TY - JOUR

T1 - On modulated ergodic theorems

AU - Eisner, Tanja

AU - Lin, Michael

N1 - Funding Information:
The authors are grateful to Guy Cohen, Christophe Cuny, M?t? Wierdl and Manfred Wolff for several helpful discussions. The second author is grateful to the University of Leipzig, where part of the research was carried out, for its support and hospitality.
Publisher Copyright:
© 2018 Journal of Nonlinear and Variational Analysis

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Averaging along the prime numbers

KW - Contractions

KW - Hartman sequences

KW - Modulated ergodic theorems

KW - Weakly almost periodic operators

UR - http://www.scopus.com/inward/record.url?scp=85080894527&partnerID=8YFLogxK

U2 - 10.23952/jnva.2.2018.2.03

DO - 10.23952/jnva.2.2018.2.03

M3 - Article

AN - SCOPUS:85080894527

VL - 2

SP - 131

EP - 154

JO - Journal of Nonlinear and Variational Analysis

JF - Journal of Nonlinear and Variational Analysis

SN - 2560-6921

IS - 2

ER -