# Inverses of generators of nonanalytic semigroups

Studia Mathematica (2009)

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

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topRalph 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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