Generalized Cesaro Vector-Valued Sequence Space Using Modulus Function

Sudhanshu Kumar, Arvind Kumar Verma

Abstract

In this paper, we introduced generalized Ces$\grave{a}$ro vector-valued sequence space $X(E,f,\Delta^m,p)$ by taking sequence $(E_k, q_k)$ of seminormed spaces, modulus function $f$, $m^{th}$-order difference operator $\Delta^m$ and bounded sequence $(p_k)$ of strictly positive real numbers. It is proved that the space $X(E,f,\Delta^m,p) $ is complete paranormed space if $(E_k, q_k)$ is  a sequence of complete seminormed spaces. Some inclusion relations on the space are obtained. By using composite function $f^v$,  space $X(E,f^v,\Delta^m,p)$ is studied for any $v\in \mathbb{N}$. A result on multiplier space of $X(E,f,\Delta^m,p)$ is also obtained, if $m=0$.