Operator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operator

Abstract

In this study, we introduce the vector valued multiplier spaces Mf∞(∑kTk) and Mwf∞(∑kTk) by means of almost summability and weak almost summability, and a series of bounded linear operators. Since these multiplier spaces are equipped with the sup norm and are subspaces of ℓ∞(X), we obtain the completeness of a normed space via the multiplier spaces which are complete for every c0(X)-multiplier Cauchy series. We also characterize the continuity and (weakly) compactness of the summing operator S from the multiplier spaces Mf∞(∑kTk) or Mwf∞(∑kTk) to an arbitrary normed space Y through c0(X)-multiplier Cauchy and ℓ∞(X)-multiplier convergent series, respectively. Finally, we show that if ∑kTk is ℓ∞(X)-multiplier Cauchy, then the multiplier spaces of almost convergence and weak almost convergence are identical. These results are more general than the corresponding consequences given by Swartz [20], and are analogues given by Altay and Kama [6].