Spectral resolution for integro-differential equations

A spectral theory is developed for the solution semigroup of a scalar Volterra integro-differential equation with completely monotonic kernel. Under certain conditions, the state space decomposes in a finite-dimensional subspace and a closed complement, where the generator is self-adjoint with respect to a suitable inner product, and hence has a spectral resolution. The results rely on the fact that the generator is Hermitian with respect to an indefinite inner product that makes the state space a Pontryagin space H₁. The study can be considered an investigation of Hermitian operators in H₁