Operators representable as multiplication-conditional expectation operators

In this paper a unified approach is presented to the study of some classes of operators, such as kernel operators and partial integral operators, between ideals of measurable functions. In particular it is shown that if the underlying measure spaces are non-atomic, then the kernel operators and partial integral operators are mutually disjoint, and these operators are disjoint to all weighted composition operators. Moreover, if the ideals of measurable functions are Banach function spaces satisfying appropriate conditions on the norms, then the partial integral operators are disjoint to all positive compact operators.