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Multiplier spaces and the summing operator for series

  • Charles Swartz EMAIL logo
Published/Copyright: August 26, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 3

Abstract

If {xj} is a sequence in a normed space X, the space of bounded multipliers for the series ∑jxj is defined to be M∞(∑xj)={{tj}∈l∞:∑j=1∞tjxjconverges} and the summing operator S:M∞∑xj→X is defined to be S({tj})=∑j=1∞tjxj. We show that if X is complete, the series ∑jxj 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
Key words: multiplier space; summing operator; (weak) compactness; series; convergence; weak sequential continuity; complete continuity

(Communicated by Werner Timmermann)

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Received: 2013-3-1
Accepted: 2013-10-16
Published Online: 2016-8-26
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0170
Keywords for this article
multiplier space; summing operator; (weak) compactness; series; convergence; weak sequential continuity; complete continuity

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