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Ukrainian Mathematical Journal

, Volume 56, Issue 8, pp 1212–1226 | Cite as

Cayley transform of the generator of a uniformly bounded C0-semigroup of operators

  • A. M. Gomilko
Article

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 C0-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).

Keywords

Hilbert Space Bounded Operator 
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Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • A. M. Gomilko
    • 1
  1. 1.Institute of HydromechanicsUkrainian Academy of SciencesKiev

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