Inverses of generators of nonanalytic semigroups
Volume 191 / 2009
Abstract
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 $x \in {\cal D}(A^k).$
