We continue in this chapter our examination of the Cauchy problem for the equation u′(t) = Au(t) initiated in Sections 1.2 and 1.5. The main result is Theorem 2.1.1, where necessary and sufficient conditions are given in order that the Cauchy problem be well posed in t < 0. These conditions involve restrictions on the location of the spectrum of A and inequalities for the norm of the powers of the resolvent of A. The proof presented here, perhaps not the shortest or the simplest, puts in evidence nicely the fundamental Laplace transform relation between the propagator S(t) and the resolvent of A; in fact, S(t) is obtained from the resolvent using (a slight variant of) the classical contour integral for computation of inverse Laplace transforms. Section 2.2 covers the corresponding result for the Cauchy problem in the whole real line, and the adjoint equation. The adjoint theory is especially interesting in nonreflexive spaces and will find diverse applications in Chapters 3 and 4.

One of the first results in this chapter is the proof of the semigroup or exponential equations S(0) = I, S(s + t) = S(s)S(t) for the propagator S(t). We show in Section 2.3 that any strongly continuous operator-valued function satisfying the exponential equations must be the propagator of an equation u′(t) = Au(t). Finally, Section 2.4 deals with several results for the inhomogeneous equation u′(t) = Au(t) + f(t).