A version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices

Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${\mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $\hat{\otimes}_{n,s,|\pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.