Resolvent conditions and growth of powers of operators

Guy Cohen, Christophe Cuny, Tanja Eisner, Michael Lin

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1 Scopus citations

Abstract

Following Bermúdez et al. [5], we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Cesàro boundedness assumptions. In Hilbert spaces, we prove that if T satisfies the Kreiss condition, ‖Tn‖=O(n/log⁡n); if T is absolutely Cesàro bounded, ‖Tn‖=O(n1/2−ε) for some ε>0 (which depends on T); if T is strongly Kreiss bounded, then ‖Tn‖=O((log⁡n)κ) for some κ>0. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order α converge strongly when α>1.

Original languageEnglish
Article number124035
JournalJournal of Mathematical Analysis and Applications
Volume487
Issue number2
DOIs
StatePublished - 15 Jul 2020

Keywords

  • Abel boundedness
  • Cesàro boundedness
  • Kreiss resolvent condition
  • Mean ergodicity
  • Power-boundedness

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