  This paper is devoted to self-adjoint cyclically compact operators on
Hilbert--Kaplansky module over a ring of bounded measurable functions. The
spectral theorem for such a class of operators are given. We apply this result
to partial integral equations on the space with mixed norm of measurable
functions and to compact operators relative to von Neumann algebras. We will
give a condition of solvability of partial integral equations with self-adjoint
kernel. Moreover, a general form of compact operators relative to a type I von
Neumann algebra is given.
