Abstract: In the space of vector functions smooth on the unit circle, we consider the matrix operator of linear conjugation generated by the Riemann boundary-value problem. It is assumed that the coefficients of the boundary value problem are smooth matrix functions. The concept of smooth degenerate factorization of the plus and minus types of a smooth matrix function is introduced and studied. In terms of degenerate factorizations, we give necessary and sufficient conditions for the noethericity of the considered Riemann matrix operator in the space of smooth vector functions. For a function smooth on a circle having at most finitely many zeros of finite orders, the concept of a singular index is introduced and studied, generalizing the concept of the index of a non-degenerate continuous function. For the Noetherian matrix Riemann operator, a formula is obtained for calculating the index of this operator, which coincides with the well-known similar formula in the case where the coefficients of the Riemann operator are non-degenerate.
For citation: Pasenchuk, A. E. and Seregina, V. V. About Riemann Matrix Operator in the Space of Smooth Vector Functions, Vladikavkaz Math. J., 2019, vol. 21, no. 3, pp. 50-61 (in Russian). DOI 10.23671/VNC.2019.3.36461
1. Gahov, F. D. Boundary Value Problems, New York, Dover, 1990, 561 p.
2. Muskhelishvili, N. I. Singuljarnye integral'nye uravneniya [Singular Integral Equations],
Moscow, Nauka, 1968, 599 p. (in Russian).
3. Vekua, N. P. Sistemy singuljarnykh integral'nykh uravnenii [Systems of Singular Integral
Equations], Moscow, Nauka, 1970, 252 p. (in Russian).
4. Gohberg, I. C. and Fel'dman, I. A. Uravneniya v svertkakh i proektsionnye metody ikh
resheniya [Convolution Equations and Projection Methods for their Solution],
Ìoscow, Nauka, 1971, 352 p. (in Russian).
5. Gohberg, I. Ñ. and Krupnik, N. Ya. Vvedenie v teoriyu odnomernykh singulyarnykh
integral'nykh operatorov [Introduction to the Theory of One-Dimentional Singular Integral Operators],
Kishinev, Shtiintsa, 1973, 426 p. (in Russian).
6. Simonenko, I. B. Some General Questions in the Theory of the Riemann Boundary Problem,
Mathematics of the USSR-Izvestiya, 1968, vol. 2, no. 5, p. 1091-1099. DOI: 10.1070/IM1968v002n05ABEH000706.
7. Presdorf, Z. Nekotorye klassy singulyarnykh uravneniy [Some Classes of Singular Equations],
Moscow, Mir, 1979, 493 p. (in Russian).
8. Soldatov, A. P. Odnomernye singulyarnye operatory i kraevye zadachi teorii funktsii
[One-Dimensional Singular operators and Boundary Value Problems of the Theory of Functions],
Moscow, Vysshaya shkola, 1991, 210 p. (in Russian).
9. Volevich, L. Z. and Gindikin, S. G. Obobshhennye funktsii i uravneniya v svertkakh
[Generalized Convolution Functions and Equations], Moscow, Nauka, 1994, 336 p. (in Russian).
10. Dybin, V. B. and Karapetyants, N. K. Application of the Normalization Method to a Class
of Infinite Systems of Linear Algebraic Equations, Izv. Vyssh. Uchebn. Zaved. Mat.,
1967, no. 10, p. 39-49. (in Russian).
11. Zilberman, B. On Singular Operators in Spaces of Infinitely Differentiable
and Generalized Functions, Matematicheskiye Issledovaniya, Kishinev, Shtiinca,
1971, vol. 6, no. 3, p. 168-179. (in Russian).
12. Pasenchuk, A. E. Diskretnye operatory tipa svertki v klassakh posledovatel'nostei
so stepennym kharakterom povedeniya na beskonechnosti [Discrete Operators of Convolution
Type in Classes of Sequences with Power-Law Behavior at Infinity], Rostov-on-Don, SFU, 2013, 279 p. (in Russian).
