Lp-independence of spectral bounds of Schrödinger type semigroups

Let L be an m-symmetric Markov generator and μ a signed measure in the Kato class. We consider a Schrödinger type operator Hμ=−L+μ on Lp(m). We prove that under certain conditions for the Markov semigroup generated by L and the potential μ, the Lp-spectral bound of Hμ is independent of p if and only if the L2-spectral bound is non-positive.