Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification

We construct counterexamples to Rockafellar’s sum conjecture in which two maximally monotone operators satisfy the interior-domain condition but their sum is not maximally monotone. We give one counterexample on $c_0$ and another on ${\mathrm{\ell }}^1$ with its usual norm. We establish a general construction theorem that computes the entire monotone polar of a class of graphs, gives a necessary and sufficient condition for their maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum under this condition. We verify the theorem’s hypotheses and its maximality criterion on $c_0$, thereby obtaining a counterexample to the conjecture. Furthermore, we construct a bounded linear surjection from ${\mathrm{\ell }}^1$ onto $c_0$ and use it to obtain the counterexample on ${\mathrm{\ell }}^1$.