Kernels of Bounded Operators on the Classical Transfinite Banach Sequence Spaces

TL;DR

The paper investigates whether every closed subspace of classical transfinite Banach sequence spaces is the kernel of a bounded operator on the same space. Through a duality-driven framework and detailed structural analysis, it proves that for $X=\ell_p(\Gamma)$ and $X=c_0(\Gamma)$ with $1<p<\infty$, every closed subspace is the kernel of some bounded operator $X\to X$, while for uncountable $\Gamma$, $\ell_1(\Gamma)$ contains a closed subspace not realizing such a kernel. It further shows that for $c_0(\Gamma)$ this kernel-property holds universally, whereas for uncountable index sets the corresponding property fails for $\ell_1(\Gamma)$, using operator-injection obstructions. Finally, the paper demonstrates that Wark’s reflexive space $E_W$ (and its dual) also contains a closed subspace not obtainable as a kernel on itself, connecting the kernel problem to the ‘few operators’ phenomenon and highlighting a nuanced landscape across separable/non-separable and reflexive/non-reflexive spaces.

Abstract

Every closed subspace of each of the Banach spaces $X = \ell_p(Γ)$ and $X=c_0(Γ)$, where $Γ$ is a set and $1<p<\infty$, is the kernel of a bounded operator $X\to X$. On the other hand, whenever $Γ$ is an uncountable set, $\ell_1(Γ)$ contains a closed subspace that is not the kernel of any bounded operator $\ell_1(Γ)\to\ell_1(Γ)$.

Theorems & Definitions (37)

• Theorem 1.2
• Theorem 1.3
• Remark 1.5
• Lemma 2.1
• proof : Proof of Lemma \ref{['kernelreflexive']}
• Lemma 2.2
• proof
• Remark 2.3
• proof : Alternative proof of \ref{['Prop2.5d']}$\Rightarrow$\ref{['Prop2.5c']}
• Definition 2.4