New Sequence Spaces Generated via a Matrix Defined by Schröder and Catalan Numbers

This study introduces the sequence spaces c(S⁾ and c₀(S⁾, defined as the domain of the matrix S^ constructed from a hybrid structure involving Schr\"{o}der and Catalan numbers. A comprehensive investigation is conducted into the fundamental topological and structural properties of these sequence spaces, such as completeness and their relationships and embedding within classical sequence spaces. Furthermore, the dual space structures corresponding to these newly defined spaces are thoroughly characterized. In the final sections, various classes of matrix transformations and compact linear operators acting on these sequence spaces are examined, emphasizing their significance in functional analysis.