In this paper, we introduce the generalized spaces of the form ℋ(f, g, Δnm), where ℋ represents one of the spaces ℓ∞, c or c0.The köthe duals corresponding to these spaces will be computed and construction of the Schauder bases for ℋ ∈ {c, c0} will be given. Also, some matrix characterizations concerning these spaces will be computed. Moreover, the characterization of some classes of compact operators on the spaces ℓ∞(f, g, Δnm)and c0(f, g, Δnm)by employing the Hausdorff measure of non-compactness will be determined.