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Integral representation for a class of vector valued operators

Author: Lakhdar Meziani
Journal: Proc. Amer. Math. Soc. 130 (2002), 2067-2077
MSC (2000): Primary 28C05; Secondary 46G10
Published electronically: January 17, 2002
MathSciNet review: 1896043
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Abstract: Let $S$ be a compact space and let $X$, $\left\Vert \cdot \right\Vert _{X}$be a (real, for simplicity) Banach space. We consider the space $C_{X}=C\left( S,X\right) $ of all continuous $X$-valued functions on $S$, with the supremum norm $\left\Vert \cdot \right\Vert _{\infty }$.

We prove in this paper a Bochner integral representation theorem for bounded linear operators

\begin{displaymath}T:C_{X}\longrightarrow X \end{displaymath}

which satisfy the following condition:

\begin{displaymath}x^{*},y^{*}\in X^{*},f,g\in C_{X}:x^{*}\circ f=y^{*}\circ g\Longrightarrow x^{*}\circ Tf=y^{*}\circ Tg \end{displaymath}

where $X^{*}$ is the conjugate space of $X$. In the particular case where $X=\mathbb{R}$, this condition is obviously satisfied by every bounded linear operator

\begin{displaymath}T:C_{\mathbb{R}}\longrightarrow \mathbb{R} \end{displaymath}

and the result reduces to the classical Riesz representation theorem.

If the dimension of $X$ is greater than $2$, we show by a simple example that not every bounded linear $T:C_{X}\longrightarrow X$ admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

References [Enhancements On Off] (What's this?)

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Additional Information

Lakhdar Meziani
Affiliation: Department of Mathematics, Faculty of Science, University of Batna, Algeria

Keywords: Integral representation, Riesz theorem, Bochner integral
Received by editor(s): October 5, 2000
Received by editor(s) in revised form: February 10, 2001
Published electronically: January 17, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society