= Radial kernel of the Kac collision operator
{title2=$\ker(I-Q)=L^2_{\rm radial}(\mathbb R^N)$}
Zero <Dirichlet form of the Kac collision operator> forces invariance under every pair rotation; strong continuity upgrades almost every angle to every angle. Coordinate-plane rotations generate the <special orthogonal group>, whose action is transitive on each sphere for $N\geq2$. Averaging over normalized <Haar measure> therefore identifies the invariant $L^2$ functions with <radial functions>. Conversely radial functions are fixed by every pair rotation.
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