On the Index of a Bisingular Operator with an Involutive Shift

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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/a3118-8799-1112-n On the Index of a Bisingular Operator with an Involutive Shift Efimov, S. V. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 4. Abstract: In the theory of singular operators with an involutive shift, the issues of Noether (Fredholm) property and the index of an operator of the form \(A+VB\) are fully studied, where \(A\) and \(B\) are singular operators, and \(V\) is an operator of an involutive shift in the space of \(p\)-summable functions on a simple closed contour of the Lyapunov type. Together with the operator \(A+VB\), the corresponding matrix singular operator without shift \(M=\left(\begin{array}{cc}A&{VBV}\\B&{VAV}\end{array}\right)\) is considered. It is well known that the operators \(A+VB\) and \(M\) are Noetherian operators or not simultaneously, and their indices are related as \(1:2\). Similar questions about simultaneous Noetherian property and proportionality of indices arise for bisingular operators with an involutive shift \(A+WB\) and their corresponding matrix operators \(M=\left(\begin{array}{cc}A&{WBW}\\B&{WAW}\end{array}\right)\), where \(A\) and \(B\) are bisingular operators, and \(W\) is an operator of an involutive shift in the space of \(p\)-summable functions on the direct product of simple closed contours of the Lyapunov type. In this paper, we study bisingular operators with an involutive shift that decomposes into one-dimensional components. Two types of such shifts are considered - coordinate-wise and cross. In these cases, the corresponding matrix operators are matrix bisingular operators without shift. The simultaneous Noetherian property of the bisingular operator with a shift and the corresponding matrix bisingular operator without shift is obtained. The proportionality of the indices of bisingular operators with a coordinate-wise shift and the corresponding matrix operators is established. Namely: it is proved that the indices of these operators are related as 1:2. In a special case, the same result about the indices is obtained for the cross shift. Keywords: Noether operator, operator index, bisingular operator, involutive shift Language: Russian Download the full text For citation: Efimov, S. V. On the Index of a Bisingular Operator with an Involutive Shift, Vladikavkaz Math. J., 2024, vol. 26, no. 4, pp. 66-77 (in Russian). DOI 10.46698/a3118-8799-1112-n + References 1. Litvinchuk, G. S. Kraevye zadachi i singulyarnye integral'nye uravneniya so sdvigom} [Boundary Value Problems and Singular Integral Equations with Shift], Moscow, Nauka, 1977, 448 p. (in Russian). 2. Karapetyants, N. K. and Samko, S. G. Uravneniya s involyutivnymi operatorami i ikh prilozheniya [Equations with Involutive Operators and Their Applications], Rostov-on-Don, Publishing house of the Rostov University, 1988, 192 p. (in Russian). 3. Sazonov, L. I. A Bisingular Equation with Translation in the Space \(L_p\), Mathematical Notes, 1973, vol. 13, no. 3, pp. 235-239. DOI: 10.1007/BF01155662. 4. Pilidi, V. S. and Stefanidi, E. N. On an Algebra of Bisingular Operators with Shift, Izvestiya vysshikh uchebnykh zavedeniy. Matematika [Bulletin of Higher Education Institutes. Mathematics], 1981, no. 9, pp. 80-81 (in Russian). 5. Pilidi, V. S. and Stefanidi, E. N. Ob odnoy algebre bisingulyarnykh operatorov so sdvigom [On an Algebra of Bisingular Operators with Shift], Rostov-on-Don, 1981, 26 p. Submitted to VINITI, no. 3036 (in Russian). 6. Efimov, S. V. 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Calculation of the Index of Some Bisingular Operators with Shift by the Homotopy Method, Vestnik Don State Technical University, 2010, vol. 10, no. 1, pp. 22-27 (in Russian). 11. Pilidi, V. S. On a Bisingular Equation in the Space $L_p$, Matematicheskie issledovaniya. Kishinev. Shtiintsa [Mathematical Researches. Kishinev. Shtiintsa], 1972, vol. 7, issue 3, pp. 167-175 (in Russian). 12. Pilidi, V. S. Index Computation for a Bisingular Operator, Functional Analysis and Its Applications, 1973, vol. 7, no. 4, pp. 337-338. DOI: 10.1007/BF01075747. 13. Pilidi, V. S. On the Question of the Index of Bisingular Integral Operators, Matematicheskij analiz i ego prilozheniya. Rostov-na-Donu [Mathematical Analysis and Its Applications], Rostov-on-Don, 1975, vol. 7, pp. 123-136 (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. 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