This is an announcement for the paper "A class of Banach spaces with few non strictly singular operators" by S. A. Argyros, J. Lopez-Abad, and S. Todorcevic. Abstract: We construct a family $(\mathcal{X}_\al)_{\al\le \omega_1}$ of reflexive Banach spaces with long transfinite bases but with no unconditional basic sequences. In our spaces $\mathcal{X}_\al$ every bounded operator $T$ is split into its diagonal part $D_T$ and its strictly singular part $S_T$. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 46B20; 03E05 Remarks: 52 pages, 1 figure The source file(s), om1hi.tex: 252736 bytes, om1hi1.eps: 181035 bytes, is(are) stored in gzipped form as 0312522.tar.gz with size 117kb. The corresponding postcript file has gzipped size 325kb. Submitted from: jlopez@crm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0312522 or http://arXiv.org/abs/math.FA/0312522 or by email in unzipped form by transmitting an empty message with subject line uget 0312522 or in gzipped form by using subject line get 0312522 to: math@arXiv.org.