Spectral Solvability Criterion for Partial Integral Equations in Kaplansky–Hilbert Modules

Source abstract

We study eigenvalue problems for partial integral operators acting in the Kaplansky–Hilbert module associated with a measurable bundle of Hilbert spaces. Assuming the operator is self-adjoint and cyclically compact, we obtain a solvability criterion expressed via the spectra of the fibre operators: a non-trivial solution exists precisely when the spectral parameter belongs to the spectrum of the fibre operator on a measurable set of positive measure. In the case of an atomic measure the problem decomposes into a family of independent fibre equations. A discrete example on a two-atom base space with a Green-type kernel illustrates the applicability of the criterion.