Motivated by the recently introduced notion of quasi-rigidity we study, for a continuous linear operator T acting on an F-space X and for an appropriated Furstenberg family <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_448_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F} \subset \mathscr {P}(\mathbb {N}_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>⊂</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, when the respective set of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_448_Article_IEq2.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>-recurrent vectors <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_448_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}\text {Rec}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mtext>Rec</mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is lineable or even dense lineable. To this end we recall and generalize some basic but useful results on how homomorphisms, quasi-conjugacies and commutants behave with respect to Furstenberg families in pointwise recurrence.

Questions in linear recurrence II: Lineability properties

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2025年8月7日

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Sophie Grivaux,
Antoni López-Martínez,
Alfred Peris

摘要

Motivated by the recently introduced notion of quasi-rigidity we study, for a continuous linear operator T acting on an F-space X and for an appropriated Furstenberg family $\mathcal {F} \subset \mathscr {P}(\mathbb {N}_0)$ $F \subset P (N_{0})$ , when the respective set of $\mathcal {F}$ $F$ -recurrent vectors $\mathcal {F}\text {Rec}(T)$ $F Rec (T)$ is lineable or even dense lineable. To this end we recall and generalize some basic but useful results on how homomorphisms, quasi-conjugacies and commutants behave with respect to Furstenberg families in pointwise recurrence.