For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-integrable functions on the chosen manifold). Also one now defines the generalized heat-kernel corresponding to this as $K(t,f,D) = Tr_{L^2}(f\, exp(-tD))$.

Now I am faced with the following identity for which I would like to know the proof:

$$ \xi(s,f,D) = \frac{1}{\Gamma(s)}\int_0^{\infty} t^{s-1} K(t,f,D) \, dt $$

It would be helpful if someone could also elaborate on the notion of what is a "positive operator of Laplacian type" (..I have some rough idea based on specific examples..) and if someone could specify peculiarities in the above equations that are likely to come up if one goes to non-compact manifolds like hyperbolic spaces.

I am mostly interested in doing this on hyperbolic spaces.