2017
Том 69
№ 9

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Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators

Gomilko A. M.

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Abstract

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).

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 8, pp 1212–1226.

Citation Example: Gomilko A. M. Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators // Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1018-1029.

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