On the extensions of homogeneous polynomials

Abstract

We investigate the problem of the uniqueness of the extension of n-homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space X that admits contractive projection of finite rank of at least dimension 2, for every n > 3 there exists an n-homogeneous polynomial on X that has infinitely many extensions to X**. We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice E, for instance when E satisfies an upper p-estimate with constant one for some p > 2, any 2-homogeneous polynomial on E attaining its norm at x ∈ E with a finite rank band projection P x , has a unique extension to its bidual E**. We apply these results in a class of Orlicz sequence spaces.