Harmonic analysis of causal operators and their spectral properties

A. G. Baskakov ; I. A. Krishtal

Izvestiya. Mathematics , Tome 69 (2005) no. 3, pp. 439-486.

Résumé

The definition and study of causal operators are based on the representation theory of group algebras. We study the structure of the spectra of causal operators, obtain conditions for causal invertibility and state criteria for a causal operator to belong to the radical.

Bibliographie

[1] Feintuch A., Saeks R., System Theory. A Hilbert Space Approach, Academic press, N. Y. –London, 1982 | MR | Zbl

[2] Kurbatov V. G., Lineinye differentsialno-raznostnye uravneniya, Izd-vo Voronezh. un-ta, Voronezh, 1990 | MR | Zbl

[3] Kurbatov V. G., Functional Differential Operators and Equations, Kluwer Academic Publishers, Dordrech–Boston–London, 1999 | MR | Zbl

[4] Kolmanovskii V., Myshkis A., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrech–Boston–London, 1999 | MR | Zbl

[5] Azbelev N. V., Maksimov V. P., Rakhmatullina L. F., Metody sovremennoi teorii lineinykh funktsionalno-differentsialnykh uravnenii, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2000

[6] Gokhberg I Ts., Krein M G., Teoriya volterrovykh operatorov v gilbertovom prostranstve i ee prilozheniya, Nauka, M., 1967 | MR

[7] Rudin W., Fourier Analysis on Groups, Int. Publ., N. Y., 1962 | MR | Zbl

[8] Hewitt E., Ross K.-A., Abstract Harmonic Analysis, V. II, Springer-Verlag, Berlin–N. Y., 1977 | Zbl

[9] Gelfand I. M., Raikov D. A., Shilov G. E., Kommutativnye normirovannye koltsa, Nauka, M., 1960

[10] Baskakov A. G., Garmonicheskii analiz lineinykh operatorov, Izd-vo Voronezh. un-ta, Voronezh, 1987 | MR | Zbl

[11] Danford N., Shvarts Dzh. T., Lineinye operatory. Spektralnye operatory, T. 3, Mir, M., 1966

[12] Bratteli U., Robinson D., Operatornye algebry i kvantovaya statisticheskaya mekhanika, Mir, M., 1982 | MR | Zbl

[13] Domar Y., “Harmonic Analysis in Certain Commutative Banach Algebras”, Acta Math., 96 (1956), 1–66 | DOI | MR | Zbl

[14] Domar Y., Lindahl L.-A., “Three Spectral notions for representations of commutative Banach algebras”, Ann. Inst. Fourier, 25:2 (1975), 1–32 | MR | Zbl

[15] Lyubich Yu. I., Matsaev V. I., Feldman G. M., “O predstavleniyakh s otdelimym spektrom”, Funktsion. analiz i ego prilozh., 7:2 (1973), 52–61 | MR | Zbl

[16] Baskakov A. G., “K spektralnomu analizu ogranichennykh predstavlenii lokalno kompaktnykh abelevykh grupp”, Tr. matem. f-ta VGU, 1973, no. 10, 103–108 | MR

[17] Baskakov A. G., Nekotorye voprosy teorii vektornykh pochti periodicheskikh funktsii, Dis. $\dots$ kand. fiz.-mat. nauk, VGU, Voronezh, 1973

[18] Arveson W., “On Group of Automorphisms of Operator Algebras”, J. Funct. Anal., 15 (1974), 217–243 | DOI | MR | Zbl

[19] Baskakov A. G., “Neravenstva bernshteinovskogo tipa v abstraktnom garmonicheskom analize”, Sib. matem. zhurn., 20:5 (1979), 942–952 | MR | Zbl

[20] Marshall E., “On the functional calculus of non-quasianalytic groups of operators and cosine functions”, Rend. Circ. Mat. Palermo, 35 (1986), 58–81 | DOI | MR

[21] Nagel R., Huang S.-Z., “Spectral mapping theorems for $C_0$-groups satisfying non-quasianalytic growth conditions”, Math. Nachr., 169 (1994), 207–218 | MR | Zbl

[22] Baskakov A. G., Garmonicheskii analiz lineinykh operatorov, Dis. ... dokt. fiz.-mat. nauk, In-t matematiki, Kiev, 1987

[23] Baskakov A. G., “Ob obschikh ergodicheskikh teoremakh v banakhovykh modulyakh”, Funktsion. analiz i ego prilozh., 14:3 (1980), 63–64 | MR | Zbl

[24] Baskakov A. G., “Operatornye ergodicheskie teoremy i dopolnyaemost podprostranstv banakhovykh prostranstv”, Izv. vuzov. Matematika, 11 (1988), 3–11 | MR | Zbl

[25] Baskakov A. G., “Spektralnye kriterii pochti periodichnosti reshenii funktsionalnykh uravnenii”, Matem. zametki, 24:2 (1978), 195–206 | MR

[26] Lyubich Yu. I., Vvedenie v teoriyu banakhovykh predstavlenii, Vischa shkola, Kharkov, 1985 | MR | Zbl

[27] Sheffer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR

[28] Baskakov A. G., “Metody abstraktnogo garmonicheskogo analiza v teorii vozmuschenii lineinykh operatorov”, Sib. matem. zhurn., 24:1 (1983), 21–39 | MR

[29] Baskakov A. G., “Otsenki elementov obratnykh matrits i spektralnyi analiz lineinykh operatorov”, Izv. RAN. Ser. matem., 61:6 (1997), 3–26 | MR | Zbl

[30] Baskakov A. G., “Asimptoticheskie otsenki elementov matrits obratnykh operatorov i garmonicheskii analiz”, Sib. matem. zhurn., 38:1 (1997), 14–28 | MR | Zbl

[31] Diestal I., Uhl J. I., Vector measures, Mathematical Surveys and Monographs, 15, Amer. Math. Soc., 1977 | MR

[32] Burbaki N., Spektralnaya teoriya, Mir, M., 1972 | MR | Zbl

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

[34] Zabreiko P. P., “O spektralnom radiuse operatora Volterra”, Matematikos Rinkinus, 7:2 (1967), 281–287 | MR | Zbl

[35] Evgrafov M. A., Asimptoticheskie otsenki i tselye funktsii, Fizmatlit, M., 1962 | MR

[36] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR

[37] Baskakov A G., “Polugruppy raznostnykh operatorov v spektralnom analize lineinykh differentsialnykh operatorov”, Funktsion. analiz i ego prilozh., 30:3 (1996), 1–11 | MR | Zbl

[38] Chicone C., Latushkin Y., Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999 | MR | Zbl

[39] Mlak W., “Notes on Circular Operators. I; II; III; IV”, Univ. Iagel. Acta Math., 29 (1992), 145–152 ; 153–162 ; 163–170 ; 171–175 | MR | Zbl | Zbl | Zbl

[40] Baskakov A. G., Chernyshov K. I., “Spektralnyi analiz lineinykh otnoshenii i vyrozhdennye polugruppy operatorov”, Matem. sb., 193:11 (2002), 3–42 | MR | Zbl