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Volume 2, Issue 2, 1 August 2018, Pages 131-154
Abstract. Let be a weakly almost periodic (WAP) linear operator on a Banach space . A sequence of scalars modulates on if converges in norm for every . We obtain a sufficient condition for to modulate every WAP operator on the space of its flight vectors, a necessary and sufficient condition for (weakly) modulating every WAP operator 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 (where 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 on a Hilbert space and , and also for every invertible with on () and , the averages along the primes converge.
How to Cite this Article:
Tanja Eisner, Michael Lin, On modulated ergodic theorems, J. Nonlinear Var. Anal. 2 (2018), 131-154.