On the uniform Kreiss resolvent condition

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

$left| {sumlimits_{k = 0}^n {frac{{T^k }}{{lambda ^{k + 1} }}} } right| leqslant frac{L}{{|lambda | - 1}}, |lambda | > 1, n = 0,1,2, ldots .$

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.