Decomposable operators acting between distinct \(L^p\)-direct integrals of Banach spaces

The notion of decomposable operators acting between distinct \(L^p\) $L^p$ -direct integrals of Banach spaces is introduced. We show that these operators generalize the composition operator in the sense that a binary relation replaces a mapping. The necessary and sufficient conditions for the boundedness of those operators are the main results of the paper.