Spectral solvability criterion for partial integral equations in Kaplansky–Hilbert modules

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

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