关于与无条件p-紧算子和Sinha-Karn p-紧算子相关的逼近性质
On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators
AI总结：
该研究在1≤p<2情形下推导了无条件p-紧算子理想的逼近性质新结论，否定回答了Kim的p=1相关问题，并证明ℓ^q的闭子空间可对所有r≥p失效Sinha-Karn r-紧逼近性质。
摘要：
我们针对1≤p<2的情形，建立了关于巴拿赫算子理想$\u2702I=\u2702K_{up}$（无条件p-紧算子构成的理想）的$\u2702I$-逼近性质的新结果。作为结果的推论，我们对J.M. Kim（2017）提出的问题中p=1的情形给出了否定回答：即$\u2702K_{u1}$-逼近性质既不蕴含$\u2702SK_1$-逼近性质，也不蕴含（经典的）逼近性质；而$\u2702SK_1$-逼近性质既不蕴含$\u2702K_{u1}$-逼近性质，也不蕴含逼近性质。其中$\u2702SK_p$表示p≥1时的Sinha和Karn p-紧算子。我们还证明，对所有2<p,q<∞，存在$\u2113^q$的闭子空间X，它对所有r≥p都不满足$\u2702SK_r$-逼近性质。
英文摘要：
We establish new results on the $\mathcal I$-approximation property for the Banach operator ideal $\mathcal I=\mathcal{K}_{up}$ of the unconditionally $p$-compact operators in the case of $1\le p<2$. As a consequence of our results, we provide a negative answer for the case $p=1$ of a problem posed by J.M. Kim (2017). Namely, the $\mathcal K_{u1}$-approximation property implies neither the $\mathcal{SK}_1$-approximation property nor the (classical) approximation property; and the $\mathcal{SK}_1$-approximation property implies neither the $\mathcal{K}_{u1}$-approximation property nor the approximation property. Here $\mathcal{SK}_p$ denotes the $p$-compact operators of Sinha and Karn for $p\ge 1$. We also show for all $2<p,q<\infty$ that there is a closed subspace $X\subset\ell^q$ that fails the $\mathcal{SK}_r$-approximation property for all $r\ge p$.
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