Projective norm-attainments and their implications

Abstract

We show that nuclear norm-attaining operators (resp.\ polynomials) are always $w^{\ast }$-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 ${\ell }_1$. 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 $Y^{\ast }$ attains its nuclear norm. As a consequence, if $X^{\ast }$ or $Y^{\ast }$ has the approximation property, then the set of norm-attaining operators from $X^{\ast }$ to $Y^{\ast \ast }$ is dense. 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.

20 pages

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