By ()U,V, we denote the set of all sequences ϵ=()ϵn such that Σϵnan is summable V whenever Σan is summable U, where U and V are two summability methods. Recently, Sarıgöl has characterized the set ()||C,αk,||trueN‾,pn for k  >  1,α  >  −1 and arbitrary positive sequences ()pn. Now, in the present paper, we characterize the sets ()||C,−1k,||trueN‾,pn, k  >  1 and ()||trueN‾,pn,||C,−1k, k  ≥  1 for arbitrary positive sequences ()pn. Hence we extend these results to the range α≥ − 1. In this way, some open problems in this topic are also completed.

This paper deals with new series space | | (∇) introduced by using Cesàro means and difference operator. It is shown that this newly defined space | | (∇) is a-space and has Schauder basis. Furthermore, the , , and-duals of | | (∇) are computed and the characterizations of classes of matrix mappings from | | (∇) to = {ℓ ∞ , , 0 } are also given.

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By $\left( X,Y\right) ,$ we denote the set of all sequences $\epsilon =\left( \epsilon _{n}\right) $ such that $\Sigma \epsilon _{n}a_{n}$ is summable $Y$ whenever $\Sigma a_{n}$ is summable $X,$ where $X$ and $Y$ are two summability methods. In this study, we get necessary and sufficient conditions for $\epsilon \in \left( \left\vert N,q_{n},u_{n}\right\vert _{k},\left\vert \bar{N},p_{n}\right\vert \right) $ and $\epsilon \in \left( \left\vert \bar{N},p_{n}\right\vert ,\left\vert N,q_{n},u_{n}\right\vert _{k}\right) $, $k\geq 1,$ using functional analytic tecniques, where $% \left\vert \bar{N},p_{n}\right\vert $ and $\left\vert N,q_{n},u_{n}\right\vert _{k}$ are absolute weighted and N\"{o}rlund summability methods, respectively, \cite{1}, \cite{5}. Thus, in the special case, some well known results are also deduced.

Recently, many authors have focused on the studies related to sequence and series spaces. In the literature the simple and fundamental method is to construct new sequence and series spaces by means of the matrix domain of triangular matrices on the classical sequence spaces. Based on this approach, in this study, we introduce a new series space as the set of all series summable by absolute summability method , , where = denotes Euler totient matrix, = is a sequence of non-negative terms and ≥ 1. Also, we show that the series space is linearly isomorphic to the space of all-absolutely summable sequences ℓ for ≥ 1. Moreover, we determine some topological properties and , and-duals of this space and give Schauder basis for the space. Finally, we characterize the classes of the matrix operators from the space | | to the classical spaces ℓ ∞ , , 0 , ℓ 1 for 1 ≤ < ∞ and vice versa.