Let S be a non-empty set, X be a topological measure space, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\alpha _{t,1}\}_{t\in S}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mrow> <mi>t</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, ..., <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\alpha _{t,m}\}_{t\in S}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be some families of d-aperiodic bijections from X onto X, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$\{w_{t,l}\}_{t\in S,l\in {1,\dots m}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>w</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mi>l</mi> <mo>∈</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mi>m</mi> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> be a system of weights on X. In this paper, among other results, we give some equivalent condition for the families <Equation ID="Equ17"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_Equ17.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="231" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned}\{T_{\alpha _{t,1}, w_{t,1}}\}_{t\in S},\ldots , \{T_{\alpha _{t,m}, w_{t,m}}\}_{t\in S}\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mrow> <msub> <mi>α</mi> <mrow> <mi>t</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow> <mi>t</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mrow> <msub> <mi>α</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>of the weighted translation operators on a Banach function space <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Y</mi> </math></EquationSource> </InlineEquation> on X, to be d<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-transitive, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7693_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> is a Furstenberg family.

D$\mathcal {F}$-TRANSITIVITY OF SOME FAMILY OF OPERATORS ON BANACH FUNCTION SPACES

期刊论文
发表日期：

2025年4月21日

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Alireza Bagheri Salec,
Sarah Nahed Abdul Abbas,
Seyyed Mohammad Tabatabaie

摘要

Let S be a non-empty set, X be a topological measure space, $\{\alpha _{t,1}\}_{t\in S}$ ${α_{t, 1}}_{t \in S}$ , ..., $\{\alpha _{t,m}\}_{t\in S}$ ${α_{t, m}}_{t \in S}$ be some families of d-aperiodic bijections from X onto X, and $\{w_{t,l}\}_{t\in S,l\in {1,\dots m}}$ ${w_{t, l}}_{t \in S, l \in 1, \dots m}$ be a system of weights on X. In this paper, among other results, we give some equivalent condition for the families $\begin{aligned}\{T_{\alpha _{t,1}, w_{t,1}}\}_{t\in S},\ldots , \{T_{\alpha _{t,m}, w_{t,m}}\}_{t\in S}\end{aligned}$ $\begin{matrix} {T_{α_{t, 1}, w_{t, 1}}}_{t \in S}, \dots, {T_{α_{t, m}, w_{t, m}}}_{t \in S} \end{matrix}$ of the weighted translation operators on a Banach function space $\mathcal {Y}$ $Y$ on X, to be d $\mathcal {F}$ $F$ -transitive, where $\mathcal {F}$ $F$ is a Furstenberg family.

D\(\mathcal {F}\)-TRANSITIVITY OF SOME FAMILY OF OPERATORS ON BANACH FUNCTION SPACES

摘要