In this study, novel sequence spaces are introduced as the domains of the Jordan-type matrix operator within the spaces of c and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Initially, these new spaces are defined, and their fundamental properties, including completeness and other topological characteristics, are examined. Furthermore, the existence of a Schauder basis for these spaces is established, highlighting its crucial role in their structural analysis. Next, the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-duals of the newly constructed sequence spaces are determined to provide a deeper understanding of their duality relations. Subsequently, the classes of infinite matrices that map sequences from these new spaces into classical sequence spaces, such as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell _{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, c, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, are characterized, along with the reverse transformations. Finally, attention is given to compact operators acting on these spaces, and a comprehensive characterization of their behavior is provided. Necessary and sufficient conditions for compactness are explored, and their implications in functional analysis are discussed.

ON THE JORDAN-TYPE SEQUENCE SPACES WHICH INCLUDE THE SPACES c AND $c_0$ AND COMPACT OPERATORS

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2026年5月2日

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Hacer Bilgin Ellidokuzoğlu,
Sezer Erdem,
Serkan Demiriz,
Merve İlkhan Kara

摘要

In this study, novel sequence spaces are introduced as the domains of the Jordan-type matrix operator within the spaces of c and $c_0$ $c_{0}$ . Initially, these new spaces are defined, and their fundamental properties, including completeness and other topological characteristics, are examined. Furthermore, the existence of a Schauder basis for these spaces is established, highlighting its crucial role in their structural analysis. Next, the $\alpha $ $α$ -, $\beta $ $β$ -, and $\gamma $ $γ$ -duals of the newly constructed sequence spaces are determined to provide a deeper understanding of their duality relations. Subsequently, the classes of infinite matrices that map sequences from these new spaces into classical sequence spaces, such as $\ell _p$ $ℓ_{p}$ , $\ell _{\infty }$ $ℓ_{\infty}$ , c, and $c_0$ $c_{0}$ , are characterized, along with the reverse transformations. Finally, attention is given to compact operators acting on these spaces, and a comprehensive characterization of their behavior is provided. Necessary and sufficient conditions for compactness are explored, and their implications in functional analysis are discussed.

ON THE JORDAN-TYPE SEQUENCE SPACES WHICH INCLUDE THE SPACES c AND \(c_0\) AND COMPACT OPERATORS

摘要