Contractivity of split Crank-Nicolson diffusion ($\Vert C_A(h)C_B(h){\Vert }_2\le 1$)

For symmetric negative semidefinite matrices $A,B$, each factor $C_A(h)=(I-hA/2{)}^{-1}(I+hA/2)$ has norm at most one, since its eigenvalues are $(1+h\lambda /2)/(1-h\lambda /2)$ with $\lambda \le 0$. The same holds for $B$, so their ordered product is a contraction for every $h\ge 0$, whether or not $A,B$ commute. This proves unconditional discrete L2 norm stability of the split Crank-Nicolson method for directional diffusion. Its splitting error remains first order in general, because the product's quadratic defect is $h^2[A,B]/2$.