A Banach space $X$ is ***subprojective*** if every infinite dimensional closed subspace $Y$ of $X$ contains an infinite dimensional subspace $Z$ which is complemented in $X$.
I am interested in conditions on an Orlicz function $M$ implying that the Orlicz sequence space $\ell\_M$ is subprojective.


A concrete case: $M\_a(t)=t^2|\log t|^a$ $(a>0)$. In this case, $\ell\_{M\_a}$ is hereditarily $\ell\_2$. So the question is if every infinite dimensional closed subspace of $\ell\_{M\_a}$ contains a subspace isomorphic to $\ell\_2$ and complemented in $\ell\_{M\_a}$.
Moreover, the dual space $\ell\_{M\_a}^\*$ is isomorphic to $\ell\_{M\_a^\*}$, where $M\_a^\*(t)=t^2|\log t|^{-a}$ is also an Orlicz function.


Are $\ell\_{M\_a}$ and $\ell\_{M\_a}^\*$ subprojective?



Subprojective Orlicz sequence spaces
As Bill pointed out the answer is Yes. In this particular example, the situation is even simpler: every normalized block sequence $(u\_i)$ has a subsequence equivalent to either uvb of $\ell\_M$ or to of $\ell\_p$. This is easy to see since the set $C\_{M}=\{t^p\}$. Specifically, if $\|u\_i\|\_{\infty}>0$ then the sequence is equivalent to $\ell\_M$, and otherwise to $\ell\_p$. In both cases, those sequences are complemented. See Lemma 2.5 of this [paper](https://www.ams.org/journals/proc/2014-142-05/S0002-9939-2014-11903-4/S0002-9939-2014-11903-4.pdf) for the calculation. Moreover, in the paper it is shown that this space (and a class of similar Orlicz spaces called $p$-regular) is complementably homogeneous.


