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

Abstract

In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}{(}^{N}X,Y^{\ast })$, $X{\hat{\otimes }}_{\pi }\cdots {\hat{\otimes }}_{\pi }X$ and ${\hat{\otimes }}_{{\pi }_s,N}X$. Among other results, we characterize when the norms of the spaces $\mathcal{P}{(}^N{\ell }_p,{\ell }_q),\mathcal{P}{(}^N{l}_{M_1},l_{M_2})$, and $\mathcal{P}{(}^{N}d(w,p),l_{M_2})$ 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 on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X{\hat{\otimes }}_{\pi }\cdots {\hat{\otimes }}_{\pi }X$ and ${\hat{\otimes }}_{{\pi }_s,N}X$, respectively. Specifically, we prove that ${\hat{\otimes }}_{{\pi }_s,N}{\ell }_2$ and ${\ell }_2{\hat{\otimes }}_{\pi }\cdots {\hat{\otimes }}_{\pi }{\ell }_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0{\hat{\otimes }}_{{\pi }_s}{c}_0$ and $c_0{\hat{\otimes }}_{\pi }{c}_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case.

38 pages