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- Publisher
- Springer Journals
- Copyright
- Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
- Subject
- Mathematics; Analysis
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-009-0019-5
- Publisher site
- See Article on Publisher Site
Abstract
The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and $${M_{A}:=\limsup_{|\lambda| \rightarrow \infty, \lambda \in \mathbb {C}_+}\|\lambda R(\lambda , A) \|}$$ , then every closed linear operator C such that $${(\omega,\infty)\subset\rho(C)}$$ for some $${\omega\in \mathbb {R}}$$ and $$\limsup_{\lambda \rightarrow \infty}\|\lambda R(\lambda ,A)- \lambda R(\lambda ,C)\|< \frac{1}{2+\sqrt{3}}\left( 1+ M_{A}^2 \right)^{-\frac{1}{2}}$$ generates a holomorphic semigroup, too. Moreover, we obtain an analogue of this result for differences of semigroups. If T is a holomorphic semigroup and $${k_T:=(\limsup_{t\rightarrow 0^+}\|(T(t)+I)^{-1}\|)^{-1}}$$ , then every C 0 -semigroup S with $$\limsup\limits_{t\rightarrow 0^+}\|T(t)-S(t)\|< k_T$$ is holomorphic. We also give certain estimates for the constants M A and k T appearing in the above conditions.
Journal
Journal of Evolution Equations – Springer Journals
Published: Sep 1, 2009