The Generalized quantum difference operator in $c$ and $c_0$ and their duals

Toseef Ahmed Malik

doi:10.5644/SJM.21.01.11

Authors

Toseef Ahmed Malik Department of Mathematics, Govt. Boys Higher Secondary School, Darhal & School Education Department, Jammu and Kashmir-185135, India

DOI:

https://doi.org/10.5644/SJM.21.01.11

Keywords:

quantum sequence space, generalized quantum difference operator, Kothe duals, matrix mappings

Abstract

In this article, we present a generalization of the $k^{th}$ order of forward difference operator $\Delta^k$ using its quantum analog $\Delta^k_q$. We study its domains $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ in the spaces $c$ and $c_0$ of convergent and null sequences, respectively. Additionally, we show that the domains $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ are $BK$-spaces and linearly isomorphic to $c$ and $c_0$, respectively. Furthermore, we construct Schauder bases and examine the K\"{o}the duals of the spaces $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$. The final segment deals with the characterization of certain class of matrix mappings from the spaces $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ to the space $\nu\in\{c, c_0, \ell_\infty,~\ell_1\}$.

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