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In the present paper, we introduce the absolute Fibonacci space |F u | k , give some inclusion relations and investigate topological and algebraic structure such as BK-space, α-, β-, γ-duals and Schauder basis. Further, we characterize certain matrix and compact operators on these spaces, also determine their norms and Hausdroff meausures of noncompactness.

The main purpose of this study is to introduce the absolute Lucas series spaces and to investigate their some algebraic and topological structure such as some inclusion relations, BK− to this space, duals and Schauder basis. Also, the characterizations of matrix operators related to these space with their norms are given. Finally, by using Hausdorff measure of noncompactness, the necessary and sufficient conditions for a matrix operator on them to be compact are obtained.

In this paper, determining the operator norm, we give certain characterizations of matrix transformations from the space N φ p k , the space of all series summable by the absolute weighted mean summability method, to one of the classical sequence spaces c 0 , c, l ∞ . Also, we obtain the necessary and sufficient conditions for each matrix in these classes to be compact and establish a number of estimates or identities for the Hausdorff measures of noncompactness of the matrix operators in these classes.