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.