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Cayley transform of the generator of a uniformly bounded C 0-semigroup of operators

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Abstract

We consider the problem of estimates for the powers of the Cayley transform V = (A + I)(A - I)−1 of the generator of a uniformly bounded C 0-semigroup of operators e tA, t ≥ 0, that acts in a Hilbert space H. In particular, we establish the estimate \(\sup _{n \in \mathbb{N}} \left( {\left\| {V^n } \right\|/\ln (n + 1)} \right) < \infty\). We show that the estimate \(\sup _{n \in \mathbb{N}} \left\| {V^n } \right\| < \infty\) is true in the following cases: (a) the semigroups e tA and \(e^{tA^{ - 1}}\) are uniformly bounded; (b) the semigroup e tA uniformly bounded for t ≥ ∞ is analytic (in particular, if the generator of the semigroup is a bounded operator).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1018–1029, August, 2004.

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Gomilko, A.M. Cayley transform of the generator of a uniformly bounded C 0-semigroup of operators. Ukr Math J 56, 1212–1226 (2004). https://doi.org/10.1007/s11253-005-0053-6

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