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It is well known that all surjective isometries of $l_p$ for $p\neq 2$ are the signed permutations of the unit vector basis $(e_n)$. Is there a characterization for the linear non-surjective isometries of $l_p$ for $p\neq 2$? If $(x_n)$ is a normalized block basis of $e_n$ then $e_n\to \varepsilon_n x_{\pi(n)}$ (where $\pi$ is a permutation of $\mathbb{N}$) defines an isometry. But probably not all isometries are of this form.

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    $\begingroup$ The second time you're writing "permutation" (which means bijective self-map), you probably mean "injective self-map". $\endgroup$
    – YCor
    Commented Apr 21, 2023 at 9:34
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    $\begingroup$ No, the injective self-map $n\mapsto n+1$ of $\mathbf{N}_{\ge 0}$ is not a permutation on its range $\mathbf{N}_{\ge 1}$. $\endgroup$
    – YCor
    Commented Apr 21, 2023 at 9:38
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    $\begingroup$ Do you assume linearity? (It is for free in the surjective case by the Banach-Mazur Thm, but it is not guaranteed for non surjective isometries) $\endgroup$
    – Pietro Majer
    Commented Apr 21, 2023 at 10:14
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    $\begingroup$ Concerning the obvious isometries I think you should use sequences of normalized and disjoint supported vectors instead of block sequences. $\endgroup$
    – S Argyros
    Commented Apr 21, 2023 at 10:42
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    $\begingroup$ Indeed! If $(x_n)$ are disjointly supported (not necessarily finitely support even), then one also gets isometries. $\endgroup$
    – Markus
    Commented Apr 21, 2023 at 10:52

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