= Solution
For the <normalized velocity-reset collision operator>,
$$
\langle Lf,g\rangle=\rho(f)\overline{\rho(g)}-\langle f,g\rangle=\langle f,Lg\rangle.
$$
This bounded everywhere-defined operator is <self-adjoint>. Put $h=f/M$. Expanding the square with the <probability measure> $M(v)\,dv$ gives
$$
\int\!\!\int|h(v_*)-h(v)|^2M(v)M(v_*)\,dv\,dv_*
=2\|f\|_H^2-2|\rho(f)|^2.
$$
It follows that
$$
\boxed{\langle Lf,f\rangle=-\frac12\int\!\!\int|h(v_*)-h(v)|^2M(v)M(v_*)\,dv\,dv_*\leq0}.
$$
Replacing the difference by $(f(v_*)M(v)-f(v)M(v_*))/(M(v)M(v_*))$ gives the other printed <integral> expression. For real functions the modulus squares are ordinary squares; the modulus version also proves the complex-space statement.
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