### Abstract

Let {a_{k}}_{k≥0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1} ∑_{k=0}^{n} a_{k} T^{k}x for every contraction T on a Hilbert space H and every x ∈ H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers {k_{j}}, we study the problem of when n^{-1} ∑_{j=1}^{n} T^{kj}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Original language | English (US) |
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Pages (from-to) | 1653-1665 |

Number of pages | 13 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 22 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*22*(6), 1653-1665. https://doi.org/10.1017/S0143385702000846

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*Ergodic Theory and Dynamical Systems*, vol. 22, no. 6, pp. 1653-1665. https://doi.org/10.1017/S0143385702000846

**Modulated and subsequential ergodic theorems in Hilbert and Banach spaces.** / Berend, D.; Lin, M.; Rosenblatt, J.; Tempelman, Arkady.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Modulated and subsequential ergodic theorems in Hilbert and Banach spaces

AU - Berend, D.

AU - Lin, M.

AU - Rosenblatt, J.

AU - Tempelman, Arkady

PY - 2002/12/1

Y1 - 2002/12/1

N2 - Let {ak}k≥0 be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n-1 ∑k=0n ak Tkx for every contraction T on a Hilbert space H and every x ∈ H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers {kj}, we study the problem of when n-1 ∑j=1n Tkjx converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

AB - Let {ak}k≥0 be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n-1 ∑k=0n ak Tkx for every contraction T on a Hilbert space H and every x ∈ H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers {kj}, we study the problem of when n-1 ∑j=1n Tkjx converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

UR - http://www.scopus.com/inward/record.url?scp=0036897502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036897502&partnerID=8YFLogxK

U2 - 10.1017/S0143385702000846

DO - 10.1017/S0143385702000846

M3 - Article

VL - 22

SP - 1653

EP - 1665

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 6

ER -