Abstract
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.
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deLaubenfels, R. Inverses of Generators of Integrated or Regularized Semigroups. Semigroup Forum 75, 457–463 (2007). https://doi.org/10.1007/s00233-006-0654-x
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DOI: https://doi.org/10.1007/s00233-006-0654-x