Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels
2009 Studia Mathematica  
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
more » ... of −d/dx on the closure of its image in L 1 ([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 τ A −1 x = O(τ 1/4−k/2 ) for x ∈ D(A k ).
doi:10.4064/sm191-1-2 fatcat:aecx7bu5ezh6pi3ubcfhng5leu