On modulated ergodic theorems

Tanja Eisner, Michael Lin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Let T be a weakly almost periodic (WAP) linear operator on a Banach space X. A sequence of scalars (an)n≥1 modulates T on Y ⊂ X if n 1n k=1 akTkx converges in norm for every x ∈Y. We obtain a sufficient condition for (an) to modulate every WAP operator on the space of its flight vectors, a necessary and sufficient condition for (weakly) modulating every WAP operator T on the space of its (weakly) stable vectors, and sufficient conditions for modulating every contraction on a Hilbert space on the space of its weakly stable vectors. We study as an example modulation by the modified von Mangoldt function Λ0(n):= logn1P(n) (where P = (pk)k1 is the sequence of primes), and show that, as in the scalar case, convergence of the corresponding modulated averages is equivalent to convergence of the averages along the primes n 1n k=1 T pkx. We then prove that for any contraction T on a Hilbert space H and x ∈ H, and also for every invertible T with supnZ kTnk < ∞ on Lr(Ω, µ) (1 < r < ∞) and f ∈ Lr, the averages along the primes converge.

Original languageEnglish
Pages (from-to)131-154
Number of pages24
JournalJournal of Nonlinear and Variational Analysis
Issue number2
StatePublished - 1 Aug 2018


  • Averaging along the prime numbers
  • Contractions
  • Hartman sequences
  • Modulated ergodic theorems
  • Weakly almost periodic operators

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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