A class of Banach spaces with few non strictly singular operators

We construct a family $({\mathcal{X}}_{\text{al}}{)}_{\text{al}\le {\omega }_1}$ of reflexive Banach spaces with long transfinite bases but with no unconditional basic sequences. In our spaces ${\mathcal{X}}_{\text{al}}$ every bounded operator $T$ is split into its diagonal part $D_T$ and its strictly singular part $S_T$.