Singular integral operators. The case of an unlimited contour

V. Neaga

doi:10.33993/jnaat342-802

Authors

V. Neaga State University of Moldova, Chisinau, Republic of Moldova

DOI:

https://doi.org/10.33993/jnaat342-802

Keywords:

Lyapunov closed curves, Noetherian singular integral, piecewise Lyapunov contour, singular integral equations, singular operators with Cauchy kernel singular operators with shift

Abstract views: 230

Abstract

Let \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\not\equiv t\) for \(1\leq j<n\). In this work we study subalgebra \(\Sigma\) of algebra\(L(L_p(\Gamma,\rho))\), which contains all operators of the form\[\left (M \varphi \right) (t) = \sum_{k=0}^{n-1} \bigg \{a_k (t) \varphi (\alpha_k (t)) + \tfrac{b_k(t)}{\pi {\rm i} } \int_{\Gamma} \tfrac{\varphi ( \tau )}{\tau - \alpha_k (t)} d \tau \bigg \}\]with piecewise-continuous coefficients. The existence of such an isomorphism between \(\Sigma\) and some algebra \(\frak A\) of singular operators with Cauchy kernel that an arbitrary operator from \(\Sigma\) and its image are Noetherian or not Noetherian simultaneously is proved. It allows to introduce the concept of a symbol for all operators from \( \Sigma \) and, using the known results for algebra \( \frak A \), in terms of a symbol to receive conditions of Noetherian property.

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References

Böttcher A., Gohberg I., Karlovich Yu., Krupnik N., eds., Banach algebras generated by idempotents and applications, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 90, pp. 19-54, 1996, https://doi.org/10.1007/978-3-0348-9040-3_2 DOI: https://doi.org/10.1007/978-3-0348-9040-3_2

Gohberg I., Krupnik N., Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 17, pp. 322-337, 1993, https://doi.org/10.1007/bf01200289 DOI: https://doi.org/10.1007/BF01200289

Gohberg I., Krupnik N., Extension theorems for Fredholm and invertibility symbols, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 16, pp. 514-529, 1993, https://doi.org/10.1007/bf01205291 DOI: https://doi.org/10.1007/BF01205291

Gohberg I., Krupnik N., Extension theorems for invertibility symbols in Banach algebras, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 15, pp. 991-1010, 1992, https://doi.org/10.1007/bf01203124 DOI: https://doi.org/10.1007/BF01203124

Goluzin G.M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Moskva-Leningrad, Gos-tekh-iz-dat, 1952.

Karanetyants N.K., Samko S.G., Uravneniya s involyutivnymi operatorami i ikh prilozheniya, Rostov, Izd-vo Rostovskogo un-ta, 1988.

Khvedelidze B.V., Lineinye razryvnye granichnye zadachi teorii funktsii, singulyarnye integralinye uravneniya i nekotorye ikh prilozheniya, Trudy Tbilisskogo mat. in-ta AN Gruz. SSR, XXII, s. 3-158, 1957.

Krupnik N., Banach algebras with symbol and singular integral operators, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 26, 1987, https://doi.org/10.1007/978-3-0348-5463-4_5 DOI: https://doi.org/10.1007/978-3-0348-5463-4

Krupnik N.Ya., Nyaga V.I., O singulyarnykh operatorakh so sdvigom v sluchae kusochno-lya-pu-nov-skogo kontura, Soobshchenie AN Gruz. SSR, 76, no. 1, s. 25-28, 1974.

Litvinchuk G.S., Kraevye zadachi i singulyarnye integralinye uravneniya so sdvigom, Moskva, Nauka, 1977.

Neaga V., Asupra algebrei de operatori singulari cu coeficienţi continui pe porţiuni, Analele ştiinţifice ale USM, pp. 11-16, 1997.

Nyaga V., Singular integral operators. II. The case of a piecewise Lyapunov contour, Analele ştiinţifice ale USM, p. 202-213, 1999.

Nyaga V.I.. Singulyarnye integralinye operatory so sdvigom vdoli neogranichennogo kontura, Izvestiya vuzov, Ser. mat., no 5, s. 35-42, 1982.

Nyaga V.I., Kriterii neterovosti singulyarnykh integralinykh uravnenii s drobno-lineinym sdvigom v prostranstve Lp, Izvestiya AN MSSR, Matematika, no. 3, s. 14-18, 1990.

Nyaga V.I., O simvole singulyarnykh integralinykh operatorov s obratnym sdvigom na R Algebraicheskie struktury i geometriya. Kishinev, Shtiintsa, s. 84-92, 1991.

Nyaga V.I., Singulyarnye integralinye operatory s oboshchennym Karlemanovskim sdvigom v sluchae neogranichennogo kontura, Izvestiya AN RM, Matematika, no 2, s. 87-100, 1997.

Nyaga V.I., Laĭla, M., Ob algebre singulyarnykh integralnykh operatorov s obratnym sdvigom v prostranstve Lp(R,ρ), Izvestiya AN RM, Matematika, 3, s. 46-55, 1992.

Nyaga V., The simbol of singular integral operators with congugation the case of piecewise Lyapunov contour, American Math. Society, 27, no. 1, pp. 173-176, 1983.

Nyaga, V.I., Usloviya neterovosti singulyarnykh integralinykh operatorov s sopryazheniem v sluchae kusochno-lyapunovskogo kontura, Issledovaniya po funktsionalinomu analizu i differentsialinym uravneniyam. Kishinev, Shtiintsa, s. 90-102, 1984.

Nyaga V.I., Vozmushcheniya singulyarnykh operatorov s kusochno-nepreryvnymi koe1ffitsientami, Issledovaniya po funktsionalinomu analizu i differentsialinym uravneniyam, Kishinev, Shtiintsa, s. 64-68, 1978.

Vinogradova G.Yu., Singulyarnye integralinye operatory na osi s drobno-lineinym sdvigom v prostranstvakh s vesom, Izvestiya vuzov, Ser. mat., no. 3, s. 67-72, 1979.

Zverovich E1.I., Litvinchuk G.S. , Kraevye zadachi so sdvigom dlya analiticheskikh funktsii singulyarnye funktsionalinye uravneniya, UMN, 23, vyp. 3, s. 67-121, 1968.