Construction of Globally Semigroup-Invariant C^∞ -Domains

Abstract

In this chapter, the ‘infinitesimal’ spectral-type commutation theory on Banach spaces (Chapters 5 and 6) is connected with the global invariant-domain commutation theory on locally convex spaces (Chapter 3). In particular, we obtain from the hypotheses of Chapter 3 (domain regularity) or of Chapter 5 (graph-density) that whenever the underlying space E is Banach (or Fréchet) there exists a domain D_∞ of C^∞-vectors which is a Fréchet space with respect to a stronger C^∞- topology. This space is invariant under the action at least of the (semi) group operators V(t,A) involved in the problem and under the operational images φ(A) of A constructed via a suitably restricted semigroup/transform operational calculus. In suitable instances, it is also invariant under certain of the resolvents R(λ,A) for A as well. The net effect of these developments is to unify the different types of commutation relations, while preparing for a unified, economical treatment of smooth integrals and continuous exponentials of operator Lie algebras in Chapters 8 and 9, and laying the groundwork for our subsequent analysis of smeared (distribution) exponentials.