On the invertibility of almost periodic operators

V. G. Kurbatov

Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 367-377

Résumé

A number of results of the following type are established: {\it If $T$ is an operator with almost periodic coefficients such that $$ \inf\{\|Tx\|:\|x\|=1\}>0, $$ then $T$ is invertible}. Bibliography: 26 titles.

Bibliographie

[1] Favard J., “Sur les equations differentielles a coefficients presque-periodiques”, Acta Math., 51 (1927), 31–81 | DOI | MR | Zbl

[2] Mukhamadiev E. M., “Ob obratimosti differentsialnykh operatorov v prostranstve nepreryvnykh i ogranichennykh na osi funktsii”, DAN SSSR, 196:1 (1971), 47–49 | Zbl

[3] Mukhamadiev E. M., “Ob obratimosti funktsionalnykh operatorov v prostranstve ogranichennykh na osi funktsii”, Matem. zametki, 11:3 (1972), 269–274 | Zbl

[4] Mukhamadiev E. M., “Ob obratimosti differentsialnykh operatorov v chastnykh proizvodnykh ellipticheskogo tipa”, DAN SSSR, 205:6 (1972), 1292–1295 | Zbl

[5] Mukhamadiev E. M., Issledovaniya po teorii periodicheskikh i ogranichennykh reshenii differentsialnykh uravnenii, Dis. ... dokt. fiz.-matem. nauk, Dushanbe, 1978

[6] Kolesov Yu. S., “Neobkhodimye i dostatochnye usloviya eksponentsialnoi dikhotomii reshenii lineinykh pochti periodicheskikh uravnenii s posledeistviem”, Vestn. Yarosl. un-ta, 5 (1973), 28–62 | MR

[7] Shubin M. A., “Pochti-periodicheskie funktsii i differentsialnye operatory s chastnymi proizvodnymi”, UMN, 28:2 (1978), 3–47 | MR

[8] Slyusarchuk V. E., “Ob obratimosti pochti periodicheskikh $c$-nepreryvnykh funktsionalnykh operatorov”, Matem. sb., 116(158) (1981), 483–501 | MR | Zbl

[9] Slyusarchuk V. E., “Obratimost neavtonomnykh differentsialno-funktsionalnykh operatorov”, Matem. sb., 130(172) (1986), 86–104 | MR

[10] Slyusarchuk V. E., “K teorii obratimosti pochti periodicheskikh operatorov”, Matem. zametki, 42:1 (1987), 50–59 | MR | Zbl

[11] Lange B. V., Rabinovich V. S., Algebra operatorov obobschennoi diskretnoi svertki s ostsilliruyuschimi koeffitsientami, Dep. v VINITI 9.07.82, No 3661-92, M., 1982

[12] Coburn L. A., Moyer R. D., Singer I. M., “$C$-algebras of almost periodic pseudo-differential operators”, Acta Math., 139:3–4 (1973), 279–307 | DOI | MR

[13] Neiman Dzh. fon, Izbrannye trudy po funktsionalnomu analizu, 2, Nauka, M., 1987 | MR

[14] Naimark M. A., Normirovannye koltsa, Nauka, M., 1968 | MR | Zbl

[15] Breur M., “Fredholm theories in von Neumann algebras. I, II”, Math. Ann., 178:3 (1968), 243–254 ; 180:4 (1969), 313–325 | DOI | MR | DOI | MR

[16] Kurbatov V. G., “Ob obratimosti pochti periodicheskikh operatorov”, Funktsion. analiz i ego pril., 19:3 (1985), 71–72 | MR | Zbl

[17] Sadovskii B. N., “Predelno kompaktnye i uplotnyayuschie operatory”, UMN, 27:1 (1972), 81–146 | MR | Zbl

[18] Burbaki N., Integrirovanie. Mery na lokalno kompaktnykh prostranstvakh. Prodolzhenie mery. Integrirovanie mer. Mery na otdelimykh prostranstvakh, Nauka, M., 1977

[19] Kurbatov V. G., Algebra raznostnykh operatorov, Dep. v VINITI 11.03.82, No 1017-82, M., 1982

[20] Lutskii Ya. A., Rabinovich V. S., “Psevdodifferentsialnye operatory v prostranstvakh obobschennykh funktsii eksponentsialnogo povedeniya na beskonechnosti”, Funktsion. analiz i ego pril., 12:1 (1978), 79 | MR

[21] Levendorskii S. 3., Rabinovich V. S., “Ob eksponentsialnom ubyvanii na beskonechnosti reshenii mnogomernykh uravnenii tipa svertki v konusakh”, Izv. vuzov. Matematika, 1979, no. 3, 38–44 | MR | Zbl

[22] Shubin M. A., “Psevdoraznostnye operatory i ikh funktsii Grina”, Izv. AN SSSR. Ser. matem., 49 (1985), 652–671 | MR | Zbl

[23] Khyuitt E., Ross K., Abstraktnyi garmonicheskii analiz, no. 1, Nauka, M., 1975

[24] Kurosh A. G., Teoriya grupp, Nauka, M., 1967 | MR | Zbl

[25] Kurbatov V. G., “Ob ogranichennykh resheniyakh differentsialno-raznostnykh uravnenii”, Sib. matem. zhurn., 27:1 (1986), 86–99 | MR | Zbl

[26] Bellman R., Kuk K. L., Differentsialno-raznostnye uravneniya, Mir, M., 1967 | MR | Zbl