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A short proof for the stability theorem for positive semigroups on $L_p(\mu )$

Author: Lutz Weis
Journal: Proc. Amer. Math. Soc. 126 (1998), 3253-3256
MSC (1991): Primary 47D06
MathSciNet review: 1469440
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Abstract: We give a short proof showing that the growth bound of a positive semigroup on $L_p(\mu )$ equals the spectral bound of its generator. It is based on a new boundedness theorem for positive convolution operators on $L_p(L_q)$. We also give a counterexample, showing that Gearhart’s result does not extend from Hilbert spaces to $L_p(\mu )$-spaces.

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Lutz Weis
Affiliation: Universität Karlsruhe, Mathematisches Institut I, Postfach 69 80, Englerstrasse 2, 76128 Karlsruhe, Germany
MR Author ID: 181530

Keywords: Stability of semigroups, positive operators
Received by editor(s): July 22, 1996
Received by editor(s) in revised form: March 12, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society