Integral representation of linear operators on Orlicz-Bochner spaces (Q634825)

In this paper, \(X, Y\) are Banach spaces and \(\mathcal{L}(X, Y)\) is the space of all continuous linear operators from \(X\) to \(Y\). \( (\Omega, \Sigma, \mu) \) is a \(\sigma\)-finite measure space and \(L^{0}(\mu, X)\) the space of \(X\)-valued, strongly \(\mu\)-measurable functions with the complete metrizable topology \( \tau_{0}\) of convergence in measure for all \(A \in \Sigma\) with \(\mu(A) < \infty\); \(\mathcal{S}(\Sigma, X)\) denotes the set of all \(\Sigma\)-simple \(X\)-valued functions on \(\Omega\). \(\varphi: [0, \infty) \to [0, \infty]\) is a left continuous convex function vanishing and continuous at \(0\) such that \(\varphi(t) / t \to \infty\) as \(t \to \infty\) (this \(\varphi\) is called a Young function). For a Young function \(\varphi\), \(\varphi^{\ast}\) denotes the complementary Young function. \(L^{\varphi}(\mu, X) = \{ f \in L^{0}(\mu, X): \int \varphi( \lambda \| f(\omega) \|_{X})\, d \mu < \infty \text{ for some } \lambda > 0 \} \) is the Orlicz-Bochner space with the norm topology \(\tau_{\varphi}\), \(\| f \|_{\varphi} = \inf \{ \lambda > 0: \int (\varphi( \| f(\omega) \|_{X} / \lambda)\, d \mu \leq 1 \} \). \(\gamma_{\varphi}\) is the mixed topology \(\gamma[\tau_{\varphi}, (\tau_{0})_{| L^{\varphi}(\mu, X)}]\) on \(L^{\varphi}(\mu, X)\). For a finitely additive measure \(m: \Omega \to \mathcal{L}(X, Y)\), with \(m(A)=0\) whenever \(\mu(A)=0\), its \(\varphi^{\ast}\)-semivariation \(\tilde{m}_{\varphi^{\ast}}(A)\) is defined as \(\tilde{m}_{\varphi^{\ast}}(A)= \sup \|a_{i} m(A_{i})(x_{i}) \|_{Y}\), where the sup is taken on all finite disjoint sequences \( \{A_{i} \} \subset \Omega \), \(\{ x_{i} \}\) in the unit ball of \(X\) and \(\{ a_{i} \}, a_{i} \geq 0\), with \(\sum \varphi (a_{i}) \mu (A_{i}) \leq 1\). \(\text{fasv}_{\varphi^{\ast}, \mu}(\Sigma, \mathcal{L}(X, Y))\) denotes the set of all such measures with \(\tilde{m}_{\varphi^{\ast}}(\Omega) < \infty\). An \(m \in \text{fasv}_{\varphi^{\ast}, \mu}(\Sigma, \mathcal{L}(X, Y))\) is called \(\varphi^{\ast}\)-variationally \(\mu\)-continuous if \(\tilde{m}_{\varphi^{\ast}}(A_{n}) \to 0\) whenever \(\mu((A_{n} \cap A) \to 0\) for any sequence \( \{ A_{n} \} \subset \Sigma, \; A \in \Sigma\) with \(A_{n} \downarrow\) and \(\mu(A) < \infty\). The main result proved is: A linear operator \(T: L^{\varphi}(\mu, X) \to Y\) has a unique integral representation with respect to an \(m \in \text{fasv}_{\varphi^{\ast}, \mu}(\Sigma, \mathcal{L}(X, Y))\), which is also \(\varphi^{\ast}\)-variationally \(\mu\)-continuous, iff it is \((\gamma_{\varphi}, \| . \|_{Y})\)-continuous. As a consequence, some corollaries and some Vitali-Hahn-Saks type theorems for families of operator measures are obtained.