Exploring the Z-Spaces of Double Sequences of Scalars: Z2-Spaces

In this paper, we introduced and investigated the spaces Z2u,w;X of double sequences as natural extensions of the classical sequence spaces Zu,w;X. We characterized the β-duals of these spaces and obtained explicit representations in terms of multiplier spaces and difference-type conditions involving the operators Δkm+. Furthermore, we examined the relationships between weighted mean methods and the β-duals of certain special cases of Z2u,w;X. Several illustrative examples are provided to clarify the structure of the introduced spaces and to demonstrate the applicability of the obtained results. The results presented here generalize a number of known results on Z-spaces from the setting of ordinary sequences to double sequences.