A short proof for the stability theorem for positive semigroups on 𝐿_{𝑝}(𝜇)

Weis, Lutz

doi:10.1090/S0002-9939-98-04612-7

A short proof for the stability theorem for positive semigroups on $L_p(\mu )$
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Proc. Amer. Math. Soc. 126 (1998), 3253-3256 Request permission

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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