On modulated ergodic theorems

TL;DR

This paper investigates conditions under which sequences modulate weakly almost periodic operators on Banach spaces, including applications to prime number sequences and convergence of operator averages.

Contribution

It provides new necessary and sufficient conditions for modulation of WAP operators and extends results to averages along primes and contractions on Hilbert spaces.

Findings

Established conditions for modulation of WAP operators on Banach spaces.

Proved convergence of prime-modulated averages for contractions on Hilbert spaces.

Extended modulation results to averages along primes in various operator settings.

Abstract

Let $T$ be a weakly almost periodic (WAP) linear operator on a Banach space $X$. A sequence of scalars $(a_n)_{n\ge 1}$ {\it modulates} $T$ on $Y \subset X$ if $\frac1n\sum_{k=1}^n a_kT^k x$ converges in norm for every $x \in Y$. 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 $\Lambda'(n):=\log n1_\mathbb P(n)$ (where $\mathbb P =(p_k)_{k\ge 1}$ 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 $\frac1n\sum_{k=1}^n T^{p_k}x$. We then prove that for any contraction $T$ on a Hilbert space $H$ and $x \in H$, and also for every invertible $T$ with $\sup_{n \in \mathbb Z} \|T^n\| <\infty$ on $L^r(\Omega,\mu)$ ($1<r< \infty$) and $f \in L^r$, the averages along the primes converge.