On spectral measures and convergence rates in von Neumann’s Ergodic theorem

Abstract

We show that the power-law decay exponents in von Neumann’s Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value 1. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann’s Ergodic Theorem depend on sequences of time going to infinity.

Appendix A

Here, we present the proofs of Lemma 1 and Theorem 2.

1.1 Proof of Lemma 1

For each $K \in {\mathbb {N}}$ and each $z \in \partial {\mathbb {D}}{\setminus }\{1\}$, recall that

$$\begin{aligned} \displaystyle \sum _{j=0}^{K-1} z^j= & {} \frac{z^K -1}{z-1}. \end{aligned}$$

Note that, for each $\psi \in {\mathcal {H}}$, $\mu _{\psi - \psi ^*}^U (\{1\})=0$. Thus, by the Spectral Theorem, it follows that for each $\psi \in {\mathcal {H}}$ and each $K \in {\mathbb {N}}$,

$$\begin{aligned} \left\| {\frac{1}{K}} \displaystyle \sum _{j=0}^{K-1} U^j\psi - \psi ^* \right\| ^2= & {} \left\| {\frac{1}{K}} \displaystyle \sum _{j=0}^{K-1} U^j ( \psi - \psi ^*) \right\| ^2\\ {}= & {} \left\| {\frac{1}{K}} \displaystyle \sum _{j=0}^{K-1} \int _{ \partial {\mathbb {D}}} z^j dP^U(z)( \psi - \psi ^*)\right\| ^2\\ {}= & {} \left\| {\frac{1}{K}} \int _{ \partial {\mathbb {D}} } \frac{z^K -1}{z-1} dP^U(z)(\psi - \psi ^*)\right\| ^2\\ {}= & {} \frac{1}{K^2} \int _{ \partial {\mathbb {D}} } \left| \frac{z^K -1}{z-1} \right| ^2 d\mu _{\psi - \psi ^*}^U(z). \end{aligned}$$

1.2 Proof of Theorem 2

Since, for each $\psi \in {\mathcal {H}}$, ${{\,\textrm{supp}\,}}(\mu _{\psi - \psi ^*}^U) \subset \sigma (U) \subset \{1\} \cup \{e^{i\theta } \mid \theta \in (-\pi ,-\gamma ] \cup (\gamma ,\pi ] \}$ and $\mu _{\psi - \psi ^*}^U (\{1\})=0$, it follows from Lemma 1 that for each $\psi \in {\mathcal {H}}$ and each $K \in {\mathbb {N}}$,

$$\begin{aligned}\nonumber \left\| {\frac{1}{K}} \displaystyle \sum _{j=0}^{K-1} U^j\psi - \psi ^* \right\| ^2= & {} \frac{1}{K^2} \int _{ \partial {\mathbb {D}}} \left| \frac{z^K -1}{z-1} \right| ^2 d\mu _{\psi - \psi ^*}^U(z)\\ \nonumber\le & {} \frac{1}{ K^2} \left( \sup _{z \in \{e^{i\theta } \mid \theta \in (-\pi ,-\gamma ] \cup (\gamma ,\pi ] \}} \left| \frac{z^K -1}{z-1} \right| ^2 \right) \mu _{\psi - \psi ^*}^U(\partial {\mathbb {D}}) \\ \nonumber= & {} \frac{1}{ K^2} \left( \sup _{\theta \in (-\pi ,-\gamma ] \cup (\gamma ,\pi ] } \left| \frac{\sin (K\theta /2)}{\sin (\theta /2)} \right| ^2 \right) \Vert \psi - \psi ^*\Vert ^2 \\ \nonumber\le & {} \frac{1}{ (K\sin (\gamma /2))^2}\, \Vert (I - P^U({\{1\}}))\psi \Vert ^2\\\le & {} \frac{16}{\gamma ^2 K^2} \, \Vert \psi \Vert ^2, \end{aligned}$$

where we have used that $\gamma /4\le \sin (\gamma /2)$ for each $0<\gamma <\pi$. Hence, for every $K \in {\mathbb {N}}$,

$$\begin{aligned} \left\| {\frac{1}{K}} \displaystyle \sum _{j=0}^{K-1} U^j - P^U(\{1\}) \right\| \le \frac{4}{\gamma K}. \end{aligned}$$