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Functional Analytic Background

  • Zhi-Ming Ma
  • Michael Röckner
Chapter
  • 643 Downloads
Part of the Universitext book series (UTX)

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 L2-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.

Keywords

Linear Operator Bilinear Form Real Hilbert Space Sector Condition Dirichlet Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Zhi-Ming Ma
    • 1
  • Michael Röckner
    • 2
  1. 1.Institute of Applied MathematicsAcademia SinicaBeijingPeople’s Republic of China
  2. 2.Institut für Angewandte MathematikUniversität BonnBonn 1Germany

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