Strictly singular non-compact diagonal operators on HI spaces

TL;DR

This paper constructs a Hereditarily Indecomposable Banach space with a basis that admits strictly singular, non-compact diagonal operators, and shows its diagonal operator space contains an isomorphic copy of ll_{}(\u2113).

Contribution

It introduces a new Hereditarily Indecomposable Banach space with a basis supporting strictly singular, non-compact diagonal operators and embeds ll_{}() into its diagonal operator space.

Findings

Existence of a Hereditarily Indecomposable Banach space with specified properties.

The diagonal operator space contains an isomorphic copy of ll_{}().

Construction of operators with particular spectral properties.

Abstract

We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$.