## Abstract

We prove that if the numerical range of a Hilbert space contraction T is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ||Tn(I-T)||=O(1/nβ) with β∈[1/2,1). For normal contractions the condition is also necessary. Another sufficient condition for β=12, necessary for T normal, is that the numerical range of T be in a disk (z:|z-δ|≤1-δ} for some δ∈(0, 1). As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies ||Tn(I-T)||=O(1/nβ) with β∈(0, 1] if and only if sup_{1<|λ|<2}|λ-1|^{1/β}||R(λ, T)||<∞. When T is a contraction on L_{2} satisfying the numerical range condition, it is shown that T^{n}f/n^{1-β} converges to 0 a.e. with a maximal inequality, for every f∈L_{2}. An example shows that in general a positive contraction T on L_{2} may have an f≥0 with limsupTnf/lognn=∞ a.e.

Original language | English |
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Pages (from-to) | 1196-1213 |

Number of pages | 18 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 436 |

Issue number | 2 |

DOIs | |

State | Published - 15 Apr 2016 |

## Keywords

- Contractions on L
- Numerical range
- Quasi-sectorial operators
- Rate of convergence of powers
- Resolvent estimates
- Ritt operators