Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels ¹

¹ 1841 Drew Avenue Columbus, OH 43235, U.S.A. and Department of Mathematics Ohio State University Columbus, OH 43210, U.S.A.

Studia Mathematica, Tome 191 (2009) no. 1, pp. 11-38

Résumé

Suppose $A$ is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup $\{e^{tA}\}_{t \geq 0}.$ It is shown that $A^{-1}$ generates an $O(1 + \tau)$ $A(1 - A)^{-1}$-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 $\{e^{tA}\}_{t \geq 0},$ 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 $L^1([0, \infty))$ does not generate a strongly continuous semigroup. We also show that, for $k$ a natural number, if $\{e^{tA}\}_{t \geq 0}$ is exponentially stable, then $\|e^{\tau A^{-1}}x\| = O(\tau^{1/4 - k/2})$ for $x \in {\cal D}(A^k).$

Détail

DOI : 10.4064/sm191-1-2

Keywords: suppose injective linear operator banach space generates uniformly bounded strongly continuous semigroup geq shown generates tau regularized semigroup several equivalences generating strongly continuous semigroup given these generate sufficient conditions growth geq subspaces generating strongly continuous semigroup inverse d closure its image infty does generate strongly continuous semigroup natural number geq exponentially stable tau tau cal