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


Karakuş M., Basar F.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.484, no.1, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 484 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.1016/j.jmaa.2019.123651
  • Title of Journal : JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Abstract

In this study, we introduce the vector valued multiplier spaces M-f(infinity)(Sigma T-k(k) ) and M-wf(infinity)(Sigma(k) T-k) 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(infinity) (X), we obtain the completeness of a normed space via the multiplier spaces which are complete for every c(0) (X)-multiplier Cauchy series. We also characterize the continuity and (weakly) compactness of the summing operator S from the multiplier spaces M-f(infinity)(Sigma T-k(k) ) or M-wf(infinity)(Sigma(k) T-k) to an arbitrary normed space Y through c(0) (X)-multiplier Cauchy and too (X)-multiplier convergent series, respectively. Finally, we show that if Sigma(k) T-k is l(infinity) (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]. (C) 2019 Published by Elsevier Inc.