On the Paranormed Space of Bounded Variation Double Sequences

Abstract

In this study, as the domain of four-dimensional backward difference matrix in the space $$\mathcal {L}_u(t)$$, we introduce the complete paranormed space $$\mathcal {BV}(t)$$ of bounded variation double sequences and examine some properties of that space. Also, we determine the $$\gamma $$-dual and $$\beta (\vartheta )$$-dual of the space $$\mathcal {BV}(t)$$. Finally, we characterize the classes $$(\mathcal {BV}(t):\mathcal {M}_{u})$$, $$(\mathcal {BV}(t):\mathcal {C}_{\vartheta })$$ and $$(\mathcal {L}_u(t):\mu )$$ with $$\mu \in \{\mathcal {BS},\mathcal {CS}_{\vartheta },\mathcal {M}_{u}(\Delta ),\mathcal {C}_{\vartheta }(\Delta )\}$$, where $$\mathcal {M}_{u}(\Delta )$$ and $$\mathcal {C}_{\vartheta }(\Delta )$$ denote the spaces of all double sequences whose $$\Delta $$-transforms are in the spaces $$\mathcal {M}_{u}$$ and $$\mathcal {C}_{\vartheta }$$, respectively.