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If $T(w)=w$, the defining equation becomes $M(w)+F(w)=M(w)$, so $\boxed{F(w)=0}$.

A map is firmly nonexpansive in the $M$-inner product whenPut $p=T(v)$ and $q=T(w)$. The two implicit equations giveBecause $F$ is a monotone operator,Thereforewhich is precisely firm nonexpansiveness. In particular, the preconditioned proximal point algorithm map $T=(M+F{)}^{-1}M$ is nonexpansive in the norm induced by the positive-definite matrix $M$.

Write $w_k=(x_k,z_k)$ and $w_{k+1}=(x_{k+1},z_{k+1})$. ExpandinggivesThe off-diagonal terms in the first equation cancel. By the optimality condition for the proximal operator,The second equation then becomes the explicit linear updateThus each step of this preconditioned proximal point algorithm uses only one evaluation of the proximal operator of ${\alpha }^{-1}f$, together with applications of the linear map $A$ and its transpose $A^T$.