SURJECTIVE ISOMETRIES ON REARRANGEMENT-INVARIANT SPACES
Copyright policy )SURJECTIVE ISOMETRIES ON REARRANGEMENT-INVARIANT SPACES
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 $Tf(s)=a(s)f(σ(s))$ for a Borel function $a$ and an invertible Borel map $σ:[0,1] \to [0,1].$ 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 $σ$ must be measure-preserving.
- University of Missouri United States
Borel map, rearrangement invariant Banach function space, Lebesgue measure, Borel function, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Functional Analysis (math.FA), Mathematics - Functional Analysis, Borel measure, FOS: Mathematics, measure preserving, 46E, surjective isometry
Borel map, rearrangement invariant Banach function space, Lebesgue measure, Borel function, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Functional Analysis (math.FA), Mathematics - Functional Analysis, Borel measure, FOS: Mathematics, measure preserving, 46E, surjective isometry
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