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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 66, Issue 3

# Multiplier spaces and the summing operator for series

Charles Swartz
Published Online: 2016-08-26 | DOI: https://doi.org/10.1515/ms-2015-0170

## Abstract

If {xj} is a sequence in a normed space X, the space of bounded multipliers for the series $\sum _{j}{x}_{j}$ is defined to be ${M}^{\mathrm{\infty }}\left(\sum {x}_{j}\right)=\left\{\left\{{t}_{j}\right\}\in {l}^{\mathrm{\infty }}:\phantom{\rule{thickmathspace}{0ex}}\sum _{j=1}^{\mathrm{\infty }}{t}_{j}{x}_{j}\phantom{\rule{thickmathspace}{0ex}}\text{converges}\right\}$ and the summing operator $S:{M}^{\mathrm{\infty }}\left(\phantom{\rule{0.056em}{0ex}}\sum {x}_{j}\right)\to X$ is defined to be $S\left(\left\{{t}_{j}\right\}\right)=\sum _{j=1}^{\mathrm{\infty }}{t}_{j}{x}_{j}$. We show that if X is complete, the series $\sum _{j}{x}_{j}$ is subseries convergent iff the operator S is compact and the series is absolutely convergent iff the operator is absolutely summing. Other related results are established.

MSC 2010: Primary 40A05; 40B05; 40F05; 40J05; 46A45

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## About the article

Accepted: 2013-10-16

Published Online: 2016-08-26

Published in Print: 2016-06-01

Citation Information: Mathematica Slovaca, Volume 66, Issue 3, Pages 687–694, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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© 2016 Mathematical Institute Slovak Academy of Sciences.