A lower bound for the differences of powers of linear operators

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Abstract

Let T be a bounded linear operator in a Banach space, with σ(T) = {1}. In 1983, Esterle–Berkani’ s conjecture was proposed for the decay of differences (I − T) T n as follows: Either

lim inf n →∞ (n + 1)∥ (I − T) T n ∥ ≥ 1/e

or T = I. We prove this claim and discuss some of its consequences.
Original languageEnglish
Pages (from-to)745-748
JournalACTA MATHEMATICA SINICA : ENGLISH SERIES
Volume23
Issue number4
DOIs
Publication statusPublished - 2007
MoE publication typeA1 Journal article-refereed

Keywords

  • Esterle–Berkani’s conjecture
  • Quasi–nilpotent linear operator
  • Differences of powers
  • Decay

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