Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform

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 ${\displaystyle {\left\{e^{-sA}\right\}}_{s\le 0}}$ such that ${\displaystyle {\left\{\left(\frac{1}{s^2}\right)e^{-sA}\right\}}_{s>0}}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${\displaystyle {\left\{e^{-sA}\right\}}_{s\ge 0}}$ and ∃ M < ∞ such that ${\displaystyle \parallel H_n\left(s\right)\parallel \equiv \parallel ({\sum }_{k=0}^n\frac{s^k{A}^k}{k}!)e^{-sA}\parallel \le M}$, ∀s > 0, n ∈ ℕ ∪ {0}. (4) -A generates a strongly continuous holomorphic semigroup ${\displaystyle {\left\{e^{-zA}\right\}}_{Re\left(z\right)>0}}$ that is O(|z|) in all half-planes Re(z) > a > 0 and ${\displaystyle K\left(t\right)\equiv {ʃ}_{1+iℝ}{e}^{zt}{e}^{-zA}\frac{dz}{2\pi i{z}^3}}$ defines a differentiable function of t, with Lipschitz continuous derivative, with K'(0) = 0. We may then construct a decomposition of the identity, F, for A, from K(t) or ${\displaystyle H_n\left(s\right)}$. For ϕ ∈ X*, x ∈ X, ${\displaystyle (F\left(t\right)\varphi )\left(x\right)={\left(\frac{d}{dt}\right)}^2(\varphi \left(K\left(t\right)x\right))=\underset{n\to \infty }{lim}\varphi \left(H_n\left(\frac{n}{t}\right)x\right)}$, for almost all t.