Our main purpose in this study is to investigate the matrix domains of the 4-dimensional Euler-totient matrix operator on the classical double sequence spaces M u , C p , C bp and C r. Besides these, we examine their topological and algebraic properties and give inclusion relations about the new spaces. Also, the α−, β(ϑ)− and γ−duals of these spaces are determined and finally, some matrix classes are characterized.

No abstract

The aim of this paper is to present the new double Binomial sequence space B r,s p which consists of all sequences whose double Binomial transforms of orders r, s (r and s are nonzero real numbers with r + s 0) are in the space L p , where 0 < p < ∞. We examine its topological and algebraic properties and inclusion relations. Furthermore, the α−, β(bp)− and γ−duals of the space B r,s p are determined and finally, some 4-dimensional matrix mapping classes related to this space are characterized.

The 4 dimensional (4d) binomial matrix and its domains on the classical double sequence spaces L p , M u , C P , C bP , C r , C f and C f 0 have been described and examined by Demiriz and Erdem in the papers [1]- [3]. In this article, we describe two double sequence spaces with the aid of the aforementioned matrix and study some properties of these. After giving inclusion relations, we compute α−, β (bp)− and γ−duals and give some new matrix classes related them.

This study includes the new Banach space ℒ ˜ s q $$\tilde {\cal L}_s^q$$ designed as the domain in 𝓛 s space of the 4d (4-dimensional) q-Cesàro matrix obtained as the q-analog of the well-known 4d Cesàro matrix. After showing the completeness of the aforementioned space, giving some inclusion relations, determining the fundamental set of this space and calculating the duals, finally, some matrix transformations related to the new space were characterized.