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Characterization of a class of infinite matrices with applications

  • P. N. Natarajan (a1)

Abstract

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.

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References

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[1]Bachman, G., Introduction to p-adic numbers and valuation theory, Academic Press, 1964.
[2]Fridy, J. A., “A note on absolute summability”, Proc. Amer. Math. Soc. 20 (1969), 285286.
[3]Fridy, J. A., “Properties of absolute summability matrices”, Proc. Amer. Math. Soc. 24 (1970), 583585.
[4]Knopp, K., Lorentz, G. G., “Beiträge zur absoluten Limitierung”, Arch. Math. 2 (1949), 1016.
[5]Koskela, M., “A characterization of non-negative matrix operators on lp to lq with ∞ > pq > 1”, Pacific J. Math. 75 (1978), 165169.
[6]Moddox, I. J., Elements of Functional Analysis, Cambridge, 1977.
[7]Mears, F. M., “Absolute regularity and the Nörlund mean”, Ann. of Math. 38 (1937), 594601.
[8]Natarajan, P. N., “The Steinhaus theorem for Teoplitz matrices in non-archimedean fields”, Comment. Math. Prace Mat. 20 (1978), 417422.
[9]Rangachari, M. S., Srinivasan, V. K., “Matrix transformations in non-archimedean fields”, Indag. Math. 26 (1964), 422429.
[10]Schur, I., “Über lineare Transformationen in der Theorie der unendlichen Reihen”, J. Reine Angew. Math. 151 (1921), 79111.
[11]Steiglitz, M., Tietz, H., “Matrix transformationen von Folgenräumen eine Ergebnisübersicht”, Math. Z. 154 (1977), 116.
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