Geodesic
Fredholm operator manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 187 (2020), pp. 12-18.

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We consider special classes of operators acting in functional spaces on manifolds. We can say that our approach is an operator-geometric treatment of the well-known local principle. In an abstract form, the conditions of the fredholmness are described and it is shown how these results can be applied to the study of elliptic pseudodifferential operators on manifolds with a non-smooth boundary.
Keywords: local operator, operator manifold, Fredholm property, index. 
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V. B. Vasilev (Vasilyev). Fredholm operator manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 187 (2020), pp. 12-18. http://geodesic.mathdoc.fr/item/INTO_2020_187_a1/

[1] Vasilev V. B., “Regulyarizatsiya mnogomernykh singulyarnykh integralnykh uravnenii v negladkikh oblastyakh”, Tr. Mosk. mat. o-va., 59 (1998), 72–105

[2] Vasilev V. B., “Psevdodifferentsialnye operatory i uravneniya peremennogo poryadka”, Differ. uravn., 54:9 (2018), 1184–1195

[3] Gokhberg I. Ts., Krupnik N. Ya., Vvedenie v teoriyu odnomernykh singulyarnykh integralnykh operatorov, Shtiintsa, Kishinev, 1973

[4] Plamenevskii B. A., Algebry psevdodifferentsialnykh operatorov, Nauka, M., 1986 | MR

[5] Plamenevskii B. A., Rozenblyum G. V., “Psevdodifferentsialnye operatory s razryvnymi simvolami: K-teoriya i formuly indeksa”, Funkts. anal. prilozh., 26:4 (1992), 45–56 | MR

[6] Plamenevskii B. A., Senichkin V. N., “Razreshimye algebry operatorov”, Algebra i analiz., 6:5 (1994), 1–87 | MR

[7] Rempel Sh., Shultse B. V., Teoriya indeksa ellipticheskikh kraevykh zadach, Mir, M., 1983

[8] Simonenko I. B., Lokalnyi metod v teorii operatorov invariantnykh otnositelno sdviga i ikh ogibayuschikh, TsVVR, Rostov-na-Donu, 2007

[9] Eskin G. I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973

[10] Dynin A., “Inversion problem for singular integral operators: $C^*$-approach”, Proc. Natl. Acad. Sci. U.S.A., 75 (1978), 4668–4670 | DOI | MR | Zbl

[11] Dynin A., “Multivariable Wiener–Hopf operators. I. Representations.”, Integral Equat. Oper. Theory., 9 (1986), 537–556 | DOI | MR | Zbl

[12] Dynin A., “Multivariable Wiener–Hopf operators. II. Spectral topology and solvability”, Integr. Equat. Oper. Theory., 10 (1987), 554–576 | DOI | MR | Zbl

[13] Egorov Yu. V., Schulze B. W., Pseudo-differential operators, singularities, applications, Birkhäuser-Verlag, Basel, 1997 | MR | Zbl

[14] Mikhlin S. G., Prößdorf S., Singular integral operators, Akademie-Verlag, Berlin, 1986 | MR

[15] Nazaikinskii V. E., Savin A. Yu., Schulze B. W., Sternin B. Yu., Elliptic theory on singular manifolds, Chapman Hall/CRC, Boca Raton, 2006 | MR | Zbl

[16] Schulze B. W., Boundary value problems and singular pseudo-differential operators, Wiley, Chichester, 1998 | MR | Zbl

[17] Schulze B. W., Sternin B., Shatalov V., Differential equations on singular manifolds; semiclassical theory and operator algebras, Wiley, Berlin, 1998 | MR | Zbl

[18] Vasil'ev V. B., Wave factorization of elliptic symbols: theory and applications. Introduction to the theory of boundary value problems in non-smooth domains, Kluwer Academic Publ., Dordrecht–Boston–London, 2000 | MR | Zbl

[19] Vasil'ev V. B., “Elliptic operators and their symbols”, Demonstr. Math., 52 (2019), 361–369 | DOI | MR