Lyapunov exponents for linear delay equations in arbitrary phase spaces

A linear integral equation with infinite delay is considered where the admissible function space $$\mathcal{B}$$ of initial conditions is as usually only described axiomatically. Merely using this axiomatic description, the long time behavior of the solutions is determined by calculating the Lyapunov exponents. The calculation is based on a representation of the solution in the second dual space of $$\mathcal{B}$$ and on a connection between the asymptotic behavior of the solutions of the integral equation under consideration and its adjoint equation subject to the spectral decomposition of the space of initial functions. We apply the result to an example of a stochastic differential equation with infinite delay