A Neutral FDE Is Similar to the Product of an ODE and a Shift

In finite dimensional control theory a minimal realization of a transfer function is determined only module a similarity transform. The infinite dimensional situation is similar: There is no real reason to distinguish between two realizations that differ from each other only by a similarity transform. The purpose of this work is to show that under natural assumptions, the semigroup induced by a neutral functional differential equation with infinite delay is similar to the cross product of a semigroup induced by an ordinary differential equation and a shift semigroup. This result is expected to lead to simpler realization results for transfer functions induced by delay equations.