Inverses of Generators of Integrated or Regularized Semigroups

It is shown that, for any bounded, injective operator C, the class of injective, densely defined operators with dense range and nonempty resolvent that generate bounded holomorphic C-regularized semigroups is closed under inversion, but, for any n ∈ N, the class of injective, densely defined operators with dense range that generate bounded holomorphic n-times integrated semigroups is very far from being closed under inversion: it is shown that, if both A and A^-1 generate bounded holomorphic n-times integrated semigroups of sufficiently large angle θ, then they both generate strongly continuous bounded holomorphic semigroups of angle θ.