In the present study, we construct a new Banach series space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left| \tilde{C}_{\phi }\right| _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close="|" open="|"> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">~</mo> </mover> <mi>ϕ</mi> </msub> </mfenced> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> by using the concept of absolute Catalan summability <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left| \tilde{C},\phi _{n}\right| _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close="|" open="|"> <mover accent="true"> <mi>C</mi> <mo stretchy="false">~</mo> </mover> <mo>,</mo> <msub> <mi>ϕ</mi> <mi>n</mi> </msub> </mfenced> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> which is derived by the infinite regular matrix of the fascinating Catalan numbers. We establish that this space is linearly isomorphic to the classical space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we determine the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$,\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>- duals of this space and construct Schauder basis for the series space <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\left| \tilde{C}_{\phi }\right| _{p}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="|" open="|"> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">~</mo> </mover> <mi>ϕ</mi> </msub> </mfenced> <mi>p</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The study concludes with a complete characterization of the matrix transformation classes <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\left( \left| \tilde{C}_{\phi }\right| _{p},X\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mfenced close="|" open="|"> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">~</mo> </mover> <mi>ϕ</mi> </msub> </mfenced> <mi>p</mi> </msub> <mo>,</mo> <mi>X</mi> </mfenced> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\left( X,\left| \tilde{C}_{\phi }\right| _{p}\right) ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mi>X</mi> <mo>,</mo> <msub> <mfenced close="|" open="|"> <msub> <mover accent="true"> <mi>C</mi> <mo stretchy="false">~</mo> </mover> <mi>ϕ</mi> </msub> </mfenced> <mi>p</mi> </msub> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where X is any of classical sequence spaces <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\ell _{\infty },$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> c,  <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$c_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\ell _{1}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Characterizations of matrix operators on the absolute Catalan series space

期刊论文
发表日期：

2026年7月1日

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Özlem Girgin Atlıhan,
Canan Hazar Güleç

摘要

In the present study, we construct a new Banach series space $\left| \tilde{C}_{\phi }\right| _{p}$ ${|{\tilde{C}}_{ϕ}|}_{p}$ by using the concept of absolute Catalan summability $\left| \tilde{C},\phi _{n}\right| _{p}$ ${|\tilde{C}, ϕ_{n}|}_{p}$ which is derived by the infinite regular matrix of the fascinating Catalan numbers. We establish that this space is linearly isomorphic to the classical space $\ell _{p}$ $ℓ_{p}$ for $p\ge 1$ $p \geq 1$ . Moreover, we determine the $\alpha $ $α$ - $,\beta $ $, β$ - and $\gamma $ $γ$ - duals of this space and construct Schauder basis for the series space $\left| \tilde{C}_{\phi }\right| _{p}.$ ${|{\tilde{C}}_{ϕ}|}_{p} .$ The study concludes with a complete characterization of the matrix transformation classes $\left( \left| \tilde{C}_{\phi }\right| _{p},X\right) $ $({|{\tilde{C}}_{ϕ}|}_{p}, X)$ and $\left( X,\left| \tilde{C}_{\phi }\right| _{p}\right) ,$ $(X, {|{\tilde{C}}_{ϕ}|}_{p}),$ where X is any of classical sequence spaces $\ell _{\infty },$ $ℓ_{\infty},$ c, $c_{0}$ $c_{0}$ and $\ell _{1}.$ $ℓ_{1} .$