On an integral transform by R. S. Phillips

Sten Bjon

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Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup ${e^{- s A}}_{s \leq 0}$ such that ${(1 / s^{2}) e^{- s A}}_{s > 0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{- s A}}_{s \geq 0}$ and ∃ M < ∞ such that $∥ H_{n} (s) ∥ \equiv ∥ (\sum_{k = 0}^{n} (s^{k} A^{k}) / k!) e^{- s A} ∥ \leq M$ , ∀s > 0, n ∈ ℕ ∪ 0. (4) -A generates a strongly continuous holomorphic semigroup ${e^{- z A}}_{R e (z) > 0}$ that is O(|z|)...