Core equality of real sequences
Abstract
Given an ideal $\mathcal {I}$ on $\omega $ and a bounded real sequence $\boldsymbol x$, we denote by $\mathrm {core}_{\boldsymbol x}(\mathcal {I})$ the smallest interval $[a,b]$ such that $\{n \in \omega : x_n \notin [a-\varepsilon ,b+\varepsilon ]\} \in \mathcal {I}$ for all $\varepsilon \gt 0$ (which corresponds to the interval $[\liminf \boldsymbol x, \limsup \boldsymbol x]$ if $\mathcal {I}$ is the ideal $\mathrm{Fin}$ of finite subsets of $\omega $).
First, we characterize all the infinite real matrices $A$ such that $$ \mathrm{core}_{A\boldsymbol x}(\mathcal {J})=\mathrm{core}_{\boldsymbol x}(\mathcal {I}) $$ for all bounded sequences $\boldsymbol x$, provided that $\mathcal {J}$ is a countably generated ideal on $\omega $ 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. {321} (2006), 515–523]. Lastly, we prove that if $\mathcal {J}=\mathrm{Fin}$, the above equality holds for some matrix $A$ if and only if $\mathcal {I}=\mathrm{Fin}$ or $\mathcal {I}$ is an isomorphic copy of $\mathrm{Fin}\oplus \mathcal {P}(\omega )$ on $\omega $.
