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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {Yu. Kh. Eshkabilov and R. R. Kucharov},
     title = {Partial integral operators of {Fredholm} type on {Kaplansky{\textendash}Hilbert} module over $L_0$},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
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}
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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/

[1] Appell J., Kalitvin A. S., Zabrejko P. P., Partial Integral Operators and Integro-Differential Equations, New York–Basel, 2000, 578 pp. | MR | Zbl

[2] Eshkabilov Yu. Kh., “On a Discrete “Three-Particle” Schrodinger Operator in the Hubbard Model”, Theor. Math. Phys., 149:2 (2006), 1497–1511 | DOI | MR | Zbl

[3] Eshkabilov Yu. Kh., Kucharov R. R., “Essential and Discrete Spectra of the Three-Particle Schrödinger Operator on a Lattice”, Theor. Math. Phys., 170:3 (2012), 341–353 | DOI | MR | Zbl

[4] Eshkabilov Yu. Kh., “The Efimov Effect for a Model “Three-Particle” Discrete Schrödinger Operator”, Theor. Math. Phys., 164:1 (2010), 896–904 | DOI | MR | Zbl

[5] Eshkabilov Yu. Kh., “Spectra of Partial Integral Operators with a Kernel of Three Variables”, Central European J. Math., 6:1 (2008), 149–157 | DOI | MR | Zbl

[6] Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht etc., 2000, 445 pp. | MR | Zbl

[7] Kudaybergenov K. K., “$\nabla$-Fredholm Operators in Banach–Kantorovich Spaces”, Methods Func. Anal. Topology, 12:3 (2006), 234–242 | MR | Zbl

[8] Sarymsakov T. A., Semifields and Probability Theory, Fan, Tashkent, 1980 (in Russian) | MR

[9] Akhiezer N. I., Glazman I. M., Theory of Linear Operators in Hilbert Space, Nauka, M., 1966, 544 pp. (in Russian) | MR

[10] Kusraev A. G., “Cyclically Compact Operators in Banach Spaces”, Vladikavkaz Math. J., 2:1 (2000), 10–23 | MR | Zbl