π-性质、一致凸性与一致球覆盖性质
$π$-Properties, Uniformly Convexity and Uniform Ball Coverings Properties
AI总结：
研究包含有限秩算子的算子空间闭子空间的一致球覆盖性质，利用可分一致凸Banach空间的\(\pi_\lambda\)-性质及收缩估计构造覆盖，得出值域为特定空间的闭算子子空间有此性质的结论。
AI中文摘要：
我们证明了包含有限秩算子的算子空间的闭子空间具有一致球覆盖性质的一个充分准则。设F是一个可分一致凸Banach空间，且设\(\Lambda_F>1\)是由其凸性模确定的常数。若对于某个\(1\leq \lambda < \Lambda_F\)，F具有\(\pi_\lambda\)-性质，那么对于每个对偶可分的Banach空间E，\(\mathcal{B}(E,F)\)中每个包含\(\mathcal{F}(E,F)\)的闭子空间都具有UBCP。证明使用了一致凸空间上近距有限秩投影的收缩估计。我们用此估计为相应的算子空间构造一致球覆盖。作为应用，我们得到值域空间为向量值\(L_p\)-空间或可分一致凸\(\mathcal{L}_{p,C+}\)-空间的闭算子子空间的UBCP。
英文摘要：
We prove a sufficient criterion for closed subspaces of operator spaces containing the finite-rank operators to have the uniform ball-covering property. Let $F$ be a separable uniformly convex Banach space, and let $Λ_F>1$ be a constant determined by its modulus of convexity. If $F$ has the $π_λ$-property for some $1\leq λ< Λ_F$, then for every Banach space $E$ with separable dual, every closed subspace of $\mathcal{B}(E,F)$ containing $\mathcal{F}(E,F)$ has the UBCP. The proof uses a contraction estimate for near-metric finite-rank projections on uniformly convex spaces. We use this estimate to construct uniform ball coverings for the corresponding operator spaces. As applications, we obtain the UBCP for closed operator subspaces whose range spaces are vector-valued $L_p$-spaces, or separable uniformly convex $\mathcal{L}_{p,C+}$-spaces.
