Matrix Transformation and Compactness on q-Catalan Sequence Spaces

Abstract

This article intends to develop q-Catalan sequence spaces ${\mathcal{l}}_p(\mathcal{C}(q))$ and ${\mathcal{l}}_{\infty }(\mathcal{C}(q))$ due to q-Catalan matrix $\mathcal{C}(q)$ in ${\mathcal{l}}_p$ and ${\mathcal{l}}_{\infty },$ respectively. Apart from obtaining some basic topological properties and Schauder basis, we compute α-, β-, and γ-duals of the spaces ${\mathcal{l}}_p(\mathcal{C}(q))$ and ${\mathcal{l}}_{\infty }(\mathcal{C}(q)).$ We state and prove a theorem that characterize certain matrix classes (X, Y), where X is either of the space ${\mathcal{l}}_p(\mathcal{C}(q))$ or ${\mathcal{l}}_{\infty }(\mathcal{C}(q))$ and $Y\in \{{\mathcal{l}}_{\infty },c_0,c,{\mathcal{l}}_1\}.$ The final section is devoted to determination of certain conditions by which a matrix operator becomes compact.

KEYWORDS:

• Compact
• duals
• matrix transformation
• operator
• q-Catalan matrix
• sequence space

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 46A45
• 46B45
• 47B07

Acknowledgments

The authors would like to thank the anonymous reviewers for making helpful comments that have improved the readability of the paper to a great extent.

Data availability statement

No data and materials are associated with this research.

Disclosure statement

No conflict of interests.

Additional information

Funding

No funding.