弱极大化性质、紧扰动与对偶性
The weak maximizing property, compact perturbations and duality
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- Universitat de València（瓦伦西亚大学）
- Universidad Torcuato Di Tella（托尔夸托·迪泰拉大学）
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)（国家科学研究委员会）
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中文总结 AI 辅助
构造一族与\u2113_2同构的自反Banach空间，证明弱极大化性质不对偶稳定，且紧扰动性质不蕴含弱极大化性质。
中文摘要 AI 辅助
我们构造了一个实自反Banach空间族$(X_c)_{0<c<1}$，该空间族与$\u2113_2$同构，并满足$d_{\mathrm{BM}}(X_c,\u2113_2)\le c^{-1}$，使得$(X_c,X_c)$不具有弱极大化性质，而$(X_c^*,X_c^*)$具有该性质。$X_c$不满足弱极大化性质由一对角算子所证实，该算子不达到其范数，但存在非弱零的极大化序列。在对偶侧，分离块估计在对偶化后产生反向毕达哥拉斯不等式和Opial性质的定量形式；结合性质$(M)$，这给出了$X_c^*$的弱极大化性质。我们还证明了紧扰动性质对于自反对是自对偶的。由于弱极大化性质蕴含紧扰动性质，每个$(X_c,X_c)$都具有紧扰动性质。因此，紧扰动性质即使在任意接近希尔伯特空间的情况下也不蕴含弱极大化性质，并且弱极大化性质在自反情形下不对偶稳定。
英文摘要
We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator which does not attain its norm but admits a non-weakly null maximizing sequence. On the dual side, a separated-block estimate yields, after dualization, a reverse Pythagorean inequality and a quantitative form of the Opial property; together with property $(M)$, this gives the weak maximizing property for $X_c^*$. We also show that the compact perturbation property is self-dual for reflexive pairs. Since the weak maximizing property implies the compact perturbation property, each $(X_c,X_c)$ has the compact perturbation property. Consequently, the compact perturbation property does not imply the weak maximizing property even arbitrarily close to Hilbert space, and the weak maximizing property is not stable under duality in the reflexive setting.
