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A permanence theorem for sums of sequence spaces

Author: A. K. Snyder
Journal: Proc. Amer. Math. Soc. 93 (1985), 489-492
MSC: Primary 46A45; Secondary 40H05
MathSciNet review: 774008
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Abstract: Let $ l$ be the space of absolutely summable sequences. Using difficult functional analytic techniques Bennett proved that if $ X$ is a separable FK space containing $ {\delta ^n}$ for all $ n$ and if $ {\delta ^n} \to 0$ in $ X + l$, then $ l \subset X$. Bennett also asked whether the separability assumption can be dropped. Using an elementary invertibility criterion for Banach algebras, the present note gives a self-contained proof that if $ z$ is a null sequence, $ X$ is an $ {\text{FK}}$ space containing $ {\delta ^n}$ for all $ n$, and $ X + zl = l$, then $ X = l$. This answers Bennett's question in the affirmative.

References [Enhancements On Off] (What's this?)

  • [1] G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49-75. MR 0344846 (49:9585)
  • [2] A. K. Snyder, Universal families for conull FK spaces, Trans. Amer. Math. Soc. 284 (1984), 389-399. MR 742431 (86a:46009)
  • [3] A. Wilansky, Topics in functional analysis, Lecture Notes in Math., Vol. 45, Springer-Verlag, 1967. MR 0223854 (36:6901)
  • [4] -, Summability through functional analysis, North-Holland, Amsterdam, 1984. MR 738632 (85d:40006)

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Keywords: FK space, wedge space
Article copyright: © Copyright 1985 American Mathematical Society

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