Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification
Mathematical paper constructs counterexamples on c0 and l1 disproving Rockafellar's conjecture that sums of maximally monotone operators remain maximally monotone.
The authors build counterexamples where two maximally monotone operators satisfy the interior-domain condition yet their sum is not maximally monotone, refuting Rockafellar's sum conjecture. One counterexample is constructed on c0 and another on l1 with its usual norm. A general construction theorem computes the monotone polar of a class of graphs, gives necessary and sufficient conditions for maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum.
- Counterexamples on c0 and l1 refute Rockafellar's sum conjecture
- Theorem computes monotone polars and maximal monotonicity criteria
- Rank-one perturbations shown to yield nonmaximal sums
Full article117 words · extracted from arxiv.org · click to collapse
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 $\ell^1$ 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 $\ell^1$ onto $c_0$ and use it to obtain the counterexample on $\ell^1$.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.10487