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I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or finding details will be greatly appreciated.

The statement; a transformation $T:(X,\lambda) \rightarrow (X,\lambda)$ is weak mixing, i.e. $\frac{1}{N}\sum | \lambda(T^{-n}A \cap B) - \lambda(A)\lambda(B)|^2 \rightarrow 0$, if and only if the associated operator $U_T : L^2(X) \rightarrow L^2(X)$ has only constant eigenfunctions.

By the spectral theorem, if $\sigma$ is the spectrum of $U_T$, for any $f \in L^2(X)$ there is a measure $\mu \in \mathcal{M}(\sigma)$ for which $\int_\sigma z^n d\mu = < U_T^n f, f> $

What I'm missing; in the converse, if $U_T$ has only constant eigenvalues and $\int_X f = 0$ then the corresponding function $\mu$ is non atomic.

Thank you in advance. Diego

edited Mar 15 2012 at 2:57

asked Mar 14 2012 at 18:05

Diego
19●2

	What is your definition of weak mixing (there are several equivalent ones)? – Mark Schwarzmann Mar 14 2012 at 21:05
	Yes, actually this exercise is the last link between two of them. Considering the original $T$-invariant measure as $ \lambda \in \mathcal{M}^T (X)$ I am using $ \frac{1}{N}\sum \| \lambda(T^{-1}(A) \cap B ) - \lambda(A)\lambda(B) \|^2 \rightarrow 0$ – Diego Mar 15 2012 at 2:02
	This is not a research level question, more of a homework question. It appears in every introductory book of ergodic theory. see for example the recent Einsiedler-Ward book. – Asaf Mar 15 2012 at 8:25
	Fair enough. Although it's not in that book. – Diego Mar 15 2012 at 14:19
	See thm 2.36 in E-W book, or thm 6.1 in Petersen's book, or Halmos' book the "mixing theorem" in page 40, or thm 1.6 in Walters' book. Those are the most common and standard ref. in ergodic theory! – Asaf Mar 15 2012 at 23:28

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weak mixing and spectral theorem

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