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On modulated ergodic theorems

Profile image of Tanja EisnerTanja Eisner

2017, arXiv (Cornell University)

https://doi.org/10.48550/ARXIV.1709.05322
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26 pages

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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 1 n 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 ′ (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 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. 1 n n k=1 a k T k x, for every weakly almost periodic T and x ∈ X. Some general results are

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