Surjective isometries on rearrangement-invariant spaces

Details Surjective isometries on rearrangement-invariant spaces Surjective isometries on rearrangement-invariant spaces

1992-11-05

arxiv.org/abs/math/9211208v1

Mathematics

Functional Analysis

Scientific paper

We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel function $a$ and an invertible Borel map $\sigma:[0,1] \to [0,1].$ If $X$ is not equal to $L_p$, up to renorming, for some $1\le p\le \infty$ then in addition $|a|=1$ a.e. and $\sigma$ must be measure-preserving.

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