โ Scribed by Ola Bratteli, Palle E. T. Jรธrgensen (auth.), Ola Bratteli, Palle E. T. Jรธrgensen (eds.)
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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.
Front Matter....Pages i-vi
Positive Semigroups of Operators, and Applications: Editorsโ Introduction....Pages 213-219
Positive One-Parameter Semigroups on Ordered Banach Spaces....Pages 221-296
Asymptotic Behavior of One-Parameter Semigroups of Positive Operators....Pages 297-309
Positivity in Time Dependent Linear Transport Theory....Pages 311-331
Quantum Dynamical Semigroups, Symmetry Groups, and Locality....Pages 333-352
Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups....Pages 353-378
Order Properties of Attractive Spin Systems....Pages 379-390
Book Reviews....Pages 391-398
Back Matter....Pages 399-410
Analysis
๐ SIMILAR VOLUMES
A comprehensive account of the main theoretical aspects of linear semigroups, with examples and exercises included.
A comprehensive account of the main theoretical aspects of linear semigroups, with examples and exercises included.
<p>This book is about stability of linear dynamical systems, discrete and continuous. More precisely, we discuss convergence to zero of strongly continuous semigroups of operators and of powers of a bounded linear operator, both with respect to different topologies. The discrete and the continuous c
From the reviews: "Since E. Hille and K. Yoshida established the characterization of generators of C0 semigroups in the 1940s, semigroups of linear operators and its neighboring areas have developed into a beautiful abstract theory. Moreover, the fact that mathematically this abstract theory has man
The book is organized as follows. We concentrate our attention on three subjects of semigroup theory: characterization, spectral theory and asymptotic behavior. By characterization , we understand the problem of describing special properties of a semigroup, such as positivity, through the generator.