Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1052.47017
Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching
Biseparating linear maps between continuous vector-valued function spaces. (English)
[J] J. Aust. Math. Soc. 74, No. 1, 101-109 (2003). ISSN 1446-7887; ISSN 1446-8107/e

The object of the paper under review is to present some vector-valued Banach-Stone theorems. Let $X$, $Y$ be compact Hausdorff spaces, $E$ and $F$ two Banach spaces and $C(X,E)$ the Banach space of all continuous $E$-valued functions defined on $X$. A linear map $T: C(X,E) \to C(Y,F)$ is called separating if $\Vert f(x)\Vert \ \Vert g(x)\Vert =0$ for every $x\in E$ implies that $\Vert (Tf)(y)\Vert \ \Vert (Tg)(y)\Vert =0$ for every $y\in Y$. The authors show that every biseparating linear bijection $T$ (that is, a $T$ for which $T$ and $T^{-1}$ are separating) is a ``weighted composition operator". This means that there exists a function $h$ of $Y$ into the set of all bijective linear maps of $E$ onto $F$ and a homeomorphism $\phi$ from $Y$ onto $X$ such that $Tf(y)=h(y) (f(\phi(y))$ for every $f\in C(X,E)$ and $y\in Y$. It is also shown that $T$ is bounded if and only if for every $y\in Y$, $h(y)$ is a bounded linear operator from $E$ onto $F$. An example of an unbounded $T$ is given.
[Raymond Mortini (Metz)]

MSC 2000:: *47B33 Composition operators
47B38 Operators on function spaces
46E40 Spaces of vector-valued functions

Keywords: weighted composition operator; biseparating maps; vector-valued function spaces; Banach-Stone theorems

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]