On perturbations of differentiable semigroups

Abstract

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

• (i) If lim sup _t→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable.

• (ii) If 0<L=lim sup _t→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero

• (iii) For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsup _t→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞).

We also show that if lim sup _t→0+t ^p ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim sup _t→0+t ^p‖S′(t)‖<∞; it was known previously that if limsup _t→0+t ^p‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsup _t→0+t ^2p–1‖S′(t)‖<∞.