Resolvent conditions and powers of operators (Q2717568)

Having no answer to the question of J.Zemánek who asked whether there are quasinilpotent operators \(Q\) such that \(A=I+Q\) would satisfy the Ritt resolvent conditions, i.e. NEWLINE\[NEWLINE \|(\lambda I-A)^{-1}\|{c\over |\lambda-1|} \quad \text{for } |\lambda|>1, NEWLINE\]NEWLINE the author proves that this condition is equivalent to the following one: there exist constants \(M_0\) and \(M_1\) such that for all natural \(n\) NEWLINE\[NEWLINE \|A^n \|\leq M_0 NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \|A^n(A-I)\|\leq{M_1\over{n+1}}. NEWLINE\]NEWLINE The author discusses other relations between the growth of the resolvent near the unite circle and bounds for the powers of the operator.