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This article is cited in 9 scientific papers (total in 9 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
$$
\|(T-\lambda I)^{-k}\|\le\frac{M}{(|\lambda|-1)^k},\qquad|\lambda|>1,\ k=1,2,\dots,
$$
implies the uniform Kreiss resolvent condition
$$
\bigg\|\sum_{k=0}^n \frac{T^k}{\lambda^{k+1}}\bigg\|\le\frac{L}{|\lambda|-1},\qquad|\lambda|>1,\ n=0,1,2,\dotsc.
$$
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.
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
Linking options:
https://www.mathnet.ru/eng/faa2918https://doi.org/10.4213/faa2918 https://www.mathnet.ru/eng/faa/v42/i3/p81
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Abstract page: | 519 | Full-text PDF : | 208 | References: | 73 | First page: | 5 |
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