 Solution

Let ${\hat{\Omega }}^{(k)}$ solve the $k$th diagonal-block problem and setIts inverse is block diagonal. On each diagonal block, the Graphical-Lasso Karush-Kuhn-Tucker conditions hold by the definition of ${\hat{\Omega }}^{(k)}$. On the off-diagonal blocks chooseThe assumed inequalities $|S_{ij}|\le \lambda$ ensure that every entry lies in $[-1,1]$, exactly the allowed subgradient at a zero entry of $\tilde{\Omega }$.

Thus $-{\tilde{\Omega }}^{-1}+S+\lambda Z=0$ on every block. The KKT conditions and the fact that the objective is strictly convex prove that $\tilde{\Omega }=\hat{\Omega }$, giving the claimed block decomposition.