Integral representation for a class of vector valued operators

Abstract:

Let $S$ be a compact space and let $X$, $\left \| \cdot \right \| _{X}$ 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 $\left \| \cdot \right \| _{\infty }$. We prove in this paper a Bochner integral representation theorem for bounded linear operators \begin{equation*} T:C_{X}\longrightarrow X \end{equation*} which satisfy the following condition: \begin{equation*} x^{*},y^{*}\in X^{*},f,g\in C_{X}:x^{*}\circ f=y^{*}\circ g\Longrightarrow x^{*}\circ Tf=y^{*}\circ Tg \end{equation*} 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{equation*} T:C_{\mathbb {R}}\longrightarrow \mathbb {R} \end{equation*} 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.