Normattaining nuclear operators
Abstract
Given two Banach spaces $X$ and $Y$, we introduce and study a concept of normattainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_\pi Y$. We exhibit positive and negative examples where both previous normattainment hold. We also study the problem of whether the class of elements which attain their norms in $\mathcal{N}(X,Y)$ and in $X\widehat{\otimes}_\pi Y$ is dense or not. We prove that, for both concepts, the density of normattaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces $X$ and $Y$ failing the approximation property in such a way that the class of elements in $X\widehat{\otimes}_\pi Y$ which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of normattaining operators throughout the paper.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.09871
 Bibcode:
 2020arXiv200609871D
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 We have modified the old version for this one with some significant corrections in Section 5