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

Abstract

We study the space of p-compact operators, Kp, using the theory of tensor norms and operator ideals. We prove that Kp is associated to /dp, the left injective associate of the Chevet-Saphar tensor norm dp (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 K p(E; F) is equal to Kq(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 Kp. For instance, we show that Kp is regular, surjective, and totally accessible, and we characterize its maximal hull Kmax p as the dual ideal of p-summing operators, Πdual p . Furthermore, we prove that Kp coincides isometrically with QNdual p , the dual to the ideal of the quasi p-nuclear operators.