Characterizations of Unconditionally Convergent Andweakly Unconditionally Cauchy Series Via Wrp -Summability, Orlicz-Pettis Type Theorems and Compact Summing Operator

Abstract

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 s-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`aro summability. Among other results, we obtain a new (weak) compactness criteria for the summing operator.