A Note on Integrability of the Norm of a Perturbed Semigroup

摘要

Recently, Michael Gil' has provided a condition for integrability of norms of semigroups generated by operators of the form A+B where B is bounded, in terms of the norm of the commutator of A and B. We show that his argument, when slightly refined, leads to a bit more general result than the one presented in his paper. We also argue, by examining an example, that the size of the commutator may not be a good indicator of whether the perturbed semigroup has integrable norm or not.