Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators
We consider the problem of estimates for the powers of the Cayley transform $V = (А + I)(А - I)^{-1}$ of the generator of a uniformly bounded $C_0$-semigroup of operators $e^{tA} , t \geq 0$, that acts in a Hilbert space $H$. In particular, we establish the estimate $\sup_{n \in N}\left(||V^n||/\ln...
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| Datum: | 2004 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2004
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3819 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We consider the problem of estimates for the powers of the Cayley transform $V = (А + I)(А - I)^{-1}$ of the generator of a uniformly bounded $C_0$-semigroup of operators $e^{tA} , t \geq 0$, that acts in a Hilbert space $H$. In particular, we establish the estimate $\sup_{n \in N}\left(||V^n||/\ln(n + 1)\right) < \infty$.
We show that the estimate $\sup_{n ∈ N} ∥V^n∥ < ∞$ is true in the following cases: (a) the semigroups $e^{tA}$ and $e^{tA^{−1}}$ are uniformly bounded; (b) the semigroup etA uniformly bounded for $t ≥ ∞$ is analytic (in particular, if the generator of the semigroup is a bounded operator). |
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