On modulated ergodic theorems

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_kT^kx 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 Λ⁰(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^kx. 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 kTⁿk < ∞ on L^r(Ω, µ) (1 < r < ∞) and f ∈ L^r, the averages along the primes converge.