On the uniform Kreiss resolvent condition

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.