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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Operators on Banach spaces taking compact sets inside ranges of vector measures


Author: Cándido Piñeiro
Journal: Proc. Amer. Math. Soc. 116 (1992), 1031-1040
MSC: Primary 47B99; Secondary 28B05, 46G10
DOI: https://doi.org/10.1090/S0002-9939-1992-1110552-X
MathSciNet review: 1110552
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Abstract: Let $ X$ and $ Y$ be real Banach spaces. We prove that an operator $ T$ from $ X$ into $ Y$ maps compact subsets of $ X$ into subsets of $ Y$ that lie inside ranges of $ Y$-valued measures if and only if its dual operator $ {T^ * }$ factors through a subspace of an $ {L^1}(\mu )$-space. In fact, we prove that every compact is taken into a subset of a compact range. We also prove that $ {T^ * }$ is $ 1$-summing if and only if $ T$ maps compact subsets into subsets of $ Y$ lying inside ranges of vector measures with bounded variation.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1110552-X
Article copyright: © Copyright 1992 American Mathematical Society