Geodesic
On a class of unconditional bases in Hilbert spaces and on the problem of similarity of dissipative Volterra operators
Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 93-126 
G. M. Gubreev. On a class of unconditional bases in Hilbert spaces and on the problem of similarity of dissipative Volterra operators. Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 93-126. http://geodesic.mathdoc.fr/item/SM_1994_77_1_a6/
@article{SM_1994_77_1_a6,
     author = {G. M. Gubreev},
     title = {On a class of unconditional bases in {Hilbert} spaces and on the~problem of similarity of dissipative {Volterra} operators},
     journal = {Sbornik. Mathematics},
     pages = {93--126},
     year = {1994},
     volume = {77},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_77_1_a6/}
}
TY  - JOUR
AU  - G. M. Gubreev
TI  - On a class of unconditional bases in Hilbert spaces and on the problem of similarity of dissipative Volterra operators
JO  - Sbornik. Mathematics
PY  - 1994
SP  - 93
EP  - 126
VL  - 77
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1994_77_1_a6/
LA  - en
ID  - SM_1994_77_1_a6
ER  - 
%0 Journal Article
%A G. M. Gubreev
%T On a class of unconditional bases in Hilbert spaces and on the problem of similarity of dissipative Volterra operators
%J Sbornik. Mathematics
%D 1994
%P 93-126
%V 77
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1994_77_1_a6/
%G en
%F SM_1994_77_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $B$ be a completely nonselfadjoint dissipative Volterra operator acting in a separable Hilbert space $\mathfrak Y$ whose resolvent $(I-\lambda B)^{-1}$ has finite exponential type. Further, let $\mathfrak{L}=(B-B^*)\mathfrak Y$, $y\in\mathfrak{L}$, and $y(\lambda)=(I-\lambda B)^{-1}y$. In this article conditions are determined on the operator $B$, the vector $y$, and the sequence $\Lambda=\{\lambda_k\}_{-\infty}^{+\infty}$ under which the family $$ \{y(\lambda_k):\lambda_k\in \Lambda\}, \qquad \inf_{\lambda_k}\operatorname{Im}\lambda_k>0, $$ forms an unconditional basis in the space $\mathfrak Y$. Moreover, a new approach is considered for the problem of similarity of dissipative Volterra operators, based on a study of the basis properties of this system of vectors.

[1] Brodskii M. S., Treugolnye i zhordanovy predstavleniya lineinykh operatorov, Nauka, M., 1969 | MR

[2] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[3] Sekefalvi-Nad B., Foyash Ch., Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M, 1970 | MR

[4] Pavlov B. S., “Bazisnost sistem eksponent i uslovie Makenkhaupta”, DAN SSSR, 247:1 (1979), 37–40 | MR | Zbl

[5] Hruščev S. V., Nikol'skii N. K., Pavlov B. S., “Unconditional base of exponentials and reproducing kernels”, Lect. Notes in Math., 864, 1981, 214–335 | MR

[6] Nikolskii N. K., “Bazisy iz eksponent i znachenii vosproizvodyaschikh yader”, DAN SSSR, 252:6 (1980), 1316–1320 | MR | Zbl

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

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

[9] Hruščev S. V., “Unconditional basis in $L_2(0,a) $”, Proc. Amer. Math. Soc., 99:4 (1987), 651–656 | DOI | MR | Zbl

[10] Gubreev G. M., “Spektralnyi analiz biortogonalnykh razlozhenii, porozhdaemykh vesami Makenkhaupta”, Zapiski nauch. sem. LOMI, ed. N. K. Nikolskii, Nauka, L., 1991, 34–80

[11] Gubreev G. M., “Integralnye preobrazovaniya tipa Dzhrbashyana i interpolyatsiya tselymi funktsiyami konechnogo poryadka”, Izv. AN ArmSSR. Matematika, 25:1 (1990), 91–98 | MR

[12] Gubreev G. M., “Obobschennye preobrazovaniya Dzhrbashyana i ikh primenenie”, Izv. AN ArmSSR. Matematika, 21:3 (1986), 306–310 | MR | Zbl

[13] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavlenie funktsii v kompleksnoi oblasti, Nauka, M., 1966

[14] Sakhnovich L. A., “O dissipativnykh volterrovskikh operatorakh”, Matem. sb., 76(118) (1968), 323–343 | MR | Zbl

[15] Sakhnovich L. A., “O podobii operatorov”, Sib. matem. zhurn., 13:4 (1972), 868–883 | MR | Zbl

[16] Gubreev G. M., “Bazisnost semeistv funktsii tipa Mittag-Lefflera, preobrazovaniya Dzhrbashyana i uslovie Makenkhaupta”, Funktsion. analiz i ego pril., 21:4 (1987), 71–72 | MR

[17] Gubreev G. M., “Bazisnost semeistv funktsii tipa Mittag-Lefflera, preobrazovaniya Dzhrbashyana i vesovye otsenki integralov tipa Koshi”, Izv. AN ArmSSR. Matematika, 23:3 (1988), 237–269 | MR | Zbl

[18] Gubreev G. M., “Spektralnyi analiz biortogonalnykh razlozhenii funktsii v ryady eksponent”, Izv. AN SSSR. Ser. matem., 53:6 (1989), 1236–1268

[19] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[20] Hunt R. A., Muckenhoupt B., Wheeden R. L., “Weighted norm inequalities for the conjugate function and Hilbert transform”, Trans. Amer. Math. Soc., 176 (1973), 227–251 | DOI | MR | Zbl

[21] Kisilevskii G. E., “O kriteriyakh odnokletochnosti dissipativnykh volterrovykh operatorov”, Funktsion. analiz i ego pril., 1:3 (1967), 90–91 | MR | Zbl

[22] Sakhnovich L. A., “O privedenii volterrovskikh operatorov k prosteishemu vidu i obratnykh zadachakh”, Izv. AN SSSR. Ser. matem., 21:2 (1957), 235–262 | MR

[23] Sakhnovich L. A., “Spektralnyi analiz operatorov vida $Kf=\int ^x_0k(x-t)f(t)\,dt $”, Izv. AN SSSR. Ser. matem., 22:2 (1958), 299–308

[24] Malamud M. M., “O vozmuscheniyakh operatora drobnogo integrirovaniya”, Funktsion. analiz i ego pril., 13:2 (1979), 85–86 | MR | Zbl

[25] Malamud M. M., “Spektralnyi analiz volterrovykh operatorov s yadrom, zavisyaschim ot raznosti argumentov”, Ukr. matem. zhurn., 32:5 (1980), 601–609 | MR | Zbl

[26] Malamud M. M., “O spektre odnomernykh vozmuschenii volterrovykh operatorov”, DAN USSR. Ser. A, 1980, no. 7, 26–28 | MR | Zbl

[27] Malamud M. M., “Dva zamechaniya o podobii volterrovykh operatorov”, Matem. fizika i nelineinaya mekhanika, 1988, no. 9(43), 19–25 | MR

[28] Beitmen G., Erdeii A., Tablitsy integralnykh preobrazovanii, T. I, Nauka, M., 1969

[29] Adamyan V. M., Arov D. Z., “Ob unitarnykh stsepleniyakh poluunitarnykh operatorov”, Matem. issledovaniya, 1:2 (1966), 3–63 | MR