On the strong subdifferentiability of homogeneous polynomials and (symmetric) tensor products

Abstract

We study the (uniform) strong subdifferentiability of norms of Banach spaces P(N X, Y ∗) of all continuous N-homogeneous polynomials and tensor products of Banach spaces, namely X⊗b π· · · ⊗b πX and ⊗b πs,NX. Among other results, we characterize when the norms of spaces P(N'p, 'q), P(N lM1, lM2), and P(N d(w, p), lM2) are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results in [38, 48, 49] (in the spirit of Pitt's compactness theorem) on the reflexivity of spaces of N-homogeneous polynomials and N-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results by considering the subsets U and Us of elementary tensors on the unit spheres of X⊗b π · · · ⊗b πX and ⊗b πs,N X, respectively. Specifically, we prove that the norms of ⊗b πs,N '2 and '2⊗b π · · · ⊗b π'2 are uniformly strongly subdifferentiable on Us and U, and that the norms of c0⊗b πs c0 and c0⊗b πc0 are strongly subdifferentiable on Us and U in the complex case.