Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem
Abstract
It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space M∞R(Σk Tk) of Riesz summability which is a generalization of the Ces\`aro summability. Therefore, we give the completeness criteria of these spaces with c0(X)-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through M∞R(Σk Tk) which will be assumed that is complete for every c0(X)-multiplier Cauchy operator series. Then, we characterize the continuity and the (weakly) compactness of the summing operator S from the multiplier space M∞R(Σk Tk) to an arbitrary normed space Y through c0(X)-multiplier Cauchy and ∞(X)-multiplier convergent series, respectively. We also prove that if ΣkTk is ∞(X)-multiplier Cauchy, then the multiplier space of weakly Riesz-convergence associated to the operator valued series M∞wR(Σk Tk) is subspace of M∞R(Σk Tk). Among other results, finally, we obtain a new version of the well-known Orlicz-Pettis theorem by using Riesz-summability.