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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Isometries of $ L\sp 1\cap L\sp p$


Author: Ryszard Grząślewicz
Journal: Proc. Amer. Math. Soc. 93 (1985), 493-496
MSC: Primary 46E30; Secondary 47B38
MathSciNet review: 774009
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Abstract: Denote by $ {L^1} \cap {L^p}$ the Banach space of all functions $ f$ such that $ f \in {L^1}$ and $ f \in {L^p}$ with norm $ \vert\vert\vert f\vert\vert\vert = \max (\vert\vert f\vert{\vert _1},\vert\vert f\vert{\vert _p})$. We give a characterization of isometries of $ {L^1} \cap {L^p}$ and show that $ T$ is an isometry if and only if $ T$ is of the form $ Tf = h\Phi (f)$, where $ \Phi $ is an operator generated by a regular set isomorphism and $ h$ is a suitable function.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0774009-4
PII: S 0002-9939(1985)0774009-4
Keywords: Isometry, $ {L^1} \cap {L^p}$ space
Article copyright: © Copyright 1985 American Mathematical Society