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

Ralph deLaubenfels

Studia Mathematica (1992)

  • Volume: 103, Issue: 2, page 143-159
  • ISSN: 0039-3223

Abstract

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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 ≤ 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 ≥ 0 and ∃ M < ∞ such that ∥ H n ( s ) ∥ ≡ ∥ ( ∑ k = 0 n ( s k A k ) / k ! ) e - s A ∥ ≤ 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|) in all half-planes Re(z) > a > 0 and K ( t ) ≡ ʃ 1 + i ℝ e z t e - z A d z / ( 2 π 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 H n ( s ) . For ϕ ∈ X*, x ∈ X, ( F ( t ) ϕ ) ( x ) = ( d / d t ) 2 ( ϕ ( K ( t ) x ) ) = l i m n → ∞ ϕ ( H n ( n / t ) x ) , for almost all t.

How to cite

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deLaubenfels, Ralph. "Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform." Studia Mathematica 103.2 (1992): 143-159. <http://eudml.org/doc/215942>.

@article{deLaubenfels1992,
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.},
author = {deLaubenfels, Ralph},
journal = {Studia Mathematica},
keywords = {Laplace transform of a Lipschitz continuous family of operators; strongly continuous differentiable semigroup; decomposition of the identity},
language = {eng},
number = {2},
pages = {143-159},
title = {Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform},
url = {http://eudml.org/doc/215942},
volume = {103},
year = {1992},
}

TY - JOUR
AU - deLaubenfels, Ralph
TI - Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 2
SP - 143
EP - 159
AB - 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.
LA - eng
KW - Laplace transform of a Lipschitz continuous family of operators; strongly continuous differentiable semigroup; decomposition of the identity
UR - http://eudml.org/doc/215942
ER -

References

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