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Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 3, Pages 81–84 (Mi faa2918)  

This article is cited in 2 scientific papers (total in 2 papers)

Brief communications

On the Uniform Kreiss Resolvent Condition

A. M. Gomilkoa, Ya. Zemanekb

a Institute of Hydromechanics of NAS of Ukraine
b Institute of Mathematics of the Polish Academy of Sciences

Abstract: Let $B$ be a Banach space with norm ${\|\cdot\|}$ and identity operator $I$. We prove that, for a bounded linear operator $T$ in $B$, the strong Kreiss resolvent condition
$$ \|(T-\lambda I)^{-k}\|\le\frac{M}{(|\lambda|-1)^k},\qquad|\lambda|>1, k=1,2,…, $$
implies the uniform Kreiss resolvent condition
$$ \|\sum_{k=0}^n \frac{T^k}{\lambda^{k+1}}\|\le\frac{L}{|\lambda|-1},\qquad|\lambda|>1, n=0,1,2,\dotsc. $$
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 $m\ge 2$.

Keywords: Banach space, bounded linear operator, Kreiss resolvent condition

DOI: https://doi.org/10.4213/faa2918

Full text: PDF file (141 kB)
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English version:
Functional Analysis and Its Applications, 2008, 42:3, 230–233

Bibliographic databases:

UDC: 517.9
Received: 19.03.2007

Citation: A. M. Gomilko, Ya. Zemanek, “On the Uniform Kreiss Resolvent Condition”, Funktsional. Anal. i Prilozhen., 42:3 (2008), 81–84; Funct. Anal. Appl., 42:3 (2008), 230–233

Citation in format AMSBIB
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\by A.~M.~Gomilko, Ya.~Zemanek
\paper On the Uniform Kreiss Resolvent Condition
\jour Funktsional. Anal. i Prilozhen.
\yr 2008
\vol 42
\issue 3
\pages 81--84
\mathnet{http://mi.mathnet.ru/faa2918}
\crossref{https://doi.org/10.4213/faa2918}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2454481}
\zmath{https://zbmath.org/?q=an:1155.47004}
\transl
\jour Funct. Anal. Appl.
\yr 2008
\vol 42
\issue 3
\pages 230--233
\crossref{https://doi.org/10.1007/s10688-008-0034-2}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Lyubich Yu., “The power boundedness and resolvent conditions for functions of the classical Volterra operator”, Studia Math., 196:1 (2010), 41–63  crossref  mathscinet  zmath  isi
    2. Gomilko A., Zemanek J., “On the Strong Kreiss Resolvent Condition”, Complex Anal. Oper. Theory, 7:2, SI (2013), 421–435  crossref  mathscinet  zmath  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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