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 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 484 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.1016/j.jmaa.2019.123651
  • Journal Name: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, INSPEC, MathSciNet, zbMATH

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.