Rockafellar’s sum conjecture

Rockafellar’s conjecture asks whether the usual interior-domain condition guarantees that the sum of two maximal monotone operators is maximal monotone in general Banach spaces. Earlier work established important special cases, while the general nonreflexive question remained open.

Known results

• Linear maximal monotone operators under a closed domain-difference condition: maximality of the sum (2006).
• A maximal monotone linear relation plus a normal-cone operator satisfies the qualification (2009).
• A maximal monotone linear relation summed with any maximal monotone operator satisfies the qualification (2012).
• The classical theorem holds in reflexive Banach spaces (2019).

September 2026 counterexamples

A preprint claims counterexamples on $c_0$ and $\ell^1$, together with a transfer construction between these spaces, showing that the interior-domain condition does not suffice in general. This contradicts a 2015 note that claimed the full result, but the new preprint is explicitly unrefereed and its counterexamples remain unverified.

Current status (as of September 2026): The conjecture is claimed to be false by counterexamples on $c_0$ and $\ell^1$, while the counterexamples and their implications remain unverified; the reflexive-space and other special cases are settled.