  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 by O.
I. Zhukova 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
$\|\cdot\|_{\Upsilon_{p}}$-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.
