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Functional Analysis and Its Applications

, Volume 42, Issue 3, pp 230–233 | Cite as

On the uniform Kreiss resolvent condition

  • A. M. Gomilko
  • J. Zemánek
Article
  • 87 Downloads

Abstract

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition
$$ \parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M} {{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots , $$
implies the uniform Kreiss resolvent condition
$$ \left\| {\sum\limits_{k = 0}^n {\frac{{T^k }} {{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L} {{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots . $$
We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T m for each integer m ⩾ 2.

Key words

Banach space bounded linear operator Kreiss resolvent condition 

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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Hydromechanics of NAS of UkraineKyivUkraine
  2. 2.Instytut Matematyczny Polskiej Akademii NaukWarszawaPoland

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