Abstract

We give a short summary of recent development of spectral analysis of autonomous or periodic linear functional differential equations with infinite delay on the abstract phase space with uniformly fading memory property. The considered functional differential equation is transformed into an operator equation on function spaces. The admissibility theory of function spaces is reduced to the problem of the solvability of this operator equation. The spectra of bounded solutions are studied from the decomposition technique of solutions. The method of commuting operators are effectively used by combining the general result about the distribution of the spectrum of the generator of solution semigroup of autonomous equation.