Laplace transforms and operator families in locally convex spaces

Summary

Section 1.1 collects the basic facts which one needs to know about Laplace transforms. Among others are three inversion formulas and two representation theorems (Theorems 1.5, 1.7 and 1.8, Theorems 1.11 and 1.12); Theorem 1.11 will be widely used later in the treatment of perturbation problems.

Section 1.2 is devoted to the statement of an integrated version of the classical Widder’s representation theorem of Laplace transforms and its proof in a sequentially complete locally convex space (in short SCLCS). This theorem (Theorem 2.1) is useful in the treatment of operator families and operator differential equations. In particular, it will be used in the proof of Theorem 2.2.2.

In Section 1.3, we introduce (exponentially equicontinuous) r-times integrated, C-regularized semigroups in SCLCS for any r ≥ 0, and characterize their generators in terms of the estimates of the resolvents using Theorem 2.1. The resulting theorems are generalizations of the corresponding results for strongly continuous semigroups in SCLCS. Also, some elementary properties about these semigroups are given.

Section 1.4 is an analogue of Section 1.3 for r-times integrated, C-regularized cosine functions in SCLCS for any r ≥ 0.

We consider in Section 1.5 a large class of differential operators on certain function spaces L ^p _l (R ⁿ) (1 < p < ∞, l = 0, ···, n; L ^p₀ (R ⁿ) is just the usual Banach space L ^p(R ⁿ)), and show that they generate integrated or regularized semigroups. This is meaningful in considering that very few of these operators generate the classical strongly continuous semigroups. Actually, as made clear by Hörmander [1] in 1960, the Schrödinger operator iΔ generates a strongly continuous semigroup on L ^p(R ⁿ) (1 ≤ p ≤ ∞) only if p = 2.

The study of integrated, regularized semigroups and other operator families provides us with unified techniques for dealing with both wellposed and illposed Cauchy problems. In Section 1.6 finally, we exhibit the connection between integrated, regularized semigroups (resp. cosine functions) and the abstract Cauchy problems.