Inverses of generators of nonanalytic semigroups

Volume 191 / 2009

Ralph deLaubenfels Studia Mathematica 191 (2009), 11-38 MSC: Primary 47A60, 47D03; Secondary 47D60, 47D62. DOI: 10.4064/sm191-1-2


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 $x \in {\cal D}(A^k).$


  • Ralph deLaubenfels1841 Drew Avenue
    Columbus, OH 43235, U.S.A.
    Department of Mathematics
    Ohio State University
    Columbus, OH 43210, U.S.A.

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