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