Graph-Density Applied to Semigroup Commutation Relations

Abstract

In this chapter, Laplace inversion techniques of HilleYosida type are used to transform the C¹-vector invariance and commutation properties of resolvents (obtained in Chapter 5) into comparable results for semigroups and groups. Modulo the last section, we thus obtain equivalence of infinitesimal ‘zero-deficiency’ graph-density conditions with the property that the (semi-)groups {V(t,A)} restrict to C₀ (semi-)groups {V₁(t,A)} acting upon the C¹-vector Banach space (D₁,‖ · ‖ ₁) · The results obtained here are applied both in constructing globally invariant C^∞ domains in Chapter 7 and in obtaining the graph-density exponentiation Theorem 9.2 in Chapter 9.