Kernels of operators on Banach spaces induced by almost disjoint families

TL;DR

This paper investigates the kernels of bounded operators on certain Banach spaces related to almost disjoint families, revealing limitations on their structure and undecidability results in set theory.

Contribution

It establishes new non-existence results for kernels of bounded operators on Banach spaces constructed from almost disjoint families, extending prior work and highlighting set-theoretic independence.

Findings

Certain subspaces cannot be kernels of bounded operators on these Banach spaces.

The specific subspace of functions with small support is not a kernel of any bounded operator.

The question of whether all operators vanishing on c₀(ω₁) have a particular form is undecidable in ZFC.

Abstract

Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$\Gamma$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(\Gamma)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$. We show that if~$\mathcal{A}$ has cardinality greater than~$\Gamma$, then the closed subspace of~$X_{\mathcal{A}}$ spanned by the indicator functions of sets of the form $\bigcap_{j=1}^{n+1}A_j$, where $n\in\N$ and $A_1,\ldots,A_{n+1}\in\mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\rightarrow \ell_{\infty}(\Gamma)$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{\infty}(\Gamma) : \text{the set}\ \{\gamma \in \Gamma : \lvert x(\gamma)\rvert > \varepsilon \}\ \text{has cardinality smaller than}\ \Gamma\ \text{for every}\ \varepsilon>0\bigr\} \] of~$\ell_\infty(\Gamma)$ is not the kernel of any bounded operator on~$\ell_\infty(\Gamma)$; this generalises results of Kalton and of Pe\l{}czy\'{n}ski and Sudakov. The situation is more complex for the Banach space~$\ell_\infty^c(\Gamma)$ of countably supported, bounded functions defined on an uncountable set~$\Gamma$. We show that it is undecidable in \textsf{ZFC} whether every bounded operator on~$\ell_\infty^c(\omega_1)$ which vanishes on~$c_0(\omega_1)$ must vanish on a subspace of the form~$\ell_\infty^c(A)$ for some uncountable subset~$A$ of~$\omega_1$.