Orhan Tug | Searcharxiv

arXiv2021

On the Domain of Four-Dimensional Forward Difference Matrix in Some Double Sequence Spaces

In this paper, we introduce some new double sequence spaces $\mathcal{M}_u(Δ)$ and $\mathcal{C}_{\vartheta}(Δ)$, where $\vartheta\in\{bp,bp0,r,r0\}$ as the domains of the four-dimensional forward difference matrix in the double sequence spaces $\mathcal{M}_u$ and $\mathcal{C}_{\vartheta}$, respectively. Then we investigate some topological and algebraic properties. Moreover, we determine the $α-$, $β(\vartheta)-$, and $γ-$duals of the new spaces $\mathcal{M}_u(Δ)$ and $\mathcal{C}_{\vartheta}(Δ)$. Finally, we characterize four-dimensional matrix classes $(λ(Δ),μ)$ and $(μ,λ(Δ))$, where $λ=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$ and $μ=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$.

math.FA↗

arXiv2019

On the Generalized Difference Matrix Domain on Strongly Almost Convergent Double Sequence Spaces

Most recently, some new double sequence spaces $B(\mathcal{M}_{u})$, $B(\mathcal{C}_{\vartheta})$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $B(\mathcal{L}_{q})$ for $0<q<\infty$ have been introduced as four-dimensional generalized difference matrix $B(r,s,t,u)$ domain on the double sequence spaces $\mathcal{M}_{u}$, $\mathcal{C}_{\vartheta}$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $\mathcal{L}_{q}$ for $0<q<\infty$, and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tuğ and Başar and Tuğ (see \cite{OT,OT2,Orhan,Orhan 2}). In this present paper, we introduce new strongly almost null and strongly almost convergent double sequence spaces $B[\mathcal{C}_f]$ and $B[\mathcal{C}_{f_0}]$ as domain of four-dimensional generalized difference matrix $B(r,s,t,u)$ in the spaces $[\mathcal{C}_f]$ and $[\mathcal{C}_{f_0}]$, respectively. Firstly, we prove that the new double sequence spaces $B[\mathcal{C}_f]$ and $B[\mathcal{C}_{f_0}]$ are Banach spaces with its norm. Then, we give some inclusion relations including newly defined strongly almost convergent double sequence spaces. Moreover, we calculate the $α-$dual, $β(bp)-$dual and $γ-$dual of the space $B[\mathcal{C}_f]$. Finally, we characterize new four-dimensional matrix classes $([\mathcal{C}_{f}];\mathcal{C}_{f})$, $([\mathcal{C}_{f}];\mathcal{M}_{u})$, $(B[\mathcal{C}_{f}];\mathcal{C}_{f})$, $(B[\mathcal{C}_{f}];\mathcal{M}_{u})$ and we complete this work with some significant results.

math.FA↗