## Abstract

Let

For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of ∑

For

For

*T*be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series ∑^{∞}k=_{0}β_{k}T^{k}x when {β_{k}} is a Kaluza sequence with divergent sum such that*β*_{k}→0 and ∑^{∞}k=_{0}β_{k}z^{k}converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also sup_{n}∥∑^{n}_{k=0}β_{k}T^{k}x∥<∞ is equivalent to the convergence of the series. The last assertion is proved also when*T*is a mean ergodic contraction of*L*_{1}.For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of ∑

^{∞}_{n=0}β_{n}*T*^{n}*x*, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary.For

*T*Dunford–Schwartz of a σ-finite measure space or a positive contraction of Lp, 1<p<∞, we prove that when {β_{k}} is also completely monotone (i.e. a Hausdorff moment sequence) and*β*=_{k}*O*(1/k), the norm convergence of ∑^{∞}_{k=0}*β*_{k}*T*^{k}ƒimplies a.e. convergence.For

*T*a positive contraction of*L*_{p},*p*>1,*ƒ*∈*L*_{p}and*β*∈R, we show that if the series ∑^{∞}_{n=0}(log(n+1))^{β }/(n+1)^{1−1}/r*T*^{n}*ƒ*converges in*L*_{p}-norm for some r∈(p/p−1,∞], then it converges a.e.Original language | English |
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Title of host publication | Études opératorielles |

Editors | Jaroslav Zemánek, Yuri Tomilov |

Publisher | Banach Center Publications |

Pages | 53-86 |

Volume | 112 |

ISBN (Print) | 978-83-86806-36-2 |

DOIs | |

State | Published - 2017 |