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We give a short proof showing that the growth bound of a positive semigroup on Lp(p) equals the spectral bound of its generator. It is based on a new boundedness theorem for positive convolution operators on Lp(Lq). We also give a counterexample, showing that Gearhart's result does not extend from Hilbert spaces to Lp(,u)-spaces.

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