Strictly singular operators on the Baernstein and Schreier spaces

Every composition of two strictly singular operators is compact on the Baernstein space $B_p$ for $1<p<\mathrm{\infty }$ and on the $p$-convexified Schreier space $S_p$ for $1\le p<\mathrm{\infty }$. Furthermore, every subsymmetric basic sequence in $B_p$ (respectively, $S_p$) is equivalent to the unit vector basis for ${\mathrm{\ell }}_p$ (respectively, $c_0$), and the Banach spaces $B_p$ and $S_p$ contain block basic sequences whose closed span is not complemented.