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Licensed Unlicensed Requires Authentication Published by De Gruyter August 26, 2016

Multiplier spaces and the summing operator for series

  • Charles Swartz EMAIL logo
From the journal Mathematica Slovaca

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=1tjxjconverges} and the summing operator S:MxjX is defined to be S({tj})=j=1tjxj. 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.


(Communicated by Werner Timmermann)


References

[1] Aizpuru, A.—Pérez-Fernandez, J.: Characterizations of series in Banach spaces, Acta Math. Univ. Comenian. 2 (1999), 337–344.Search in Google Scholar

[2] Aizpuru, A.—Pérez-Fernandez, J.: Spaces of S-bounded multiplier convergent series, Acta Math. Hungar. 87 (2000), 135–146.10.1023/A:1006781218759Search in Google Scholar

[3] Diestel, J.—Jarchow, J.—Tonge, A.: Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.10.1017/CBO9780511526138Search in Google Scholar

[4] Dunford, N.—Schwartz, J.: Linear Operators I, Interscience, New York, 1958.Search in Google Scholar

[5] Edwards, D. A.: On the continuity properties of functions satisfying a condition of Sirvant, Quart. J. Math. Oxford (2) 8 (1957), 58–67.10.1093/qmath/8.1.58Search in Google Scholar

[6] Hille, E.—Phillips, P.: Functional Analysis and Semigroups, Amer. Math. Soc., Providence, RI, 1957.Search in Google Scholar

[7] Jarchow, J.: Locally Convex Spaces, Tuebner, Stuttgart, 1981.10.1007/978-3-322-90559-8Search in Google Scholar

[8] Mohsen, A.: Weak-norm sequentially continuous operators, Math. Slovaca 50 (2000), 357–363.Search in Google Scholar

[9] Pérez-Fernandez, J.—Benítez-Trujillo, F.—Aizpuru, A.: Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series, Czechoslovak Math. J. 50(125) (2000), 889–896.10.1023/A:1022481016009Search in Google Scholar

[10] Robertson, A.: Unconditional convergence and the Vitali-Hahn-Saks theorem, Bull. Soc. Math. France Supp. Mem. 31-32 (1972), 335–341.10.24033/msmf.98Search in Google Scholar

[11] Robertson, A.: On Unconditional Convergence in Topological Vector Spaces, Proc. Roy. Soc. Edinburgh 68 (1969), 145–157.10.1017/S0080454100008335Search in Google Scholar

[12] Swartz, C.: Introduction to Functional Analysis, Marcel Dekker, New York, 1994.Search in Google Scholar

[13] Swartz, C.: Multiplier Convergent Series, World Sci. Publ., Singapore, 2009.10.1142/6977Search in Google Scholar

[14] Treves, F.: Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.Search in Google Scholar

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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