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Positive Semigroups of Operators, and Applications (Hardcover, Reprinted from `Acta Applicandae Mathematicae', Volume 2:3,4)
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Positive Semigroups of Operators, and Applications (Hardcover, Reprinted from `Acta Applicandae Mathematicae', Volume 2:3,4)
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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.
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