On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators

We establish new results on the I$\mathcal {I}$-approximation property for the Banach operator ideal I=Kup$\mathcal {I}=\mathcal {K}_{up}$ of the unconditionally p-compact operators in the case of 1 & LE;p<2$1\le p<2$. As a consequence of our results, we provide a negative answer for the case p=1$p=1$ of a problem posed by Kim. Namely, the Ku1$\mathcal {K}_{u1}$-approximation property implies neither the SK1$\mathcal {SK}_1$-approximation property nor the (classical) approximation property; and the SK1$\mathcal {SK}_1$-approximation property implies neither the Ku1$\mathcal {K}_{u1}$-approximation property nor the approximation property. Here, SKp$\mathcal {SK}_p$ denotes the p-compact operators of Sinha and Karn for p & GE;1$p\ge 1$. We also show for all 2<p,q<& INFIN;$2<p,q<\infty$ that there is a closed subspace X & SUB;lq$X\subset \ell <^>q$ that fails the SKr$\mathcal {SK}_r$-approximation property for all r & GE;p$r\ge p$.