Let \((\Omega ,\Sigma ,\mu )\) be a finite measure space and \((X,\Vert \cdot \Vert _x)\) and \((Y,\Vert \cdot \Vert _Y)\) be Banach spaces. A linear operator \(T:L^\infty (\mu ,X)\rightarrow Y\) is said to be dominated if there exists a positive function v in \(L^1(\mu )\) so that \(\Vert T(f)\Vert _Y\le \int _\Omega \Vert f(\omega )\Vert _X v(\omega )\,d\mu \) for every \(f\in L^\infty (\mu ,X)\) . It is shown that a bounded linear operator \(T:L^\infty (\mu ,X)\rightarrow Y\) is dominated if and only if its representing measure \(m_T\) is \(\mu \) -absolutely continuous and has the finite variation \(|m_T|\) . Moreover, the relationship between dominated operators \(T:L^\infty (\mu ,X)\rightarrow Y\) and their associated operators \(T^\#:L^\infty (\mu )\rightarrow \mathcal {L}(X,Y)\) is established.
