Characterizations of unconditionally convergent and weakly unconditionally Cauchy series via $w_p^R$-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 $\sigma$-family and a natural family $\mathcal{F}$ with the separation property $S_1$ through $w_p^R$-summability which may be considered as a generalization of the well-known strong $p$-Ces\`{a}ro summability. Among other results, we obtain a new (weak) compactness criteria for the summing operator.