Core equality of real sequences

Abstract

Given an ideal $\mathcal{I}$ on $\ddot{I}‰$ and a bounded real sequence $\text{x}$, we denote by ${\text{core}}_{\text{x}}(\mathcal{I})$ the smallest interval $[a,b]$ such that $\{n\in \ddot{I}‰:x_n\notin [a-\epsilon ,b+\epsilon ]\}\in \mathcal{I}$ for all $\epsilon >0$ (which corresponds to the interval $[\mathrm{liminf}\text{x},\mathrm{limsup}\text{x}]$ if $\mathcal{I}$ is the ideal $\text{Fin}$ of finite subsets of $\ddot{I}‰$). First, we characterize all the infinite real matrices $A$ such that $${\text{core}}_{A\text{x}}(\mathcal{J})={\text{core}}_{\text{x}}(\mathcal{I})$$ for all bounded sequences $\text{x}$, provided that $\mathcal{J}$ is a countably generated ideal on $\ddot{I}‰$ 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 $\mathcal{I}=\text{Fin}\oplus \mathcal{P}(\ddot{I}‰)$.

To appear on Fundamenta Mathematicae