Bounded Koopman operator criterion

For a nonsingular transformation $F$ of a probability space, let $h=d(F_{\ast }\mu )/d\mu$ be its Radon-Nikodym derivative. The Koopman operator extends to a bounded operator on $L^2(\mu )$ exactly when $h\in L^{\infty }(\mu )$, and thenIndeed, change of variables gives $\Vert K_{F}g{\Vert }_2^2=\int |g{|}^{2}hd\mu \le \Vert h{\Vert }_{\infty }\Vert g{\Vert }_2^2$. Conversely, if the squared operator norm is at most $C$, test $g=1_A$ to obtain ${\int }_{A}hd\mu \le C\mu (A)$ for every measurable $A$. The sets $\{h>C+ϵ\}$ must therefore be null, proving $h\le C$ almost everywhere and equality of the optimal constants. Nonsingularity supplies a well-defined composition on equivalence classes, while the essential bound supplies the norm estimate.