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Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous operators from $X$ to $\ell^2_2$? Is $N\!A(X,\ell^2_2)$ nontrivially nonempty, i.e., is there a surjective norm-attaining operator from $X$ to $\ell_2^2$?

The problem was posed on 29.06.2019 by Dirk Werner (Berlin) on page 137 of Volume 2 of the Lviv Scottish Book.

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  • $\begingroup$ Have a look at this paper: arxiv.org/abs/1905.08272 Some new sufficient conditions were obtained there. $\endgroup$
    – Johann Langemets
    Commented Nov 25, 2019 at 19:36
  • $\begingroup$ It seems to me that $NA(X, \ell^2_2)$ is always nontrivial: just take a norm-attaining linear functional on $X$ (using Hahn-Banach) and multiply it by a nonzero vector in $\ell^2_2$. $\endgroup$
    – Ruy
    Commented Nov 30, 2019 at 2:05
  • $\begingroup$ Checking the original question in the Lviv book I see that the existence part refers to ONTO maps. Therefore my previous example does not stand. $\endgroup$
    – Ruy
    Commented Nov 30, 2019 at 12:52
  • $\begingroup$ @Ruy Thank you for your comment. I corrected "to" to "onto" in the formulation of the question. $\endgroup$
    – Lviv Scottish Book
    Commented Dec 4, 2019 at 8:23
  • $\begingroup$ My comment referred to the last occurrence of "to", rather than the first. It is my impression that the density question still regards all (not necessarily onto) norm attaining operators. $\endgroup$
    – Ruy
    Commented Dec 4, 2019 at 12:27

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