Decompositions for weakly compact operators on the space of totally measurable functions (Q441359)

Let \(X, Y\) be real Banach spaces and let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a nonempty set. Let \(B(\Sigma, X)\) denote the Banach space of totally \(\Sigma\)-measurable \(X\)-valued functions on the nonempty set. The following Yosida-Hewitt type decomposition for a weakly compact operator (w.c.) \(T : B(\Sigma,X) \rightarrow Y\) is proved: Theorem 3.2. \(T\) can be decomposed as \(T_1 + T_2,\) where \(T_1, T_2\) are w.c. operators, \(T_1\) is \(\sigma\)-smooth and \(T_2\) is purely non \(\sigma\)-smooth. Moreover, for each \(y^{\ast} \in Y^{\ast}\) and \(A \in \Sigma\), we have \(\| y^{\ast} \circ T_A\|=\| y^{\ast} \circ (T_1)_A\|+\| y^{\ast} \circ (T_2)_A\|\); and, for each \(y^{\ast} \in Y^{\ast}\) and \(\epsilon > 0\), there exists \(A \in \Sigma\) such that \(\| y^{\ast} \circ (T_1)_{\Omega \setminus A}\|+\| y^{\ast} \circ (T_2)_A\| \leq \epsilon.\)NEWLINENEWLINEAs an application, a Yosida-Hewitt type decomposition for operator-valued measures is obtained.