非自反性之外的Haraux函数的严格下界
Sharp Lower Bounds on the Haraux Function Beyond Reflexivity
AI总结：
该研究将自反空间上极大单调算子Haraux函数的严格下界扩展到任意实Banach空间的(NI)型极大单调算子，通过能量分解证明常数1/2仍成立，并形式化了相关层级机制。
AI中文摘要：
近期研究在自反Banach空间上的极大单调算子的Haraux函数中建立了严格的图距离下界，并提出了其向非自反性扩展的问题。我们证明，严格常数1/2对任意实Banach空间上的每个(NI)型极大单调算子以及每个正权重均成立。该结果由精确能量分解生成：图点的Haraux贡献及其加权拟密度残差是互补项，二者之和为加权平方图位移的一半。此分解为每个非空图的算子生成了缺陷校正的Haraux界，并给出了零缺陷 regime 的精确Haraux侧刻画。对于极大单调算子，(NI)型与拟密度的已知等价性将该 regime 精确识别为(NI)型，因此近似证书可在无需达到的情况下恢复严格几何。我们进一步形式化了从达到的零残差到零残差下确界再到严格界的层级，一个非自反示例将其前两个层级分离，且Fenchel-Young图距离结果阐明了相同机制。
英文摘要：
We prove that the sharp $\frac{1}{2}$ lower bound for the Haraux function holds for every maximally monotone operator of type~(NI) on an arbitrary real Banach space. This extends the result established in reflexive Banach spaces to arbitrary real Banach spaces. We also establish an exact decomposition at each graph point, showing that the local contribution to the Haraux function and a nonnegative residual sum to one half of the weighted squared displacement. Due to the equivalence between type~(NI) and quasidensity, this decomposition also yields the sharp bound without requiring a graph point at which the residual vanishes. Moreover, for every operator with a nonempty graph, this decomposition yields a lower bound involving the residual infimum, and for maximally monotone operators it further yields a new characterization of type~(NI) in terms of the Haraux function. Finally, on $c_0$, we give a maximally monotone operator of type~(NI) for which the residual infimum is zero at some target but is not attained. This shows that the existence of a graph point at which the residual vanishes is strictly stronger than the vanishing of the residual infimum required in our proof.
