Partial Integral Operators on Banach–Kantorovich Spaces

In this paper, we study partial integral operators on Banach–Kantorovich spaces over a ring of measurable functions. We obtain a decomposition of the cyclic modular spectrum of a bounded modular linear operator on a Banach–Kantorovich space in the form of a measurable bundle of the spectrum of bounded operators on Banach spaces. The classical Banach spaces with mixed norm are endowed with the structure of Banach–Kantorovich modules. We use such representations to show that every partial integral operator on a space with a mixed norm can be represented as a measurable bundle of integral operators. In particular, we show the cyclic compactness of such operators and, as an application, prove the Fredholm $\nabla$-alternative. We also give an example of a partial integral operator with a nonempty cyclically modular discrete spectrum, while its modular discrete spectrum is an empty set.