Multiple completeness of root vectors of a Keldysh pencil perturbed by an operator-valued function analytic in a disc

G. V. Radzievskii

Sbornik. Mathematics, Tome 20 (1973) no. 3, pp. 323-347

Résumé

In this paper a formulation of the problem of multiple completeness is considered that generalizes the earlier well-known formulation of this problem in differential equations, and also in mechanics. To within a finite-dimensional subspace, multiple completeness tests are given for a Keldysh pencil perturbed by an analytic operator-valued function. In several cases the domain of multiple completeness is found. Bibliography: 26 titles.

Bibliographie

[1] M. V. Keldysh, “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh klassov nesamosopryazhennykh uravnenii”, DAN SSSR, 77:1 (1951), 11–14 | Zbl

[2] M. V. Keldysh, “O polnote sobstvennykh funktsii nekotorykh klassov nesamosopryazhennykh lineinykh operatorov”, Uspekhi matem. nauk, XXVI:4 (160) (1971), 15–41

[3] M. G. Krein, G. K. Langer, “K teorii kvadratichnykh puchkov samosopryazhennykh operatorov”, DAN SSSR, 154:6 (1964), 1258–1261 | MR | Zbl

[4] M. G. Krein, G. K. Langer, “O nekotorykh matematicheskikh printsipakh lineinoi teorii dempfirovannykh kolebanii kontinuumov”, Trudy mezhdunarodnogo simpoziuma po primeneniyu teorii funktsii kompleksnogo peremennogo v mekhanike sploshnoi sredy, izd-vo «Nauka», Moskva, 1965 | MR

[5] M. G. Krein, “Vvedenie v geometriyu indefinitnykh $J$-prostranstv i teoriyu operatorov v etikh prostranstvakh”, Vtoraya letnyaya matematicheskaya shkola (Katsiveli, iyun–iyul 1964 g.), izd-vo «Naukova dumka», Kiev, 1965

[6] I. V. Goryuk, “Odna teorema o polnote sistemy sobstvennykh i prisoedinennykh vektorov operatornogo puchka $L(\lambda)=\lambda^2C+\lambda B+E$”, Vestnik MGU, seriya matem., mekhan., 1970, no. 1, 55–60 | Zbl

[7] I. V. Goryuk, “O faktorizatsii kvadratichnogo operatornogo puchka”, Vestnik MGU, seriya matem., mekhan., 1970, no. 5, 28–35 | Zbl

[8] I. Ts. Gokhberg, “Zadacha faktorizatsii operator-funktsii”, Izv. AN SSSR, seriya matem., 28:5 (1964), 1055–1082 | MR | Zbl

[9] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, Moskva, 1962

[10] E. Z. Mogulskii, “Teorema polnoty sistemy sobstvennykh i prisoedinennykh vektorov ratsionalnogo operatornogo puchka”, Izv. AN Arm.SSR, seriya matem., 3:6 (1968), 427–442 | MR

[11] G. A. Voropaeva, V. P. Maslov, “Kratkaya polnota po M. V. Keldyshu i edinstvennost resheniya sootvetstvuyuschei zadachi Koshi”, Funkts. analiz, 4:2 (1970), 10–17 | MR | Zbl

[12] N. K. Askerov, S. G. Krein, G. I. Laptev, “Ob odnom klasse nesamosopryazhennykh zadach”, DAN SSSR, 155:3 (1964), 499–502 | MR | Zbl

[13] S. G. Krein, “O kolebaniyakh vyazkoi zhidkosti v sosude”, DAN SSSR, 159:2 (1964), 262–265 | MR

[14] N. K. Askerov, S. G. Krein, G. I. Laptev, “Zadacha o kolebaniyakh vyazkoi zhidkosti i svyazannye s nei operatornye uravneniya”, Funkts. analiz, 2:2 (1968), 21–31 | MR | Zbl

[15] D. E. Allakhverdiev, “O polnote sistemy sobstvennykh i prisoedinennykh elementov odnogo klassa nesamosopryazhennykh operatorov, zavisyaschikh ot parametra $\lambda$”, DAN SSSR, 160:6 (1965), 1231–1234 | Zbl

[16] G. V. Radzievskii, “O faktorizatsii trigonometricheskoi operator-funktsii”, Vestnik MGU, seriya matem., mekh., 1971, no. 4, 38–45 | MR | Zbl

[17] A. I. Markushevich, Teoriya analiticheskikh funktsii, tom II, izd-vo «Nauka», Moskva, 1968

[18] V. I. Matsaev, “Ob odnom metode otsenki rezolvent nesamosopryazhennykh operatorov”, DAN SSSR, 154:5 (1964), 1034–1037 | Zbl

[19] A. S. Markus, “Nekotorye priznaki polnoty sistemy kornevykh vektorov lineinogo operatora v banakhovom prostranstve”, Matem. sb., 70 (112) (1966), 526–561 | MR

[20] A. S. Markus, “K spektralnoi teorii polinomialnykh operatornykh puchkov v banakhovom prostranstve”, Sib. matem. zh., VIII:6 (1967), 1346–1369 | MR

[21] S. Bochner, R. S. Phillips, “Absolutely convergent Fourier expansion for noncommutative normed rings”, Ann. Math., 43:3 (1942), 409–418 | DOI | MR | Zbl

[22] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatoro, izd-vo «Nauka», Moskva, 1965

[23] R. J. Duffin, “A minimax theory for overdamped networks”, J. Rat. Mech. Anal., 4:2 (1955), 221–233 | MR | Zbl

[24] A. S. Markus, “O polnote chasti sobstvennykh i prisoedinennykh vektorov dlya nekotorykh nelineinykh spektralnykh zadach”, Funkts. analiz, 5:4 (1971), 78–79 | MR | Zbl

[25] A. Fridman, M. Shinbrot, “Nonlinear eigenvalue problems”, Acta math., 121:1–2 (1968), 77–125 | DOI | MR

[26] I. Ts. Gokhberg, M. G. Krein, Teoriya volterovykh operatorov v gilbertovom prostranstve i ee prilozheniya, izd-vo «Nauka», Moskva, 1967 | MR