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Some summability invariants

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Abstract

For summability matrices A, Macphail has introduced a condition (E′): for sequences t∈ℓ and x in the summability field of A we have t(Ax)=(tA) x, whenever (tA) x exists. He showed that, if the A-limit is the nullfunctional, the condition (E′) is equivalent to the invariance of the equation\(I = \wedge ^ \bot \). We prove that the condition (E′) invariance of the equation\(I = \wedge ^ \bot \) also in the more general case of non-μ-unique matrices. Moreover, we give some partial solutions to the open question in [3] whether the invariance of\( \wedge ^ \bot \) implies that of I.

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Authors and Affiliations

  1. Fachbereich Mathematik, Fernuniversität-Gesamthochschule, Postfach 940, 5800, Hagen

    Wolfgang Beekmann

  2. Department of Mathematics, Brock University, L2S 3A1, St. Catharines, Ontario, Canada

    Shao-Chien Chang

Authors
  1. Wolfgang Beekmann
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  2. Shao-Chien Chang
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The paper was prepared while the second author was on DAAD grant

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Beekmann, W., Chang, SC. Some summability invariants. Manuscripta Math 31, 363–378 (1980). https://doi.org/10.1007/BF02320700

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