On the New Generalized Hahn Sequence Space h d p

Abstract

In this article, we define the new generalized Hahn sequence space $h_d^p$ , where $d={\left(d_k\right)}_{k=1}^{\infty }$ is monotonically increasing sequence with $d_k\ne 0$ for all $k\in \mathbb{N}$ , and $1<p<\infty$ . Then, we prove some topological properties and calculate the $\alpha -$ , $\beta -$ , and $\gamma -$ duals of $h_d^p$ . Furthermore, we characterize the new matrix classes $\left(h_d,\lambda \right)$ , where $\lambda =\left\{bv,b{v}_p,b{v}_{\infty },bs,cs,\right\}$ , and $\left(\mu ,h_d\right)$ , where $\mu =\left\{bv,b{v}_0,bs,c{s}_0,cs\right\}$ . In the last section, we prove the necessary and sufficient conditions of the matrix transformations from $h_d^p$ into $\lambda =\left\{{\ell }_{\infty },c,c_0,{\ell }_1,h_d,bv,bs,cs\right\}$ , and from $\mu =\left\{{\ell }_1,b{v}_0,bs,c{s}_0\right\}$ into $h_d^p$ .