发表机构
Central South University（中南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对库彻提出的每个超自反巴拿赫空间E对应的ℓ_∞(E)是否为格罗滕迪克空间的问题，通过证明ℓ_∞(E)具有佩尔钦斯基性质(V)给出肯定回答，还推导了该空间上算子的若干推论。
AI 中文摘要
库彻询问，对于每个超自反巴拿赫空间$E$，$\ell_{\infty}(E)$是否为格罗滕迪克空间。我们通过证明更强的结果——$\ell_{\infty}(E)$具有佩尔钦斯基性质$(V)$，肯定地回答了这个问题。结合$\ell_{\infty}(E)$的自然对偶表示，这得出：$\ell_{\infty}(E)$上的每个非弱紧算子都固定了一个$\ell_{\infty}$的副本。主要要素是关于零维紧豪斯多夫空间上函数模的一般性质$(V)$定理，其茎具有共同的一致凸性模。证明基于吉尔斯的泛函积分表示及相关典范测度的弱紧性。在$\ell_{\infty}(E)$的情形下，相关茎是$E$的超幂，而超自反性提供了定理所需的一致几何控制。我们还推导了$\ell_{\infty}(E)$上算子的若干推论，包括到可分空间的弱紧性、可分商空间的自反性，以及到舒尔空间的算子的紧性。
英文摘要
Kucher asked whether $\ell_{\infty}(E)$ is a Grothendieck space for every super-reflexive Banach space $E$. We answer this question affirmatively by proving the stronger result that $\ell_{\infty}(E)$ has Pełczyński's property $(V)$. Combined with the natural dual representation of $\ell_{\infty}(E)$, this yields that every non-weakly compact operator on $\ell_{\infty}(E)$ fixes a copy of $\ell_{\infty}$. The main ingredient is a general property-$(V)$ theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz's integral representation of functionals and the weak compactness of the associated canonical measures. In the case of $\ell_{\infty}(E)$, the relevant stalks are ultrapowers of $E$, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on $\ell_{\infty}(E)$, including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.
