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Positive Semigroups of Operators, and Applications

  • Ola Bratteli
  • Palle E. T. Jørgensen

Table of contents

  1. Front Matter
    Pages i-vi
  2. Ola Bratteli, Palle E. T. Jørgensen
    Pages 213-219
  3. Charles J. K. Batty, Derek W. Robinson
    Pages 221-296
  4. R. L. Hudson, K. R. Parthasarathy
    Pages 353-378
  5. William G. Faris, E. B. Davies
    Pages 391-398
  6. Back Matter
    Pages 399-410

About this book

Introduction

This means that semigroup theory may be applied directly to the study of the equation I'!.f = h on M. In [45] Yau proves that, for h ~ 0, there are no nonconstant, nonnegative solutions f in [j' for 1 < p < 00. From this, Yau gets the geometric fact that complete noncom pact Riemannian manifolds with nonnegative Ricci curvature must have infinite volume, a result which was announced earlier by Calabi [4]. 6. Concluding Remarks In several of the above results, positivity of the semigroup plays an important role. This was also true, although only implicitly, for the early work of Hille and Yosida on the Fokker-Planck equation, i.e., Equation (4) with c = O. But it was Phillips [41], and Lumer and Phillips [37] who first called attention to the importance of dissipative and dispersive properties of the generator in the context of linear operators in a Banach space. The generation theorems in the Batty-Robinson paper appear to be the most definitive ones, so far, for this class of operators. The fundamental role played by the infinitesimal operator, also for the understanding of order properties, in the commutative as well as the noncommutative setting, are highlighted in a number of examples and applications in the different papers, and it is hoped that this publication will be of interest to researchers in a broad spectrum of the mathematical sub-divisions.

Keywords

Banach Space Convexity banach spaces calculus functional analysis manifold operator

Editors and affiliations

  • Ola Bratteli
    • 1
  • Palle E. T. Jørgensen
    • 2
  1. 1.Mathematics InstituteUniversity of TrondheimNorway
  2. 2.Dept. of Mathematics/E1University of PennsylvaniaPhiladelphiaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-009-6484-6
  • Copyright Information Springer Science+Business Media B.V. 1984
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-009-6486-0
  • Online ISBN 978-94-009-6484-6
  • Buy this book on publisher's site