2026/07/30 by Manwook Han, Sun Kwang Kim, Miguel Martín +1
Mathematics · #math.FA #msc:46B04 #msc:46B20 #msc:46B25 #msc:46B28 #msc:46G25
22 pages, no figures
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We show that nuclear norm-attaining operators (resp. polynomials) are always w^*-dense in the space of integral operators (resp. polynomials). Besides, the denseness is in norm if the predual space does not contain any isomorphic copy of ℓ1. We also show that there are reflexive spaces for which the set of projective norm-attaining elements does not coincide with the whole projective tensor product (which is indeed also reflexive here). Next, we show that if Y is a II-polyhedral space, then every nuclear operator from an arbitrary space X to Y^* attains its nuclear norm. As a consequence, if X^* or Y^* has the approximation property, then the set of norm-attaining operators from X^* to Y** is dense. The argument extends to the multilinear and polynomial settings. Finally, we study proximinality results of a natural subspace of the projective tensor product and obtain an application to integral projective norm-attaining tensors which solves a proposed open question.