Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.
Is $\ell_\phi$ isomorphic to $\ell_2$?
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2$\begingroup$ I would try to prove that if $1<p<2$ and $q$ is the conjugate index to $p$, then there is an Orlicz sequence space $X$ that is complementably universal for all Orlicz sequence spaces that are $p$ convex and $q$ concave. Such an $X$ would be isomorphic to $X^*$. $\endgroup$– Bill JohnsonAug 9, 2021 at 17:11
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3$\begingroup$ Why the vote to close? $\endgroup$– Bill JohnsonAug 9, 2021 at 17:13
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2$\begingroup$ Bill's suggestion indeed works. Lindenstrauss-Tzafriri notes this in the book. See the construction in Theorem 4.b.12 and the remark after. $\endgroup$– Bunyamin SariAug 9, 2021 at 19:10
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1$\begingroup$ Is there a Banach space $X$ with a symmetric basis $(e_n, e_n^*)$, non-equivalent to an orthonormal basis for a Hilbert space, s.t. the map $e_n\mapsto e_n^*$ extends to an isomorphism from $X$ onto $X^*$? $\endgroup$– Bill JohnsonAug 11, 2021 at 0:44
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1$\begingroup$ @Bunyamin: But the $x_n^*$ are not a priori biorthogonal to $x_n$. You don't know that $\langle x_n^*, x_n \rangle =\sum_{I\in F_n} a_n^2$ is bounded away from zero unless you assume what you are trying to prove. $\endgroup$– Bill JohnsonAug 11, 2021 at 15:45
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1 Answer
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For a given $1<p$ and $\frac{1}{p}+\frac{1}{q}=1$ you can construct a universal Orlicz sequence space $\ell_M$ so that every Orlicz function $N$ in between $p$ and $q$ is equivalent to a function in $E_M$ which corresponds to the Orlicz subspaces spanned by constant block bases of $\ell_M$ thus complemented. Such a space is unique up to isomorphism (depending only to $p$), and $E_{M^*}$ has the same property so $\ell_{M^*}$ is isomorphic to $\ell_{M}$ by the uniqueness. This construction is given in [LT, Theorem 4.b.12]. See also the remark after.
This answer was first given by Bill Johnson in comments, I just added the reference.
Lindenstrauss, Joram; Tzafriri, Lior, Classical Banach spaces. 1: Sequence spaces. 2. Function spaces., Classics in Mathematics. Berlin: Springer-Verlag. xx, 432 p. (1996). ZBL0852.46015.
