On the Banach-Stone problem for $L^p$ spaces

• Main Authors: Wu-Lang Day, 戴武郎
• Other Authors: Ngai-Ching Wong
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/38855059175674524085

碩士 === 國立中山大學 === 應用數學研究所 === 82 === A map $T: L^p(X) \to L^q(Y)$ $( 1 \le p, q \le \infty )$ is called disjointness-preserving if $ f \cdot g = 0$ a.e. implies $Tf \cdot Tg =0$ a.e. We say that a measure space $(X,{\cal B}, \mu) $ solves the Banach-Stone problem for $L^p$-spaces if for an arbitrary measure space $ ( Y, {\cal A}, nu ) $ and $1 \le p < \infty $ , $ 1 \le q \le \infty $ or $p = q = \infty$ , every ( surjective when $ p = q = \infty $ ) disjointness-preserving bounded linear operator ( or linear isometry when $p \not = 2 , q \not = 2$) $T:L^p(X) \to L^q(Y)$ is of the Banach-Stone type, i.e. $Tf = h \cdot (f \circ \varphi) $ for all $f$ in $L^p(X)$, where $h$ is a scalar-valued measurable function defined on $Y$ and $ \varphi: Y \to X $ a measurable mapping. In this paper, a rather complete answer of this problem is obtained. In particu- lar, it is shown that every $\sigma$-finite measure space $( X, {\cal B}, \mu)$, with $X$ a Dedekind complete separable totally ordered space and $\cal B$ the order $\sigma$-algebra of $X$ , solves the Banach-Stone problem. And so do all their measurable subspaces. Our results can be applied to show that ${\cal R}^n$ , separable Hilbert spaces, separable Banach spaces , separable complete metrizable spaces, $\cdots$ , and all their measurable subspaces solve the Banach-Stone problem. This extends a famous result of Banach, which dealt with the case $1 \le p < \infty$, $q = p \not=2$ and $X=Y=[0,1]$ for linear isometries.