Core equality of real sequences
Paolo Leonetti
TL;DR
Given ideals $\mathcal{I},\mathcal{J}$ on $\omega$, the paper characterizes exactly when an infinite matrix $A$ preserves the $\mathcal{I}$-core of every bounded sequence under the transform to the $\mathcal{J}$-core, i.e., $\mathrm{core}_{A\bm{x}}(\mathcal{J})=\mathrm{core}_{\bm{x}}(\mathcal{I})$ for all $\bm{x}\in \ell_\infty$. The main criterion is that $A$ be $(\mathcal{I},\mathcal{J})$-regular and satisfy $\mathcal{J}\text{-}\limsup_n \sum_{k\in E}|a_{n,k}|=1$ for all $E\in \mathcal{I}^+$. The results extend the Silverman–Toeplitz framework to the case of countably generated $\mathcal{J}$, show existence of such $A$ for distinct $\mathcal{I},\mathcal{J}$, and fully characterize the Fin-case (existence iff $\mathcal{I}=\mathrm{Fin}$ or $\mathcal{I}$ is isomorphic to $\mathrm{Fin}\oplus \mathcal{P}(\omega)$). Counterexamples are provided for $\mathcal{I}=\mathcal{J}=\mathcal{Z}$, and the Rudin–Keisler ordering is used to construct witnessing matrices. These results connect core convergence notions with matrix summability methods and ideal-structure constraints.
Abstract
Given an ideal $\mathcal{I}$ on $ω$ and a bounded real sequence $\textbf{x}$, we denote by $\text{core}_{\textbf{x}}(\mathcal{I})$ the smallest interval $[a,b]$ such that $\{n \in ω: x_n \notin [a-\varepsilon,b+\varepsilon]\} \in \mathcal{I}$ for all $\varepsilon>0$ (which corresponds to the interval $[\,\liminf \textbf{x}, \limsup \textbf{x}\,]$ if $\mathcal{I}$ is the ideal $\text{Fin}$ of finite subsets of $ω$). First, we characterize all the infinite real matrices $A$ such that $$ \text{core}_{A\textbf{x}}(\mathcal{J})=\text{core}_{\textbf{x}}(\mathcal{I}) $$ for all bounded sequences $\textbf{x}$, provided that $\mathcal{J}$ is a countably generated ideal on $ω$ and $A$ maps bounded sequences into bounded sequences. Such characterization fails if both $\mathcal{I}$ and $\mathcal{J}$ are the ideal of asymptotic density zero sets. Next, we show that such equality is possible for distinct ideals $\mathcal{I}, \mathcal{J}$, answering an open question in [J.~Math.~Anal.~Appl.~\textbf{321} (2006), 515--523]. Lastly, we prove that, if $\mathcal{J}=\text{Fin}$, the above equality holds for some matrix $A$ if and only if $\mathcal{I}=\text{Fin}$ or $\mathcal{I}=\text{Fin}\oplus \mathcal{P}(ω)$.