论文标题
关于与无条件P-C-CACT运算符和Sinha-Karn P-Compact运算符有关的近似属性
On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators
论文作者
论文摘要
我们在Banach操作员的$ \ Mathcal I $ -Approximation属性上建立了新的结果，理想的$ \ Mathcal i = \ Mathcal {K} _ {up} $在$ 1 \ le P <2 $的情况下，无条件$ p $ -compact Operators的$ p $ -compact Operators。由于我们的结果，我们为J.M. Kim（2017）提出的问题的案例提供了负面答案。也就是说，$ \ MATHCAL K_ {U1} $ - 近似属性都不意味着$ \ MATHCAL {SK} _1 $ -Approximation属性也不意味着（经典）近似属性； $ \ mathcal {sk} _1 $ -approximation属性都不意味着$ \ Mathcal {k} _ {u1} $ - 近似属性也不意味着近似属性。这里$ \ MATHCAL {SK} _p $表示Sinha和Karn的$ P $ -COMPACT运营商，以$ p \ ge 1 $。我们还以所有$ 2 <p，q <\ infty $显示了一个封闭的子空间$ x \ subset \ ell^q $，使所有$ r \ ge p $ able $ \ mathcal {sk} _r $ -approximation属性。
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 $r\ge p$.
