On perturbations of differentiable semigroups
| Autor: | Bogdan D. Doytchinov, Stephen J. Watson, William J. Hrusa |
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| Rok vydání: | 1997 |
| Předmět: | |
| Zdroj: | Semigroup Forum. 54:100-111 |
| ISSN: | 1432-2137 0037-1912 |
| DOI: | 10.1007/bf02676591 |
| Popis: | LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx eX. 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 supt→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable. (ii) If 0 |
| Databáze: | OpenAIRE |
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