The ideal of unconditionally $p$-compact operators

Abstract

We investigate the ideal $\mathcal K_{\rm up}$, $1 \leq p \leq \infty $, of unconditionally $p$-compact operators. We obtain the isometric identities $\mathcal K_{\rm up}=\mathcal K_{\rm up}\circ \mathcal K_{\rm up}$, $\mathcal K^{\max }_{\rm up}=\mathcal L^{\rm sur}_{p^*}$, $\mathcal K^{\min }_{\rm up}=\widehat {\otimes }_{/w_{p^*}}$ and $\mathcal K_{\rm up}=\mathcal N_{\rm up}^{\rm Qdual}$ and prove that, if $X^*$ has the approximation property or $Y$ has the $\mathcal K_{\rm up}$-approximation property, then $\mathcal K_{\rm up}(X, Y)$ is isometrically equal to $\mathcal K^{\min }_{\rm up}(X, Y)$, and the dual space $\mathcal K_{\rm up}(X, Y)^*$ is isometric to $(\mathcal L_{p}^{\rm inj})^*(X^*, Y^*)$. As a consequence, for every Banach space $X$, we obtain the isometric identities $\mathcal K_{\rm up}^{\max }(\ell _{1}(\Gamma ), X) =\mathcal L_{p^*}(\ell _{1}(\Gamma ), X)$, $\mathcal K_{\rm up}^{\min }(\ell _{1}(\Gamma ), X) =\ell _{\infty }(\Gamma )\widehat {\otimes }_{w_{p^*}} X$ and $\mathcal K_{\rm up}(\ell _{1}(\Gamma ), X)^* =\mathcal D_{p^*} (\ell _{\infty }(\Gamma ), X^*)$.