Spectral characterization of weak mixing

Statement

For an invertible probability-preserving transformation $T$ , the following are equivalent:

$T$ is weakly mixing.
Its reduced Koopman operator has no nonzero eigenvectors.
Its reduced Koopman representation has no nonzero finite-dimensional invariant subspaces.
Every spectral measure of a mean-zero observable is atomless.

For a countable discrete group action, weak mixing is likewise equivalent to the absence of nonzero finite-dimensional invariant subspaces of the reduced Koopman representation. For a nonabelian group, checking only invariant lines is insufficient.

Mechanism

If $Uf=\lambda f$ , then $f(x)\overline{f(y)}$ is invariant for $T\times T$ . Conversely, the Hilbert tensor-product description of the product representation turns an invariant vector on the square into a compact intertwining operator; a nonzero eigenspace of its positive square yields a finite-dimensional invariant subspace. For one unitary operator, such a subspace has an eigenvector.

The equivalence with atomless spectral measures follows from the spectral projection at each singleton. The correlation formulation follows from Wiener's Fourier criterion for atoms.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §§1.7 and 2.2.2.