Core equality of real sequences
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915278706704384 |
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| author | Leonetti, Paolo |
| author_facet | Leonetti, Paolo |
| contents | 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 $\mathcal{I}=\text{Fin}\oplus \mathcal{P}(ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Core equality of real sequences Leonetti, Paolo Functional Analysis Classical Analysis and ODEs 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 $\mathcal{I}=\text{Fin}\oplus \mathcal{P}(ω)$. |
| title | Core equality of real sequences |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2401.01136 |