool isusing the approach based on subordination and semi-groups by Stinga and Torrea.Assuming more on the Dirichlet space, we derive several applications to PDEssuch as Harnack and Boundary Harnack principles.

Powers Of Generators On Dirichlet Spaces And Applications To Harnack Principles