The ideal of $p$-compact operators: a tensor product approach
Volume 211 / 2012
Abstract
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 $/d_p$, the left injective associate of the Chevet–Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators to that of $p$-compact operators. Using the results known for the former class and appropriate hypotheses 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 show that $\mathcal K_p$ is regular, surjective, and totally accessible, and we characterize its maximal hull $\mathcal K_p^{\max}$ as the dual ideal of $p$-summing operators, $\varPi_p^{\rm dual}$. Furthermore, we prove that $\mathcal K_p$ coincides isometrically with $\mathcal {QN}_p^{\rm dual}$, the dual to the ideal of the quasi $p$-nuclear operators.
