On the Dichotomy of Linear Autonomous Systems of Functional-Differential Equations

We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form $x_{n}=\Gamma x_{n-1}$ with a compact operator $\Gamma$, in Banach space. The spectrum of $\Gamma$ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.