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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strong convergence of functions on Köthe spaces

Author: Gerald Silverman
Journal: Trans. Amer. Math. Soc. 165 (1972), 27-35
MSC: Primary 46A45; Secondary 46E30
MathSciNet review: 0291790
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Abstract: Let $ \Lambda $ be a rearrangement invariant Köthe space over a nondiscrete group G with Haar measure $ \mu $. For a function $ f \in \Lambda $ and relatively compact 0-neighborhood U in G the function

$\displaystyle {T_U}f(x) = \frac{1}{{\mu (U)}} \cdot \int_{U + x} {f\,d\mu } $

is continuous and also belongs to $ \Lambda $. The convergence $ {T_U}f \to f$ (as $ U \to 0$) for the strong Köthe topology on $ \Lambda $ is involved in establishing compactness criteria for subsets of a Köthe space. The main result of this paper is a necessary and sufficient condition for convergence $ {T_U}f \to f$ in the strong topology on $ \Lambda $.

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Keywords: Köthe spaces, monotonic rearrangements, equimeasurability, nonatomic measure, strong Köthe topology
Article copyright: © Copyright 1972 American Mathematical Society

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