Spectral measure of an observable

Let $T$ be invertible and probability preserving, and let $E$ be the spectral resolution of the forward Koopman operator $U_T$ . The spectral measure of $f\in L^2$ is the finite positive Borel measure

\sigma_f(B)=\langle E(B)f,f\rangle=\|E(B)f\|_2^2 \qquad(B\subseteq\mathbb S^1\text{ Borel}).

Its total mass is $\|f\|_2^2$ , and its Fourier coefficients are

\int_{\mathbb S^1}z^n\,d\sigma_f(z)=\langle U_T^nf,f\rangle.

Atoms and fixed vectors

An atom at $\lambda$ records a nonzero component of $f$ in the $\lambda$ -eigenspace. In particular, mass at $1$ is the squared norm of its invariant component. A unit-norm eigenfunction has spectral measure $\delta_\lambda$ .

Mixing

Weak mixing makes $\sigma_f$ atomless for every mean-zero $f$ . Strong mixing requires its nonzero-time Fourier coefficients to tend to zero, which is a stronger condition than atomlessness.