Partial integral operators of Fredholm type on Kaplansky-Hilbert module over $L_0$

The article studies some characteristic properties of self-adjoint partiallyintegral operators of Fredholm type in the Kaplansky-Hilbert module$L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Somemathematical tools from the theory of Kaplansky-Hilbert module are used. In theKaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over$ L_{0} \left (\Omega _ {2} \right)$ we consider the partially integral operator ofFredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ areclosed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$$\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). Theexistence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjointpartially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ hasfinite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues.In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is orderconvergent to the zero function. It is also established that theoperator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.