Remarks on rates of convergence of powers of contractions

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Abstract

We prove that if the numerical range of a Hilbert space contraction T is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ||Tn(I-T)||=O(1/nβ) with β∈[1/2,1). For normal contractions the condition is also necessary. Another sufficient condition for β=12, necessary for T normal, is that the numerical range of T be in a disk (z:|z-δ|≤1-δ} for some δ∈(0, 1). As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies ||Tn(I-T)||=O(1/nβ) with β∈(0, 1] if and only if sup1<|λ|<2|λ-1|1/β||R(λ, T)||<∞. When T is a contraction on L2 satisfying the numerical range condition, it is shown that Tnf/n1-β converges to 0 a.e. with a maximal inequality, for every f∈L2. An example shows that in general a positive contraction T on L2 may have an f≥0 with limsupTnf/lognn=∞ a.e.

Original languageEnglish
Pages (from-to)1196-1213
Number of pages18
JournalJournal of Mathematical Analysis and Applications
Volume436
Issue number2
DOIs
StatePublished - 15 Apr 2016

Keywords

  • Contractions on L
  • Numerical range
  • Quasi-sectorial operators
  • Rate of convergence of powers
  • Resolvent estimates
  • Ritt operators

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