Injective Tauberian operators on $L_1$ and operators with dense range on $\ell_\infty$

TL;DR

The paper constructs examples of injective Tauberian operators on $L_1(0,1)$ and operators on $\,ell_ty$ with dense, non-closed range, revealing new properties of these operators and subspaces.

Contribution

It demonstrates the existence of injective, non surjective operators with dense range on classical Banach spaces, and identifies non complementary subspaces of $\,ell_ty$ that are isometric to $\,ell_ty$.

Findings

Existence of injective Tauberian operators on $L_1(0,1)$ with dense, non closed range

Construction of injective, non surjective operators on $\,ell_ty$ with dense range

Identification of non complementary subspaces of $\,ell_ty$ isometric to $\,ell_ty$

Abstract

There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, non complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$.