Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

V. M. Deundyak

Informatics and Automation, Differential equations and dynamical systems, Tome 278 (2012), pp. 59-67.

Résumé

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in $L_p(\mathbb R^n)$, $1$. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the $C^*$-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator $K$-theory.

Bibliographie

[1] Mikhailov L.G., “Novyi klass osobykh integralnykh uravnenii”, Math. Nachr., 76 (1977), 91–107 | DOI

[2] Karapetyants N.K., “O neobkhodimykh usloviyakh ogranichennosti operatora s neotritsatelnym kvaziodnorodnym yadrom”, Mat. zametki, 30:5 (1981), 787–794 | MR | Zbl

[3] Karapetiants N., Samko S., Equations with involutive operators, Birkhäuser, Boston, 2001 | MR | Zbl

[4] Avsyankin O.G., Karapetyants N.K., “Ob algebre mnogomernykh integralnykh operatorov s odnorodnymi yadrami s peremennymi koeffitsientami”, Izv. vuzov. Matematika, 2001, no. 1, 3–10 | MR | Zbl

[5] Avsyankin O.G., “Ob algebre parnykh integralnykh operatorov s odnorodnymi yadrami”, Mat. zametki, 73:4 (2003), 483–493 | DOI | MR | Zbl

[6] Avsyankin O.G., Deundyak V.M., “Ob indekse mnogomernykh integralnykh operatorov s biodnorodnymi yadrami i peremennymi koeffitsientami”, Izv. vuzov. Matematika, 2005, no. 3, 3–12 | MR | Zbl

[7] Avsyankin O.G., Deundyak V.M., “Ob algebre mnogomernykh integralnykh operatorov s odnorodnymi $SO(n)$-invariantnymi yadrami i radialno slabo ostsilliruyuschimi koeffitsientami”, Mat. zametki, 82:2 (2007), 163–176 | DOI | MR | Zbl

[8] Deundyak V.M., Stepanyuchenko E.A., “Ob integralnykh operatorakh s odnorodnymi yadrami posloino singulyarnogo tipa v prostranstve $L_p(R^2)$”, Vestn. Donsk. gos. tekhn. un-ta, 7:2 (2007), 161–168

[9] Deundyak V.M., “Mnogomernye integralnye operatory s odnorodnymi yadrami kompaktnogo tipa i multiplikativno slabo ostsilliruyuschimi koeffitsientami”, Mat. zametki, 87:5 (2010), 704–720 | DOI | MR | Zbl

[10] Deundyak V.M., Miroshnikova E.I., “Mnogomernye multiplikativnye svertki i ikh prilozheniya k teorii operatorov s odnorodnymi yadrami”, Trudy nauchnoi shkoly I.B. Simonenko, Izd-vo YuFU, Rostov-na-Donu, 2010, 67–78

[11] Pilidi V.S., Sazonov L.I., “Lokalnyi metod v teorii operatorov tipa bisingulyarnykh”, Izv. vuzov. Sev.-Kavk. reg. Estestv. nauki, 2005, Spets. vypusk: Psevdodifferentsialnye uravneniya i nekotorye problemy matematicheskoi fiziki, 100–106

[12] Deundyak V.M., “Gomotopicheskaya klassifikatsiya i vychislenie indeksa semeistv mnogomernykh bisingulyarnykh integralnykh operatorov”, Izv. vuzov. Matematika, 1996, no. 5, 34–46 | MR | Zbl

[13] Deundyak V.M., “On the index theorem for multidimensional bisingular integral operators on Hilbert modules”, J. Nat. Geom., 20 (2001), 1–20 | MR | Zbl

[14] Deundyak V.M., Simonenko I.B., “O gomotopicheskikh svoistvakh i indekse semeistv singulyarnykh i bisingulyarnykh operatorov s RS-koeffitsientami”, Sovr. matematika i ee pril., 9 (2003), 99–106 | MR

[15] Merfi Dzh., $C^*$-algebry i teoriya operatorov, Faktorial, M., 1997

[16] Wegge-Olsen N.E., K-theory and $C^*$-algebras: A friendly approach, Oxford Univ. Press, Oxford, 1993 | MR | Zbl