About Riemann Matrix Operator in the Space of Smooth Vector Functions

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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: 10.23671/VNC.2019.3.36461 About Riemann Matrix Operator in the Space of Smooth Vector Functions Pasenchuk, A. E. , Seregina, V. V. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3. 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. Keywords: operator, Riemann, Noetherian, smooth, index, formula. Language: Russian Download the full text 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 + References 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], Мoscow, Nauka, 1971, 352 p. (in Russian). 5. Gohberg, I. С. 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). 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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