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

Abstract

We give a short proof showing that the growth bound of a positive semigroup on L p ( μ ) 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 ) L_p(L_q) . We also give a counterexample, showing that Gearhart’s result does not extend from Hilbert spaces to L p ( μ ) L_p(\mu ) -spaces.