Geodesic
On the Uniform Kreiss Resolvent Condition
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 3, pp. 81-84

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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. 
@article{FAA_2008_42_3_a9,
     author = {A. M. Gomilko and Ya. Zemanek},
     title = {On the {Uniform} {Kreiss} {Resolvent} {Condition}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {81--84},
     publisher = {mathdoc},
     volume = {42},
     number = {3},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_3_a9/}
}
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A. M. Gomilko; Ya. Zemanek. On the Uniform Kreiss Resolvent Condition. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 3, pp. 81-84. http://geodesic.mathdoc.fr/item/FAA_2008_42_3_a9/