Hereditarily Indecomposable Banach algebras of diagonal operators

TL;DR

This paper characterizes Banach spaces with Schauder bases where the dual is isomorphic to diagonal operators, constructs a Hereditarily Indecomposable space with this property, and shows all diagonal operators are of the form λI plus compact.

Contribution

It introduces a new class of Hereditarily Indecomposable Banach spaces with duals isomorphic to diagonal operator algebras, expanding understanding of their structure.

Findings

Dual space is isomorphic to diagonal operators for certain Banach spaces.

Constructed a Hereditarily Indecomposable Banach space with this property.

Every diagonal operator on the constructed space is a scalar multiple of the identity plus a compact operator.

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 $T=\lambda I+K$, where $K$ is a compact operator.