| Summary: | 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.
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