Characterizations of unconditionally convergent and weakly 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 ?-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.