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Matthew Tarbard

Publications and source records attributed to Matthew Tarbard.

2 recordsLinked to original sources
arXiv2013

Operators on Banach Spaces of Bourgain-Delbaen Type

We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $δ> 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $δ$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $\ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $\ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $\ell_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant $\ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.

math.FA↗
arXiv2010

Hereditarily indecomposable, separable L_\infty spaces with \ell_1 dual having few operators, but not very few operators

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 $\sum_{i=0}^{k-1} λ_i S^i +K$ where the $λ_i$ 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").

math.FA↗