The Hahn sequence space generated by the Cesàro mean of order m

Abstract

Hahn (Math Phys 32:3–88, 1922) defined the sequence space h. The main purpose of this study is to introduce the new Hahn sequence space \(h(C_{m})\) as the domain of Cesàro mean of order m and give some topological properties of the space \(h(C_{m})\). Moreover, we determine the alpha-, beta- and gamma-duals of the space \(h(C_{m})\) and characterize the classes \((\ell _1:h)\), \((h:\ell _p)\), \((h(C_m):V_{1})\) and \((V_{2}:h(C_{m}))\) of matrix transformations, where \(1<p<\infty \), \(V_{1}\in \{\ell _{\infty },c,c_{0},\ell _p\}\) and \(V_{2}\) is any given sequence space. Finally, we compute the norm of the operators belonging to \({\mathcal {B}}(\ell _1,h(C_m))\) and determine the Hausdorff measure of noncompactness of the operators in \({\mathcal {B}}(\ell _1,h(C_m))\).