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I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.

Taking $n \le m$, one can show that $A \in \mathbb{R}^{m \times n}$ is an isometry from $\ell_p^n(\mathbb{R}) \to \ell_p^m(\mathbb{R})$ if and only if the columns of $A$ have unit $\ell_p$-norm and only one nonzero entry per column. This characterises the subspaces isometric to the subspace $\ell_p^n(\mathbb{R}) \subset \ell_p^m(\mathbb{R})$.

Also, for $p$ not an even integer, I believe all the isometries between subspaces of $L_p([0,1])$ have been classified, and this can likely be extended to $\ell_p^n(\mathbb{R})$.

Does anyone know any references in this direction for even $p$?

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  • $\begingroup$ I am somewhat confused by your notation. What do you mean by $\mathbb{R}^n$? Do you mean the $\ell_2$-norm on it? It would be nice if you could fix the various typos in your question. About a reference, I am not sure if this is what you are looking for but in one of my old papers I considered surjective isometries of related spaces, possibly you can deduce something for your situation from it. The paper is available at arxiv.org/pdf/math/9604216.pdf. $\endgroup$
    – Beata Randrianantoanina
    Commented Jan 20, 2018 at 6:20
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    $\begingroup$ Apologies for my mistake; I mean $\ell_p^n(\mathbb{R})$ instead of $\ell_p^n(\mathbb{R}^n)$. The $n$-dimensional real vector space equipped with the $\ell_p$-norm. $\endgroup$
    – Dave
    Commented Jan 20, 2018 at 14:08
  • $\begingroup$ I am still not sure what you mean exactly, especially in your second paragraph. Or what do you want your subspaces to be isometric to? A lot is known about isometries of $\ell_p$ and $\ell_p^n$. In particular any subspace spanned by disjointly supported vectors is isometric to $\ell_p$ of appropriate dimension. $\endgroup$
    – Beata Randrianantoanina
    Commented Jan 23, 2018 at 18:54
  • $\begingroup$ Unfortunately I cannot edit my original question, but I would like to know which subspaces of $\ell_p^n$ are isometric, for $p$ an even integer. My second paragraph describes a special case of the result you just mentioned. To explain: $\ell_p^n$ can be identified as a subspace of $\ell_p^m$, and the subspaces isometric to this subspace are given as the range of $A$, where $A$ is described in my question. What about subspaces that are not spanned by vectors with disjoint support? We can characterise all such isometries for any $p$ that is not an even integer. What is known for even $p$? $\endgroup$
    – Dave
    Commented Jan 23, 2018 at 20:05
  • $\begingroup$ If you mean how different can the subspaces be that isometric to each other, then I recommend that you look at my paper available on Arxiv math.FA/9709214, where I show an example of two subspaces of L_p, where p\ne 2 is an even integer, so that they are isometric to each other and one subspace is complemented in L_p, while the other one is not complemented. You should check if and how this can be adapted for the finite dimensional case. $\endgroup$
    – Beata Randrianantoanina
    Commented Jan 23, 2018 at 23:53

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