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

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.