On operators that are almost isometric on the positive cones of $L^ p$-spaces, $1<p<+\infty$
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- by Sen Zhong Huang PDF
- Proc. Amer. Math. Soc. 106 (1989), 1039-1047 Request permission
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 $\left ( {1 - \varepsilon } \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq \left \| f \right \|$ 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 ${a_p}\left ( \varepsilon \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq {b_p}\left ( \varepsilon \right )\left \| f \right \|$ 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.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- 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