Matrix Transformation and Compactness on q-Catalan Sequence Spaces

Abstract

This article intends to develop q-Catalan sequence spaces l p ( C ( q ) ) and l ∞ ( C ( q ) ) due to q-Catalan matrix C ( q ) in l p and l ∞ , respectively. Apart from obtaining some basic topological properties and Schauder basis, we compute α-, β-, and γ-duals of the spaces l p ( C ( q ) ) and l ∞ ( C ( q ) ) . We state and prove a theorem that characterize certain matrix classes (X, Y), where X is either of the space l p ( C ( q ) ) or l ∞ ( C ( q ) ) and Y ∈ { l ∞ , c 0 , c , l 1 } . The final section is devoted to determination of certain conditions by which a matrix operator becomes compact.