This is an announcement for the paper "Hereditarily indecomposable, separable L_\infty spaces with \ell_1 dual having few operators, but not very few operators" by Matthew Tarbard.

Abstract: Given a natural number $k \geq 2$, we construct a hereditarily indecomposable, $\mathscr{L}_{\infty}$ space, $X_k$ with dual isomorphic to $\ell_1$. We exhibit a non-compact, strictly singular operator $S$ on $X_k$, with the property that $S^k = 0$ and $S^j (0 \leq j \leq k-1)$ is not a compact perturbation of any linear combination of $S^l, l \neq j$. Moreover, every bounded linear operator on this space has the form $\sum_{i=0}^{k-1} \lambda_i S^i +K$ where the $\lambda_i$ are scalars and $K$ is compact. In particular, this construction answers a question of Argyros and Haydon ( "A hereditarily indecomposable space that solves the scalar-plus-compact problem").

Archive classification: math.FA

Submitted from: matthew.tarbard@sjc.ox.ac.uk

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

http://front.math.ucdavis.edu/1011.4776

or

http://arXiv.org/abs/1011.4776