Weighted Hilbert structure of velocity-reset relaxation ($H=L^2(M^{-1}dv),\Pi f=\left(\int f\right)M$)

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.

For the normalized velocity-reset collision operator on the weighted Hilbert space, $L=\Pi -I$ vanishes on the equilibrium direction and equals $-I$ on its orthogonal complement. The spectral gap is exactly one. Conservation of the projected component and the displayed energy identity give exact exponential decay of the remaining component; the full solution need not decay to zero.