Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-18T01:12:57.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of a class of infinite matrices with applications

Published online by Cambridge University Press:  17 April 2009

P. N. Natarajan
P. N. Natarajan
Affiliation:
Department of Mathematics, Vivekananda College, Madras – 600 004, India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, K denotes a complete, non-trivially valued, non-archimedean field. The class (lα, lα) of infininite matrices transforming sequences over K in lα to sequences in lα is characterized. Further a Mercerian theorem is proved in the context of the Banach algebra (lα, lα), α ≥ 1 and finally a Steinhaus type result is proved for the space lα. In the case of ℝ or ℀, on the other hand, the best known result so far seems to be a characterization of positive matrix transformations of the class (lα, lβ), ∞ > α ≥ β > 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 2 , October 1986 , pp. 161 - 175
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Bachman, G., Introduction to p-adic numbers and valuation theory, Academic Press, 1964.Google Scholar
[2]Fridy, J. A., “A note on absolute summability”, Proc. Amer. Math. Soc. 20 (1969), 285–286.Google Scholar
[3]Fridy, J. A., “Properties of absolute summability matrices”, Proc. Amer. Math. Soc. 24 (1970), 583–585.Google Scholar
[4]Knopp, K., Lorentz, G. G., “Beiträge zur absoluten Limitierung”, Arch. Math. 2 (1949), 10–16.Google Scholar
[5]Koskela, M., “A characterization of non-negative matrix operators on lp to lq with ∞ > p ≥ q > 1”, Pacific J. Math. 75 (1978), 165–169.CrossRefGoogle Scholar
[6]Moddox, I. J., Elements of Functional Analysis, Cambridge, 1977.Google Scholar
[7]Mears, F. M., “Absolute regularity and the Nörlund mean”, Ann. of Math. 38 (1937), 594–601.Google Scholar
[8]Natarajan, P. N., “The Steinhaus theorem for Teoplitz matrices in non-archimedean fields”, Comment. Math. Prace Mat. 20 (1978), 417–422.Google Scholar
[9]Rangachari, M. S., Srinivasan, V. K., “Matrix transformations in non-archimedean fields”, Indag. Math. 26 (1964), 422–429.Google Scholar
[10]Schur, I., “Über lineare Transformationen in der Theorie der unendlichen Reihen”, J. Reine Angew. Math. 151 (1921), 79–111.Google Scholar
[11]Steiglitz, M., Tietz, H., “Matrix transformationen von Folgenräumen eine Ergebnisübersicht”, Math. Z. 154 (1977), 1–16.Google Scholar