Linear Semigroups

In this chapter we describe the basic notions of a linear C₀-semigroup and the related concepts of an infinitesimal generator. These concepts form infinite dimensional versions of solutions of the finite dimensional linear ordinary differential equation ə _t x = Ax.In particular, the C₀-semigroup corresponds to the solution operator or the fundamental solution matrix, and the infinitesimal generator corresponds to the linear coefficient matrix A. We will see that the C₀-semigroups are linear prototypes of the semiflows described in Chapter 2.