Spectral gap of a probability-preserving action

A probability-preserving action of a discrete group $G$ has a spectral gap if there are a finite nonempty set $F\subseteq G$ and $\varepsilon>0$ such that

\max_{g\in F}\|\kappa_0(g)f-f\|_2\geq\varepsilon\|f\|_2 \qquad(f\in L^2_0(X,\mu)).

Equivalently, the reduced Koopman representation has no almost-invariant unit vectors. This condition in particular implies ergodicity.

Quantitative averaging

Enlarge $F$ to a finite symmetric set of size $m$ , and set

M=\frac1m\sum_{g\in F}\kappa_0(g),\qquad B=\tfrac12(I+M).

Then $B$ is a positive self-adjoint contraction and

\langle(I-M)f,f\rangle=\frac1{2m}\sum_{g\in F}\|\kappa_0(g)f-f\|_2^2 \geq\frac{\varepsilon^2}{2m}\|f\|_2^2.

Hence $\|B^n f\|_2\leq(1-\varepsilon^2/(4m))^n\|f\|_2$ . This gives a rate for this averaging operator, not automatically for arbitrary orbit averages.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.2.1, discussion after Theorem 2.2.8.