On a vectorial Riemann boundary value problem with infinite defect numbers, and related factorization of matrix-valued functions

I. M. Spitkovsky

Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 521-538

Résumé

The author presents a method for the factorization of a matrix-valued function, specified on a closed contour $\Gamma$, in which functions meromorphic inside and outside of $\Gamma$ (with, generally speaking, an infinite number of zeros and poles), and products of such functions, are allowed as the diagonal elements of the central factor. The concept of partial indices of such a factorization is introduced, and their invariance is established. Conditions for solvability are obtained, as well as the general form of the solution, and criteria for the closure of the image of a vectorial Riemann boundary value problem whose matrix coefficient is factored in the sense indicated. Bibliography: 33 titles.

Bibliographie

[1] Khvedelidze B. V., “Metod integralov tipa Koshi v razryvnykh granichnykh zadachakh teorii golomorfnykh funktsii odnoi kompleksnoi peremennoi”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 7, VINITI, M., 1975, 5–162

[2] Danilyuk I. I., Neregulyarnye granichnye zadachi na ploskosti, Nauka, M., 1975 | MR

[3] Litvinchuk G. S., Spitkovskii I. M., Faktorizatsiya matrits-funktsii, Odessa, 1984, 460 pp., (Dep. v VINITI 17.04.84, No 2410–84)

[4] Spitkovskii I. M., “Faktorizatsiya izmerimykh matrits-funktsii. Svyazannye s nei voprosy teorii sistem singulyarnykh integralnykh uravnenii i vektornoi kraevoi zadachi Rimana. I”, Differents. uravneniya, 17:4 (1981), 697–709 | MR | Zbl

[5] Spitkovskii I. M., “Obobschennaya faktorizatsiya matrits-funktsii i kraevaya zadacha Rimana s beskonechnymi chastnymi indeksami”, DAN SSSR, 286:3 (1986), 559–562 | MR | Zbl

[6] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[7] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M.-L., 1950

[8] Tumarkin G. Ts., Khavinson S. Ya., “Klassy analiticheskikh funktsii v mnogosvyaznykh oblastyakh”, Issledovaniya po sovremennym problemam TFKP, Fizmatgiz, M., 1960, 45–77 | MR

[9] Khovin V. P., “Granichnye svoistva integrala tipa Koshi i garmonicheski sopryazhennykh funktsii v oblastyakh so spryamlyaemoi granitsei”, Matem. sb., 68(110) (1965), 499–517

[10] Dynkin E. M., Osilenker B. P., “Vesovye otsenki singulyarnykh integralov i ikh prilozheniya”, Itogi nauki i tekhniki. Matematicheskii analiz, 21, VINITI, M., 1983, 42–129 | MR

[11] Calderón A. P., “Cauchy integrals on Lipschitz curves and related operators”, Proc. Nat. Acad. Sci. USA, 1977, no. 4, 1324–1327 | DOI | MR | Zbl

[12] Vekua N. P., Sistemy singulyarnykh integralnykh uravnenii, Nauka, M., 1970 | MR | Zbl

[13] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1973 | MR

[14] Golubev V. V., Odnoznachnye analiticheskie funktsii. Avtomorfnye funktsii, Fizmatgiz, M., 1961 | MR

[15] Simonenko I. B., “O faktorizatsii i lokalnoi faktorizatsii izmerimykh funktsii”, DAN SSSR, 250:5 (1980), 1063–1066 | MR | Zbl

[16] Simonenko I. B., “O globalnoi i lokalnoi faktorizuemosti izmerimoi matritsy-funktsii i neterovosti porozhdennogo eyu singulyarnogo operatora”, Izv VUZov. Matematika, 1983, no. 4, 81–87 | MR | Zbl

[17] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR

[18] Gumarkin G. Ts., “Svoistva analiticheskikh funktsii, predstavimykh integralami tipa Koshi–Stiltesa i tipa Koshi–Lebega”, Izv. AN ArmSSR. Fiz.-matem. nauki, 16:5 (1963), 23–45 | MR

[19] Spitkovskii I. M., “Nekotorye otsenki dlya chastnykh indeksov izmerimykh matrits-funktsii”, Matem. sb., 111(153) (1980), 227–248 | MR

[20] Coburn L. A., “Weyl's theorem for non-normal operators”, Michigan Math. J., 13 (1966), 285–286 | DOI | MR

[21] Grudskii S. M., “Singulyarnye integralnye uravneniya i kraevaya zadacha Rimana s beskonechnym indeksom v prostranstve $L_p(\Gamma,\omega)$”, Izv. AN SSSR. Ser. matem., 49 (1985), 55–80 | MR | Zbl

[22] Grudskii S. M., Dybin V. B., “Kraevaya zadacha Rimana v prostranstve $L_p(\Gamma,\rho)$ s pochti-periodicheskimi razryvami u ee koeffitsienta”, Matem. issled., 1980, no. 54, 36–49 | MR | Zbl

[23] Gordadze E. G., “O singulyarnom integrale s yadrom Koshi”, Soobsch. AN GSSR, 27:3 (1965), 521–526 | MR

[24] Gordadze E. G., “O singulyarnykh integralakh s yadrom Koshi”, Tr. Tbilissk. matem. in-ta AN GSSR, 42 (1972), 5–17 | MR | Zbl

[25] Gordadze E. G., “O singulyarnykh integralakh na negladkikh liniyakh”, Tr. simp. po mekhanike sploshnoi sredy i rodstvennym problemam analiza, Metsniereba, Tbilisi, 1974, 74–85 | MR

[26] Spitkovskii I. M., Yatsko S. I., “O kraevoi zadache Rimana, u matrichnogo koeffitsienta kotoroi dopuskayutsya beskonechnye chastnye indeksy”, Izv. VUZov. Matematika, 1985, no. 6, 45–53 | MR | Zbl

[27] Kokilashvili V. M., “O singulyarnykh integralakh i maksimalnykh operatorakh s yadrom Koshi”, DAN SSSR, 223:3 (1975), 553–558

[28] Paatashvili V. A., Razryvnaya granichnaya zadacha lineinogo sopryazheniya i svyazannye s nei singulyarnye integralnye uravneniya, Diss. $\dots$ doktora fiz.-mat. nauk, Tbilisk. in-t matem. AN GSSR, Tbilisi, 1984

[29] Simonenko I. B., “Nekotorye obschie voprosy teorii kraevoi zadachi Rimana”, Izv. AN SSSR. Ser. matem., 32 (1968), 1138–1146 | MR | Zbl

[30] Spitkovskii I. M., “O mnozhitelyakh, ne vliyayuschikh na faktorizuemost”, Matem. zametki, 27:2 (1980), 291–299 | MR | Zbl

[31] Zinsmeister M., “Representation conforme et courbes presque Lipschittziennes”, Ann. Ints. Fourier, 34:2 (1984), 29–44 | MR | Zbl

[32] Zinsmeister M., “Domaines réguliers du plan”, Ann. Inst. Fourier, 35:1 (1985), 49–55 | MR | Zbl

[33] Paatashvili V. A., “O razryvnoi zadache lineinogo sopryazheniya”, Soobsch. AN GSSR, 36:3 (1964), 539–540