Geodesic
Partial integral operators of Fredholm type on Kaplansky–Hilbert module over $L_0$
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 3, pp. 80-90 
Yu. Kh. Eshkabilov; R. R. Kucharov. Partial integral operators of Fredholm type on Kaplansky–Hilbert module over $L_0$. Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 3, pp. 80-90. http://geodesic.mathdoc.fr/item/VMJ_2021_23_3_a6/
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     title = {Partial integral operators of {Fredholm} type on {Kaplansky{\textendash}Hilbert} module over $L_0$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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