Reduced Koopman representation

Given a Koopman representation $\kappa$ , its reduced Koopman representation is the restriction $\kappa_0=\kappa|_{L^2_0}$ , where

L^2_0(X,\mu)=\left\{f\in L^2(X,\mu):\int_X f\,d\mu=0\right\}, \qquad L^2=\mathbb C1\oplus L^2_0.

Both summands are invariant because a probability-preserving action fixes constants and preserves integrals.

Ergodicity and irreducibility

Ergodicity means $\kappa_0$ has no nonzero fixed vectors. It does not mean that $\kappa_0$ is irreducible: an irrational rotation has many invariant Fourier lines and is still ergodic.

Terminology

Some sources use “Koopman representation” for this restriction. In these knowls, the unqualified representation acts on all of $L^2$ , and “reduced” explicitly removes constants.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.2.