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 1 ∑n 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)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 pkx. We then prove that for any contraction T on a Hilbert space H and x ∈ H, and also for every invertible T with supn∈Z kTnk < ∞ on Lr(Ω, µ) (1 < r < ∞) and f ∈ Lr, the averages along the primes converge.
- Averaging along the prime numbers
- Hartman sequences
- Modulated ergodic theorems
- Weakly almost periodic operators
ASJC Scopus subject areas
- Applied Mathematics