Rockafellar 和猜想通过显式非 (FPV) 算子的失败

中文摘要 AI 辅助

基于杨伟峰的构造，我们在 $c_0(\N_0\times\N)$ 上给出了 Rockafellar 和猜想的一个简化反例。核心观察是几何性的：极大单调算子的定义域可以具有非凸的范数闭包。在我们的例子中，这为算子不是 (FPV) 型提供了简短证明，进而蕴含和猜想的失败。我们利用几何级数定义了一个显式算子，并给出了其极大单调性的详细证明。该例子是在 GPT-6 Astra 的协助下开发的。其目的是简化和解释该构造，而非声称独立的反例机制。

英文摘要

Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on $c_0(\N_0\times\N)$. The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.

发表机构

• Faculty of Mathematics, University of Vienna（维也纳大学数学学院）

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