= Bounded Koopman operator criterion
For a <nonsingular transformation> $F$ of a <probability space>, let $h=d(F_*\mu)/d\mu$ 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 then
$$
\|K_F\|^2=\|h\|_\infty.
$$
Indeed, <change of variables> gives $\|K_Fg\|_2^2=\int|g|^2h\,d\mu\leq\|h\|_\infty\|g\|_2^2$. Conversely, if the squared operator norm is at most $C$, test $g=\mathbf1_A$ to obtain $\int_Ah\,d\mu\leq C\mu(A)$ for every measurable $A$. The sets $\{h>C+\epsilon\}$ must therefore be null, proving $h\leq C$ 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.
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