Let 0 < s < ∞. In this study, we introduce the double sequence space R qt (L s) as the domain of four dimensional Riesz mean R qt in the space L s of absolutely s-summable double sequences. Furthermore, we show that R qt (L s) is a Banach space and a barrelled space for 1 ≤ s < ∞ and is not a barrelled space for 0 < s < 1. We determine the αand β(ϑ)-duals of the space L s for 0 < s ≤ 1 and β(bp)-dual of the space R qt (L s) for 1 < s < ∞, where ϑ ∈ {p, bp, r}. Finally, we characterize the classes (L s : M u), (L s : C bp), (R qt (L s) : M u) and (R qt (L s) : C bp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ≤ s < ∞ together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Maddox defined the spaceℓ(p)of the sequencesx=(xk)such that∑k=0∞‍|xk|pk<∞, in Maddox, 1967. In the present paper, the Nörlund sequence spaceNt(p)of nonabsolute type is introduced and proved that the spacesNt(p)andℓ(p)are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the spaceNt(p)are computed and the basis of the spaceNt(p)is constructed. The classes(Nt(p):μ)and(μ:Nt(p))of infinite matrices are characterized. Finally, some geometric properties of the spaceNt(p)are investigated.

Let C f 0 and C f denote the spaces of almost null and almost convergent double sequences, respectively. We show that C f 0 and C f are BDK-spaces, barreled and bornological, but they are not monotone and so not solid. Additionally, we establish that both of the spaces C f 0 and C f include the space BS of bounded double series.

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces 0 and c. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator Λ on the sequence spaces 0 and c. Finally, we present a Mercerian theorem. Since the matrix Λ is reduced to a regular matrix depending on the choice of the sequence ( ) having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.

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