Hereditarily Indecomposable Banach algebras of diagonal operators

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 $X^{\ast }$ 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 ${\mathfrak{X}}_D^{\ast }$ 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.

35 pages, submitted for publication to Israel J. Math

References in corpus (1)

• Strictly singular non-compact diagonal operators on HI spaces