The article studies some characteristic properties of self-adjoint partially
integral operators of Fredholm type in the Kaplansky-Hilbert module
$L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Some
mathematical tools from the theory of Kaplansky-Hilbert module are used. In the
Kaplansky-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 of
Fredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ are
closed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$
$\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). The
existence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjoint
partially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ has
finite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues.
In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is order
convergent to the zero function. It is also established that the
operator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.
The change in the spectrum of the multipliers H 0 f (x, y) = x α + y β f (x, y) and H 0 f (x, y) = x α y β f (x, y) for perturbation with partial integral operators in the spaces L 2 [0, 1] 2 is studied. Precise description of the essential spectrum and the existence of simple eigenvalue is received. We prove that the number of eigenvalues located below the lower edge of the essential spectrum in the model is finite.
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