On modulated ergodic theorems

Abstract

Let T be a weakly almost periodic (WAP) linear operator on a Banach space X.A sequence of scalarsx 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 (n) := log n1 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 1 n ∑ 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 T n < ∞ on L r (Ω, µ) (1 < r < ∞) and f ∈ L r , the averages along the primes converge.

Locations

• Journal of Nonlinear and Variational Analysis - View - PDF
• arXiv (Cornell University) - View - PDF