Partial Integral Operators of Fredholm Type on Kaplansky-Hilbert Module over \(L_0\)

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/w5172-0182-0041-c Partial Integral Operators of Fredholm Type on Kaplansky-Hilbert Module over \(L_0\) Eshkabilov, Y. Kh. , Kucharov, R. R. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 3. Abstract: 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. Keywords: partial integral operator, Kaplansky-Hilbert module, \(L_0\)-eigenvalue Language: English Download the full text For citation: Eshkabilov, Yu. Kh. and Kucharov, R. R. Partial Integral Operators of Fredholm Type on Kaplansky-Hilbert Module over \(L_0\), Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 80-90. DOI 10.46698/w5172-0182-0041-c + References 1. Appell, J., Kalitvin, A. S. and Zabrejko, P. P. Partial Integral Operators and Integro-Differential Equations, New York, Basel, 2000, 578 p. 2. Eshkabilov, Yu. Kh. On a Discrete ``Three-Particle'' Schrodinger Operator in the Hubbard Model, Theor. Math. Phys., 2006, vol. 149, no. 2, pp. 1497-1511. DOI: 10.1007/s11232-006-0133-2. 3. Eshkabilov, Yu. Kh. and Kucharov, R. R. Essential and Discrete Spectra of the Three-Particle Schrodinger Operator on a Lattice, Theor. Math. Phys., 2012, vol. 170, no. 3, pp. 341-353. DOI: 10.1007/s11232-012-0034-5. 4. Eshkabilov, Yu. Kh. The Efimov Effect for a Model ``Three-Particle'' Discrete Schrodinger Operator, Theor. Math. Phys., 2010, vol. 164, no. 1, pp. 896-904. DOI: 10.1007/s11232-010-0071-x. 5. Eshkabilov, Yu. Kh. Spectra of Partial Integral Operators with a Kernel of Three Variables, Central European J. Math., 2008, vol. 6, no. 1, pp. 149-157. DOI: 10.2478/s11533-008-0010-3. 6. Kusraev, A. G. Dominated Operators, Dordrecht etc., Kluwer Academic Publishers, 2000, 445p. 7. Kudaybergenov, K. K. \(\nabla\)-Fredholm Operators in Banach-Kantorovich Spaces, Methods Func. Anal. Topology, 2006, vol. 12, no. 3, pp. 234-242. 8. Sarymsakov, T. A. Polupolya i teoriya veroyatnostej [Semifields and Probability Theory], Tashkent, Fan, 1980 (in Russian). 9. Akhiezer, N. I. and Glazman, I. M. Teoriya linejnyh operatorov v gilbertovom prostranstve [Theory of Linear Operators in Hilbert Space], Moskva, Nauka, 1966, 544 p. (in Russian). 10. Kusraev, A. G. Cyclically Compact Operators in Banach Spaces. Vladikavkaz Math. J., 2000, vol. 2, no. 1, pp. 10-23. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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