Functional Analytic Background

Abstract

In this chapter we present some background from functional analysis. The proofs of a few specific facts, not necessary right away for understanding the text, are however, postponed to the Appendix (cf. A. Sections 1,2). We start in Section 1 with the relationship between strongly continuous contraction resolvents (G _α ) _α>0 , strongly continuous contraction semigroups (T _t )_t>0 and their generators (L,D(L)) on an arbitrary Banach space. In particular, we prove the Hille-Yosida theorem. These results are summarized in the diagram on p. 14. In Section 2 we study coercive closed forms (ε, D(ε)) on a Hilbert space H and their relation first with strongly continuous contraction resolvents and their generators on H and subsequently their relation with the corresponding semigroups (cf. the diagram on p. 27). Section 3 is devoted to the crucial notion of closability which is important for applications and the examples to be treated in this book. In Section 4 we specialize to the case where H is an L ²-space over an arbitrary measure space and study the relations between respective “contraction properties” of the four corresponding objects (ε,D(ε)), (G _α ) _α>0 (T _t ) _t>0 and (L, D(L)) from Section 2.