Riesz completeness of the eigenelements and associated elements of linear selfadjoint pencils

S. G. Pyatkov

Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 343-361

Résumé

Methods of the theory of interpolation of Banach spaces are applied to the study of the following questions: Riesz completeness for linear pencils; selection of maximal semidefinite invariant subspaces for a given operator defined in a Krein space; boundedness of the Riesz projections corresponding to the unbounded component of the spectrum of a positive operator.

Bibliographie

[1] Azizov T. Ya., Iokhvidov I. S., Osnovy teorii lineinykh operatorov v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR

[2] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Nauk. dumka, Kiev, 1965 | MR

[3] Berg I., Lëfstrem I., Interpolyatsionnye prostranstva, Mir, M., 1980 | MR

[4] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra differentsialnykh uravnenii”, Matematicheskii analiz, VINITI, M., 1977, 5–59 | MR

[5] Beals R., “Indefinite Sturm–Liouville problems and half-range completeness”, J. Diff. Equat., 56 (1985), 391–408 | DOI | MR

[6] Grisvard P., “Commutative de deux functeurs d'interpolation et applications”, J. Math. Pures et Appliq., 45 (1966), 144–206

[7] Grisvard P., “An approach to the singular solutions of elliptic problems via the theory of differential equations in Banach Spaces”, Lect. Notes in Math., 1223, 1986, 131–156 | MR

[8] Gokhberg I. I., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[9] Dore G., Venni A., “Separation of two (possibly unbounded) components of the spectrum of a linear operator”, Equat. Operator theory, 12 (1989), 470–470 | DOI | MR | Zbl

[10] Kislov N. V., “Neodnorodnye kraevye zadachi dlya differentsialno-operatornogo uravneniya smeshannogo tipa i ikh prilozheniya”, Matem. sb., 125 (167) (1984), 19–37 | MR

[11] Markus A. S., Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtiintsa, Kishinev, 1986 | MR | Zbl

[12] Pyatkov S. G., “Nekotorye svoistva sobstvennykh funktsii lineinykh puchkov”, Sib. matem. zhur., 30:4 (1989), 111–124 | MR | Zbl

[13] Pyatkov S. G., “Svoistva sobstvennykh funktsii odnoi spektralnoi zadachi i nekotorye ikh prilozheniya”, Nekotorye prilozheniya funktsion. analiza k zadacham matem. fiziki, Inst. matem. SO AN SSSR, Novosibirsk, 1986, 65–84

[14] Pyatkov S. G., “O svoistvakh sobstvennykh funktsii odnoi spektralnoi zadachi i ikh prilozheniya”, Korrektnye kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki, Inst matem. SO AN SSSR, Novosibirsk, 1984, 115–130 | MR

[15] Pyatkov S. G., “O razreshimosti odnoi kraevoi zadachi dlya parabolicheskogo uravneniya s menyayuschimsya napravleniem vremeni”, DAN SSSR, 285:6 (1985), 1327–1329 | MR

[16] Pyatkov S. G., “Razreshimost kraevykh zadach dlya odnogo uravneniya smeshannogo tipa vtorogo poryadka”, Neklassicheskie differentsialnye uravneniya v chastnykh proizvodnykh, Inst. matem. SO AN SSSR, Novosibirsk, 1988, 77–90 | MR

[17] Pyatkov S. G., “O nekotorykh svoistvakh sobstvennykh funktsii lineinykh puchkov”, Matem. zametki, 51:1 (1992), 141–148 | MR | Zbl

[18] Riss F., Sëkefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979

[19] Curgus B., Langer H. A, “Krein space approach to symmetric ordinary differential operators with an indefinite weight functions”, J. Diff. Equat., 79:1 (1989), 31–61 | DOI | MR | Zbl

[20] Tribel Kh., Teoriya interpolyatsii. Funktsionalnye prostranstva. Differentsialnye operatory, Mir, M., 1980 | MR

[21] Faierman M., Roach G. F., “Full and half-range eigenfunctions expansion for an elliptic boundary value problem involving an indefinite weight”, Proc. 1987 Equadiff. Conference, Lecture Notes in Pure and Appl. Math., 118, Dekker, New York and Basel, 1989, 231–236 | MR | Zbl

[22] Faierman M., “Elliptic problems involving an indefinite weight”, Trans. A.M.S., 320 (1990), 253–279 | DOI | MR | Zbl

[23] Shkalikov A. A., “O printsipakh otbora i svoistvakh chasti sobstvennykh i prisoedinennykh elementov puchkov operatorov”, Vestn. MGU. Ser. matematika, mekhanika, 1988, no. 4, 16–25 | MR

[24] Shkalikov A. A., “O minimalnosti proizvodnykh tsepochek, otvechayuschikh chasti sobstvennykh i prisoedinennykh elementov samosopryazhennykh puchkov operatorov”, Vestn. MGU. Ser. matematika, mekhanika, 1985, no. 6, 10–19 | MR | Zbl

[25] Shkalikov A. A., Pliev V. T., “Kompaktnye vozmuscheniya silno dempfirovannykh puchkov operatorov”, Matem. zametki, 45:2 (1989), 118–128 | MR