A class of Banach spaces with few non strictly singular operators
arXiv:math/0312522
Abstract
We construct a family $(\mathcal{X}_\al)_{\al\le ω_1}$ of reflexive Banach spaces with long transfinite bases but with no unconditional basic sequences. In our spaces $\mathcal{X}_\al$ every bounded operator is split into its diagonal part and its strictly singular part .
52 pages, 1 figure