Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 3 / b / Solution

For the normalized velocity-reset collision operator,This bounded everywhere-defined operator is self-adjoint. Put $h=f/M$. Expanding the square with the probability measure $M(v)dv$ givesIt follows thatReplacing the difference by $(f(v_{\ast })M(v)-f(v)M(v_{\ast }))/(M(v)M(v_{\ast }))$ gives the other printed integral expression. For real functions the modulus squares are ordinary squares; the modulus version also proves the complex-space statement.