On perturbations of differentiable semigroups |
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Authors: | Bogdan D Doytchinov William J Hrusa Stephen J Watson |
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Institution: | (1) Department of Mathematics, Carnegie Mellon University, 15213 Pittsburgh, PA |
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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.
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(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
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(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, ∞).
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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)‖<∞. |
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Keywords: | |
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