The ideal of p-compact operators: a tensor product approach

We study the space of $p$-compact operators ${\mathcal{K}}_p$, using the theory of tensor norms and operator ideals. We prove that ${\mathcal{K}}_p$ is associated to $\mathrm{/}{d}_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p^{\mathrm{\prime }}}^{\mathrm{\prime }}$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that ${\mathcal{K}}_p(E;F)$ is equal to ${\mathcal{K}}_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We also exhibit several structural properties of ${\mathcal{K}}_p$. For instance, we obtain that ${\mathcal{K}}_p$ is regular, surjective, totally accessible and characterize its maximal hull ${\mathcal{K}}_p^{max}$ as the dual ideal of the $p$-summing operators, ${\mathrm{\Pi }}_p^{dual}$. Furthermore, we prove that ${\mathcal{K}}_p$ coincides isometrically with ${\mathcal{Q}\mathcal{N}}_p^{dual}$, the dual ideal of the quasi $p$-nuclear operators.