Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels

Studia Mathematica, Tome 192 (2009), p. 11-38 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $x ∈ (A^{k})$.

Détail

Publié le : 2009-01-01

Zbl 1167.47035

EUDML-ID : urn:eudml:doc:284694