Positively p-nuclear operators, positively p-integral operators and approximation properties

Abstract

In the present paper, we introduce and investigate a new class of positively p-nuclear operators that are positive analogues of right p-nuclear operators. One of our main results establishes an identification of the dual space of positively p-nuclear operators with the class of positive p-majorizing operators that is a dual notion of positive p-summing operators. As applications, we prove the duality relationships between latticially p-nuclear operators introduced by O. I. Zhukova and positively p-nuclear operators. We also introduce a new concept of positively p-integral operators via positively p-nuclear operators and prove that the inclusion map from $L_{p^{*}}(\mu )$ to $L_{1}(\mu )$ ($\mu$ finite) is positively p-integral. New characterizations of latticially p-integral operators and positively p-integral operators are presented and used to prove that an operator is latticially p-integral (resp. positively p-integral) precisely when its second adjoint is. Finally, we describe the space of positively p-integral operators as the dual of the $\Vert \cdot \Vert _{\Upsilon _{p}}$-closure of the subspace of finite rank operators in the space of positive p-majorizing operators. Approximation properties, even positive approximation properties, are needed in establishing main identifications.