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arXiv · 0902.1646

Hereditarily Indecomposable Banach algebras of diagonal operators

Spiros A. Argyros·Irene Deliyanni·Andreas G. Tolias
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Abstract

We provide a characterization of the Banach spaces $X$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to $\mathcal{L}_{diag}({\mathfrak X}_D)$ with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every $T\in \mathcal{L}_{diag}({\mathfrak X}_D)$ is of the form $T=λI+K$, where $K$ is a compact operator.

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Spiros A. Argyros, Irene Deliyanni, Andreas G. Tolias. 2009-02-10. Hereditarily Indecomposable Banach algebras of diagonal operators. https://arxiv.org/abs/0902.1646

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