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In this study, we give some new F K-spaces by means of an innite matrix as an operator and dene some new β-and γ-type duality of sequence spaces [4, 6]. We also introduce some new sections and investigate some properties like AB-, F AK-, SAK-and AKin an F K-space. By this way, we obtain some new distinguished subspaces of an F K-space [7]. Among other results, we prove that the sum of nite numbers of F K-spaces and the intersection of a sequence of F K-spaces which have these new properties with corresponding paranorms have also these new properties. The reader can refer to [2] and [19] for the main results and related topics in F K-space theory.

In the present paper, we give a new characterization of unconditional convergent series and give some new versions of the Orlicz-Pettis theorem via FQ ?-family and a natural family F with the separation property S1 through wRp-summability which may be considered as a generalization of the well-known strong p-Ces?ro summability. Among other results, we obtain a new (weak) compactness criteria for the summing operator.