For more information about this meeting, contact Thomas Barthelme, Anatole Katok, Federico Rodriguez Hertz, Dmitri Burago.

Title: | A variational bound on the mean ergodic theorem in a uniformly convex Banach space |

Seminar: | Dynamical systems seminar |

Speaker: | Jason Rute, Penn State |

Abstract: |

I will present joint work with Jeremy Avigad (preprint: http://arxiv.org/abs/1203.4124). Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have
sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p,
where the constant C depends only on p and the modulus of uniform convexity. We also show that our inequality is sharp in that it must depend on the modulus of uniform convexity. This leads to new notion of cotype for Banach spaces. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. |

### Room Reservation Information

Room Number: | MB114 |

Date: | 01 / 27 / 2014 |

Time: | 03:35pm - 04:35pm |