Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform
Volume 103 / 1992
Abstract
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 ${e^{-sA}}_{s≤0}$ such that ${(1/s^2)e^{-sA}}_{s>0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{-sA}}_{s≥0}$ and ∃ M < ∞ such that $∥H_n(s)∥ ≡ ∥(∑_{k=0}^n (s^k A^{k})/k!) e^{-sA}∥ ≤ M$, ∀s > 0, n ∈ ℕ ∪ {0}. (4) -A generates a strongly continuous holomorphic semigroup ${e^{-zA}}_{Re(z)>0}$ that is O(|z|) in all half-planes Re(z) > a > 0 and $K(t) ≡ ʃ_{1+iℝ} e^{zt} e^{-zA} dz/(2πiz^3)$ 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 $H_n(s)$. For ϕ ∈ X*, x ∈ X, $(F(t)ϕ)(x) = (d/dt)^2 (ϕ(K(t)x)) = lim_{n→∞} ϕ(H_n(n/t)x)$, for almost all t.
