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 $\ell\_2$?



Self-dual Orlicz sequence spaces
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](https://zbmath.org/?q=an:0852.46015).


