On the index of a bisingular operator with an involutive shift

S. V. Efimov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 66-77

Résumé

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.

Bibliographie

[1] Litvinchuk, G. S., Boundary Value Problems and Singular Integral Equations with Shift, Nauka, M., 1977, 448 pp. (in Russian)

[2] Karapetyants, N. K. and Samko, S. G., Equations with Involutive Operators and Their Applications, Publishing house of the Rostov University, Rostov-on-Don, 1988, 192 pp. (in Russian) | MR

[3] Sazonov, L. I., “A Bisingular Equation with Translation in the Space $L_p$”, Mathematical Notes, 13:3 (1973), 235–239 | DOI | MR | Zbl | Zbl

[4] Pilidi, V. S. and Stefanidi, E. N., “On an Algebra of Bisingular Operators with Shift”, Bulletin of Higher Education Institutes. Mathematics, 1981, no. 9, 80–81 (in Russian) | MR | Zbl

[5] Pilidi, V. S. and Stefanidi, E. N., Ob odnoy algebre bisingulyarnykh operatorov so sdvigom, Submitted to VINITI, no. 3036, On an Algebra of Bisingular Operators with Shift, Rostov-on-Don, 1981, 26 pp. (in Russian)

[6] Efimov, S. V., “Noethericity and Index of a Characteristic Bisingular Integral Operator with Shifts”, Siberian Mathematical Journal, 60:4 (2019), 585–591 | DOI | DOI | MR | Zbl

[7] Efimov, S. V., “Bisingular Operators with Irreducible Involutive Shift”, Russian Mathematics (Iz. VUZ), 1992, no. 2, 29–36 | Zbl

[8] Efimov, S. V., “Calculation of the Index of Some Bisingular Operators with Irreducible Involutive Shift”, Bulletin of Higher Education Institutes. North Caucasus Region. Natural Sciences. Suppl., 2004, no. 9, 7–14 (in Russian) | Zbl

[9] Efimov, S. V., “On the Index of Some Bisingular Integral Operators with Shift”, Vestnik Don State Technical University, 4:3(21) (2004), 290–295 (in Russian)

[10] Efimov, S. V., “Calculation of the Index of Some Bisingular Operators with Shift by the Homotopy Method”, Vestnik Don State Technical University, 10:1 (2010), 22–27 (in Russian)

[11] Pilidi, V. S., “On a Bisingular Equation in the Space $L_p$”, Mathematical Researches, 7, no. 3, Shtiintsa, Kishinev, 1972, 167–175 (in Russian) | MR

[12] Pilidi, V. S., “Index Computation for a Bisingular Operator”, Functional Analysis and Its Applications, 7:4 (1973), 337–338 | DOI | MR

[13] Pilidi, V. S., “On the Question of the Index of Bisingular Integral Operators”, Mathematical Analysis and Its Applications, 7, Rostov-on-Don, 1975, 123–136 (in Russian) | Zbl