The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

Abstract

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)−1‖≤C|z−1|−1, |z|>1, we show how to use the Riesz turndown collar theorem to estimate supn≥0‖Tn‖. A similar estimate is shown for lim supn ‖Tn‖ in terms of the Ritt constantM=lim supz→1‖(1−z)(zI−T)−1‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤Cq‖f‖Mult, where ‖·‖Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1.