# Inverses of generators of nonanalytic semigroups

Studia Mathematica (2009)

• Volume: 191, Issue: 1, page 11-38
• ISSN: 0039-3223

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## Abstract

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Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${{e}^{tA}}_{t\ge 0}$. It is shown that ${A}^{-1}$ generates an $O\left(1+\tau \right)A{\left(1-A\right)}^{-1}$-regularized semigroup. Several equivalences for ${A}^{-1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${{e}^{tA}}_{t\ge 0}$, on subspaces, for ${A}^{-1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly continuous semigroup. We also show that, for k a natural number, if ${{e}^{tA}}_{t\ge 0}$ is exponentially stable, then $||{e}^{\tau {A}^{-1}}x||=O\left({\tau }^{1/4-k/2}\right)$ for $x\in \left({A}^{k}\right)$.

## How to cite

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Ralph deLaubenfels. "Inverses of generators of nonanalytic semigroups." Studia Mathematica 191.1 (2009): 11-38. <http://eudml.org/doc/284694>.

@article{RalphdeLaubenfels2009,
abstract = {Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup $\{e^\{tA\}\}_\{t≥0\}$. It is shown that $A^\{-1\}$ generates an $O(1+τ) A(1 - A)^\{-1\}$-regularized semigroup. Several equivalences for $A^\{-1\}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of $\{e^\{tA\}\}_\{t≥0\}$, on subspaces, for $A^\{-1\}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly continuous semigroup. We also show that, for k a natural number, if $\{e^\{tA\}\}_\{t≥0\}$ is exponentially stable, then $||e^\{τA^\{-1\}\}x|| = O(τ^\{1/4-k/2\})$ for $x ∈ (A^\{k\})$.},
author = {Ralph deLaubenfels},
journal = {Studia Mathematica},
keywords = {strongly continuous semigroups of operators; functional calculus; regularized semigroups; -semigroups; integrated semigroups},
language = {eng},
number = {1},
pages = {11-38},
title = {Inverses of generators of nonanalytic semigroups},
url = {http://eudml.org/doc/284694},
volume = {191},
year = {2009},
}

TY - JOUR
AU - Ralph deLaubenfels
TI - Inverses of generators of nonanalytic semigroups
JO - Studia Mathematica
PY - 2009
VL - 191
IS - 1
SP - 11
EP - 38
AB - Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{tA}}_{t≥0}$. It is shown that $A^{-1}$ generates an $O(1+τ) A(1 - A)^{-1}$-regularized semigroup. Several equivalences for $A^{-1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{tA}}_{t≥0}$, on subspaces, for $A^{-1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly continuous semigroup. We also show that, for k a natural number, if ${e^{tA}}_{t≥0}$ is exponentially stable, then $||e^{τA^{-1}}x|| = O(τ^{1/4-k/2})$ for $x ∈ (A^{k})$.
LA - eng
KW - strongly continuous semigroups of operators; functional calculus; regularized semigroups; -semigroups; integrated semigroups
UR - http://eudml.org/doc/284694
ER -

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