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Perturbation theorems for holomorphic semigroups

Authors:

Sebastian Król

Institution:

(1) Faculty of Mathematics and Computer Science, Nicolas Copernicus University, UL. Chopina 12/18, 87-100 Torun, Poland

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. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr N201384834.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47D06 Secondary 47A55

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