This is an announcement for the paper "Strictly singular operators in Tsirelson like spaces" by Spiros Argyros, Kevin Beanland and Pavlos Motakis.

Abstract: For each $n \in \mathbb{N}$ a Banach space $\mathfrak{X}_{0,1}^n$ is constructed is having the property that every normalized weakly null sequence generates either a $c_0$ or $\ell_1$ spreading models and every infinite dimensional subspace has weakly null sequences generating both $c_0$ and $\ell_1$ spreading models. The space $\mathfrak{X}_{0,1}^n$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{0,1}^n$, for every $S_1,S_2,\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_1S_2\cdots S_{n+1}$ is compact. Moreover, for every subspace $Y$ as above, there exist $S_1,S_2,\ldots,S_n$ strictly singular operators on $Y$, such that the operator $S_1S_2\cdots S_n$ is non-compact.

Archive classification: math.FA

Remarks: 45 pages

Submitted from: kbeanland@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1309.4358

or

http://arXiv.org/abs/1309.4358