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Proceedings of the American Mathematical Society

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

 

 

On operators that are almost isometric on the positive cones of $ L\sp p$-spaces, $ 1<p<+\infty$


Author: Sen Zhong Huang
Journal: Proc. Amer. Math. Soc. 106 (1989), 1039-1047
MSC: Primary 47B38; Secondary 47B55
DOI: https://doi.org/10.1090/S0002-9939-1989-0975658-4
MathSciNet review: 975658
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Abstract: A linear operator $ T:{L^p}\left( \Omega \right) \to {L^p}\left( {{\Omega _1}} \right)$ is said to be almost isometric on the positive cone (a.i.p.c.) with distortion coefficient $ \varepsilon $ if there is an $ \varepsilon ,0 \leq \varepsilon \leq 1$, such that $ \left( {1 - \varepsilon } \right)\left\Vert f \right\Vert \leq \left\Vert {Tf} \right\Vert \leq \left\Vert f \right\Vert$ for all nonnegative functions $ f$. We prove that if $ p \in [2,\infty )$, then there are continuous functions $ {a_p}\left( \cdot \right)$ and $ {b_p}\left( \cdot \right)$ defined on $ [0,1)$, with $ {a_p}\left( 0 \right) = {b_p}\left( 0 \right) = 1$, so that if $ T:{L^p}\left( \Omega \right) \to {L^p}\left( {{\Omega _1}} \right)$ is an a.i.p.c. operator with distortion coefficient $ \varepsilon $, then $ {a_p}\left( \varepsilon \right)\left\Vert f \right\Vert \leq \left\Vert {Tf} \right\Vert \leq {b_p}\left( \varepsilon \right)\left\Vert f \right\Vert$ for all $ f \in {L^p}\left( \Omega \right)$. We also prove that such functions $ {a_p}\left( \cdot \right)$ and $ {b_p}\left( \cdot \right)$ exist for $ p$ in the range $ \left( {1,2} \right)$ if, in addition, $ T$ is either positive or has dense range.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0975658-4
Keywords: Almost isometry, positive operator, modulus of convexity
Article copyright: © Copyright 1989 American Mathematical Society