AI 中文总结
本文构造了Rockafellar和猜想的反例，在$c_0$和$\ell^1$上给出两个极大单调算子满足内部域条件但和非极大的例子，并建立了计算单调极的构造定理。
AI 中文摘要
我们构造了Rockafellar和猜想（sum conjecture）的反例，其中两个极大单调算子满足内部域条件，但它们的和不是极大单调的。我们在$c_0$上给出一个反例，并在$\ell^1$（赋予其通常范数）上给出另一个反例。我们建立了一个一般构造定理，该定理计算一类图的整个单调极（monotone polar），给出其极大单调性的充分必要条件，并表明在该条件下，一个正的一阶扰动如何产生非极大和。我们在$c_0$上验证了定理的假设及其极大性判据，从而获得了该猜想的一个反例。此外，我们构造了一个从$\ell^1$到$c_0$的有界线性满射，并利用它得到了$\ell^1$上的反例。
英文摘要
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, thereby providing the complete disproof of the conjecture. We establish a general construction theorem that computes the entire monotone polar of a class of graphs and characterizes their maximal monotonicity by the nonexistence of solutions to explicit equations in the continuous dual. We also prove a pullback theorem that transfers counterexamples through bounded linear surjections. These theorems provide a systematic mechanism for generating entire families of counterexamples and lead to further structural consequences for the resulting operators. Specifically, we obtain four classes of counterexample families: weighted constructions with different curves, first operators with prescribed affine value dimensions, second operators obtained by positive rescaling and norm-continuous monotone perturbation with full domain, and counterexamples on further Banach spaces. The last class yields counterexamples on every Banach space containing a closed subspace isomorphic to $c_0$ or $\ell^1$, or admitting such a quotient. We also give explicit constructions on $c_0$ and standard $\ell^1$ that realize the construction and pullback mechanisms, respectively. We further determine the domain geometry and exact radial bounds of the constructed operators, characterize reflexivity by a fixed rank-one test in two classical classes of Banach spaces, identify maximal monotone extensions under surjective pullback, and compute exact Fitzpatrick identities. The appendices further extend these constructions to additional parameter and product families, nonlinear scalar and strictly monotone second operators, normal-cone and subdifferential partners, and examples with prescribed radial bounds, and more counterexample families.
CommentsSubstantially revised and reorganized around the general construction theorem and pullback result, which form the backbone of the paper, integrating the explicit counterexamples and several results that predate the general construction theorem and those counterexamples into a unified framework
