## Abstract

Let T be a weakly almost periodic (WAP) linear operator on a Banach space X. A sequence of scalars (a_{n})_{n}≥1 modulates T on Y ⊂ X if _{n} ^{1} ∑^{n} _{k}=_{1} a_{k}T^{k}x converges in norm for every x ∈Y. We obtain a sufficient condition for (a_{n}) 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 Λ^{0}(n):= logn1_{P}(n) (where P = (p_{k})_{k}≥_{1} 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} ^{1} ∑^{n} _{k}=_{1} T p^{k}x. We then prove that for any contraction T on a Hilbert space H and x ∈ H, and also for every invertible T with sup_{n}∈_{Z} kT^{n}k < ∞ on L^{r}(Ω, µ) (1 < r < ∞) and f ∈ L^{r}, the averages along the primes converge.

Original language | English |
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Pages (from-to) | 131-154 |

Number of pages | 24 |

Journal | Journal of Nonlinear and Variational Analysis |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - 1 Aug 2018 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics