Spectral decompositions in spaces of analytic functionals

V. A. Tkachenko

Izvestiya. Mathematics , Tome 14 (1980) no. 3, pp. 597-651.

Résumé

This article deals with spaces of entire vector-valued functions determined by families of radial majorants, and their dual spaces of analytic functionals. Spectral properties of generalized boundary value problems generated by the operator of multiplication by the independent variable are investigated. Bibliography: 64 titles.

Bibliographie

[1] Birkhoff G. D., “Boundary value and expansion problems of ordinary linear diffirential equations”, Trans. Amer. Math. Soc., 9:4 (1908), 373–395 | DOI | MR | Zbl

[2] Birkhoff G. D., “Note on the expansion problems of ordinary linear differential equations”, Rendiconti Palermo, 36 (1913), 115–126 | DOI | MR | Zbl

[3] Tamarkin Ya. D., “Sur quelques points de la theorie des equations differentielles”, Rendiconti Palermo, 34 (1912), 345–382 | DOI | Zbl

[4] Tamarkin Ya. D., O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii, Petrograd, 1917 | Zbl

[5] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[6] Leontev A. F., “O posledovatelnostyakh lineinykh agregatov, obrazovannykh iz reshenii differentsialnykh uravnenii”, Izv. AN SSSR. Ser. matem., 22 (1958), 201–242 | MR

[7] Leontev A. F., “O svoistvakh posledovatelnostei lineinykh agregatov, obrazovannykh iz polinomov Yakobi, i ikh primenenii k voprosu o polnote sistemy polinomov Yakobi”, Izv. AN SSSR. Ser. matem., 23 (1959), 565–594 | MR

[8] Leontev A. F., “K voprosu o posledovatelnostyakh lineinykh agregatov, obrazovannykh iz reshenii differentsialnykh uravnenii”, Matem. sb., 48:2 (1959), 129–136 | MR

[9] Leontev A. F., “O posledovatelnostyakh lineinykh agregatov, sostavlennykh iz reshenii $y(z,\lambda)$ obyknovennogo differentsialnogo uravneniya $Dy=\lambda y$”, Issledovaniya po sovremennym problemam teorii funktsii kompleksnogo peremennogo, M., 1960, 195–206 | MR

[10] Frolov Yu. N., “O svoistvakh reshenii odnogo funktsionalnogo uravneniya”, Matem. sb., 69:4 (1966), 605–615 | Zbl

[11] Frolov Yu. N., “Ob odnom metode resheniya operatornogo uravneniya beskonechnogo poryadka”, Matem. sb., 89:3 (1972), 461–474 | Zbl

[12] Frolov Yu. N., “Ob odnoi biortogonalnoi sisteme funktsii”, Sib. matem. zh., 14:4 (1973), 835–846 | Zbl

[13] Frolov Yu. N., “K probleme edinstvennosti”, Matem. zametki, 16:6 (1974), 871–878 | Zbl

[14] Matsaev V. I., Teoremy edinstvennosti, polnoty i kompaktnosti, svyazannye s klassicheskoi kvazianalitichnostyu, Dissertatsiya, Kharkov, 1964 | MR

[15] Matsaev V. I., “O razlozhenii tselykh funktsii po sobstvennym i prisoedinennym funktsiyam obobschennoi kraevoi zadachi”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, 16 (1972), 198–206 | Zbl

[16] Khromov A. P., “Ob odnom primenenii operatorov preobrazovaniya M. K. Fage–A. F. Leonteva”, Obyknovennye differentsialnye uravneniya i ryady Fure, Saratovsk. gos. un-t, 1971, 16–31

[17] Khromov A. P., “Ob odnom predstavlenii yader rezolvent volterrovskikh operatorov i ego primeneniyakh”, Matem. sb., 89:2 (1972), 207–226 | Zbl

[18] Khromov A. P., “Konechnomernye vozmuscheniya volterrovskikh operatorov”, Dokl. AN SSSR, 209:2 (1973), 309–311 | Zbl

[19] Tkachenko V. A., “O razlozhenii tseloi funktsii konechnogo poryadka po kornevym funktsiyam odnogo differentsialnogo operatora”, Matem. sb., 89:4 (1972), 558–568 | Zbl

[20] Khërmander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[21] Ehrenpreis L., “Analytically uniform spaces and some applications”, Trans. Amer. Math. Soc., 101:1 (1961), 52–74 | DOI | MR | Zbl

[22] Ehrenpreis L., Fourier analysis in several complex variables, Wiley-Interscience, 1970 | MR | Zbl

[23] Palamodov V. P., Lineinye differentsialnye operatory s postoyannymi koeffitsientami, Nauka, M., 1967 | MR

[24] Krasnikov I. F., “Polnota v prostranstvakh kompleksnoznachnykh funktsii, opisyvaemykh povedeniem modulya”, Matem. sb., 68:1 (1965), 26–57

[25] Krasnikov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii”, Izv. AN SSSR. Ser. matem., 31 (1967), 37–60

[26] Krasnikov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii, II”, Izv. AN SSSR. Ser. matem., 32 (1968), 1024–1032

[27] Krasnikov I. F., “O zamknutykh idealakh v lokalno vypuklykh algebrakh tselykh funktsii. Algebry minimalnogo tipa”, Sib. matem. zh., 9:1 (1968), 77–86

[28] Martineau A., “Equations differentielles d'orde infini”, Bull. Soc. Math. France, 95 (1967), 109–15 | MR

[29] Taylor B. A., “Some locally convex spaces of entire functions”, Entire Funct. and Relat. Parts of Anal. (La Jolla, Calif. 1966), Proc. Symp. Pure Math., 11, 1968, 431–467 | MR | Zbl

[30] Rubel L. A., Taylor B. A., “Functional analysis proofs of some theorems in function theory”, Amer. Math. Monthly, 76:5 (1969), 483–485 | DOI | MR

[31] Ferrier I.-P., Spectral theory and complex analysis, North-Holland Publ. Comp., Amst., London, N.Y., 1970

[32] Khavin V. P., “Prostranstva analiticheskikh funktsii”, Itogi nauki. Matem. analiz 1964, VINITI, M., 1966, 76–164

[33] Gelfond A. O., Leontev A. F., “Ob odnom obobschenii ryada Fure”, Matem. sb., 29:3 (1951), 477–500 | MR

[34] Leontev A. F., “Ob odnom svoistve edinstvennosti”, Matem. sb., 72:2 (1967), 237–249 | MR

[35] Leontev A. F., “K voprosu o predstavlenii proizvolnoi tseloi funktsii ryadami Dirikhle i drugimi bolee obschimi ryadami”, Izv. AN ArmSSR, ser. matem., 2:5 (1967), 295–317 | MR

[36] Leontev A. F., “O klasse funktsii, opredelyaemykh ryadami polinomov Dirikhle”, Uspekhi matem. nauk, 3:4 (1948), 176–180 | MR

[37] Leontev A. F., “O funktsiyakh, opredelennykh ryadami polinomov Dirikhle”, Izv. AN SSSR. Ser. matem., 13 (1949), 221–230 | MR

[38] Gelfond A. O., “Lineinye differentsialnye uravneniya beskonechnogo poryadka i asimptoticheskie periody tselykh funktsii”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 38, 1951, 42–67 | MR

[39] Leontev A. F., Ryady polinomov Dirikhle i ikh obobscheniya, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 39, 1951 | MR | Zbl

[40] Leontev A. F., “O svoistvakh posledovatelnostei lineinykh agregatov, skhodyaschikhsya v oblasti, gde porozhdayuschaya lineinye agregaty sistema funktsii ne yavlyaetsya polnoi”, Uspekhi matem. nauk, 11:5 (1956), 26–37 | MR

[41] Leontev A. F., “O posledovatelnostyakh polinomov Dirikhle”, Trudy III Vsesoyuzn. matem. s'ezda, M., 1958, 218–226

[42] Leontev A. F., “Differentsialnye uravneniya beskonechnogo poryadka i ikh primeneniya”, Tr. IV Vsesoyuzn. matem. s'ezda, v. II, L., 1964, 648–660 | MR

[43] Leontev A. F., “Predstavlenie funktsii obobschennymi ryadami Dirikhle”, Uspekhi matem. nauk, 24:2 (1969), 97–164 | MR

[44] Leontev A. F., Ryady eksponent, Nauka, M., 1976 | MR

[45] Khromov A. P., “Operator differentsirovaniya i ryady tipa Dirikhle”, Matem. zametki, 6:6 (1969), 759–766

[46] Khromov A. P., “O predstavlenii tselykh funktsii ryadami tipa Dirikhle”, Issledovaniya po differentsialnym uravneniyam i teorii funktsii, no. 3, Saratov, 1971, 24–31 | Zbl

[47] Schwartz L., “Théorie génerate des fonctions moyennes periodiques”, Ann. Math., 48:4 (1947), 857–929 | DOI | MR | Zbl

[48] Rashevskii P. K., “O zamknutykh idealakh v odnoi schetno-normirovannoi algebre tselykh funktsii”, Dokl. AN SSSR, 162:3 (1965), 513–515

[49] Nikolskii N. K., “Zamknutye idealy v nekotorykh algebrakh tselykh funktsii”, Sib. matem. zh., 9:1 (1968), 211–215

[50] Kelleher J., Taylor B. A., “Closed ideals in locally convex algebras of analytic functions”, J. Reine und Angew. Math., 255 (1972), 190–209 | MR | Zbl

[51] Nikolskii N. K., “Invariantnye podprostranstva v teorii operatorov i teoriya funktsii”, Itogi nauki. Matem. analiz, 12, VINITI, M., 1974, 199–412

[52] Delsartes J., Lions J.-L., “Transmutations d'operateur differentielles dans le domain complexes”, Comm. Helv., 32:2 (1957), 113–128 | MR

[53] Fage M. K., “Operatorno-analiticheskie funktsii odnoi nezavisimoi peremennoi”, Dokl. AN SSSR, 112:6 (1957), 1008–1011 | MR | Zbl

[54] Fage M. K., “Operatorno-analiticheskie funktsii odnoi nezavisimoi peremennoi”, Tr. Mosk. matem. ob-va, 7, 1958, 227–268 | MR | Zbl

[55] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Nauka, M., 1956

[56] Leontev A. F., “Otsenka rosta resheniya odnogo differentsialnogo uravneniya pri bolshikh po modulyu znacheniyakh parametra i ee primeneniya k nekotorym voprosam teorii funktsii”, Sib. matem. zh., 1:3 (1960), 456–487 | MR

[57] Frolov Yu. N., “Ob odnom obobschenii ryada Lorana”, Matem. zametki, 9:5 (1971), 521–531 | MR | Zbl

[58] Pich A., Yadernye lokalno vypuklye prostranstva, Mir, M., 1967 | MR

[59] Leontev A. F., “Ob odnom funktsionalnom uravnenii”, Izv. AN SSSR. Ser. matem., 29 (1965), 725–756 | MR

[60] Kantorovich L. V., Akilov G. I., Funktsionalnyi analiz v normirovannykh prostranstvakh, GIFML, M., 1959 | MR

[61] Leontev A. F., “Ob odnom sposobe resheniya uravneniya beskonechnogo poryadka”, Izv. AN SSSR. Ser. matem., 32 (1968), 687–708 | MR

[62] Tkachenko V. A., “Ob operatore preobrazovaniya v prostranstve tselykh funktsii konechnogo poryadka”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, no. 22, 1975, 154–159 | Zbl

[63] Sebastiao-e-Silva J., “Sur certe classi di spazi localmente convessi importanti per le applicationi”, Rend. di Math. et del. sue Appl., Roma, 5(14) (1955), 388–410 | MR

[64] Tkachenko V. A., “O edinstvennosti razlozheniya po kornevym funktsiyam odnogo differentsialnogo operatora”, Matem. sb., 92:3 (1973), 462–471