Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts
Compositio Mathematica, Volume 57 (1986) no. 1, p. 15-60
@article{CM_1986__57_1_15_0,
     author = {Chojnacki, Wojciech},
     title = {Fonctions cosinus hilbertiennes born\'ees dans les groupes commutatifs localement compacts},
     journal = {Compositio Mathematica},
     publisher = {Martinus Nijhoff Publishers},
     volume = {57},
     number = {1},
     year = {1986},
     pages = {15-60},
     zbl = {0591.47030},
     mrnumber = {817295},
     language = {fr},
     url = {http://www.numdam.org/item/CM_1986__57_1_15_0}
}
Chojnacki, Wojciech. Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts. Compositio Mathematica, Volume 57 (1986) no. 1, pp. 15-60. http://www.numdam.org/item/CM_1986__57_1_15_0/

[1] J.A. Baker: D'Alembert's functional equation in Banach algebras. Acta Sci. Math. (Szeged) 32 (1971) 225-234. | MR 322384 | Zbl 0232.46049

[2] J.A. Baker et K.R. Davidson: Cosine, exponential and quadratic functions. Glas. Math. Ser. III 16 (1981) 269-274. | MR 653056 | Zbl 0484.39005

[3] A.O. Barut et R. Raczka: Theory of Group Representations and Applications. Warszawa: Polish Scientific Publishers (1980). | MR 495836 | Zbl 0471.22021

[4] J.J. Benedetto: Spectral Synthesis. Stuttgart: Teubner (1975). | MR 622037 | Zbl 0314.43011

[5] N. Bourbaki: Eléments de mathématique, Intégration, Chapitre V: Intégration des mesures. Paris: Hermann (1967). | MR 209424 | Zbl 0143.27101

[6] A.B. Buche: On the cosine-sine operator functional equations. Aequations Math. 6 (1971) 231-234. | MR 291872 | Zbl 0232.47044

[7] R. Chandler et H. Singh: On the measurability and continuity properties of the cosine operator. Indian J. Pure Appl. Math. 12 (1981) 81-83. | MR 617855 | Zbl 0454.47020

[8] G. Da Prato et E. Giusti: Una caratterizzazione dei generatori di funzioni coseno astratte. Boll. Un. Mat. Ital. 22 (1967) 357-362. | MR 240672 | Zbl 0186.47702

[9] M.M. Day: Ergodic theorems for abelian semigroups. Trans. Amer. Math. Soc. 51 (1942) 399-412. | JFM 68.0246.01 | MR 6614 | Zbl 0063.01057

[10] J. Dixmier: Les moyennes invariantes dans les semi-groupes et leurs applications. Acta Sci. Math. (Szeged) 12 A (1950) 213-227. | MR 37470 | Zbl 0037.15501

[11] R. Doss: Aproximation and representations for Fourier transforms. Trans. Amer. Math. Soc. 153 (1971) 211-221. | MR 268597 | Zbl 0208.37901

[12] N. Dunford et J.T. Schwartz: Linear Operators, Part III: Spectral Operators. New York: Wiley-Interscience (1971). | MR 412888 | Zbl 0128.34803

[13] W.F. Eberlein: A note on Fourier-Stieltjes transforms. Proc. Amer. Math. Soc. 6 (1955) 310-312. | MR 68030 | Zbl 0065.01604

[14] H.O. Fattorini: Ordinary differential equations in linear topological spaces. I. J. Differential Equations 5 (1968) 72-105. | Zbl 0175.15101

[15] H.O. Fattorini: Ordinary differential equations in linear topological spaces. II. J. Differential equations 6 (1969) 50-70. | Zbl 0181.42801

[16] H.O. Fattorini: Uniformly bounded cosine functions in Hilbert space. Indiana Univ. Math. J. 20 (1970) 411-425. | MR 267417 | Zbl 0185.38501

[17] H.O. Fattorini: Some remarks on second-order abstract Cauchy problems. Funkcial. Ekvac. 24 (1981) 331-344. | MR 655504 | Zbl 0488.34051

[18] G.D. Faulkner et R.W. Shonkwiler: Cosine representations of abelian *-semigroups and generalized cosine operator functions. Canad. J. Math. 30 (1978) 474-482. | MR 500282 | Zbl 0358.47021

[19] B. Fuglede: A commutativity theorem for normal operators. Proc. Amer. Math. Soc. 36 (1950) 35-40. | MR 32944 | Zbl 0035.35804

[20] E. Giusti: Funzioni coseno periodiche. Boll. Un. Mat. Ital. 22 (1967) 478-485. | MR 233237 | Zbl 0182.19304

[21] J.A. Goldstein: Semigroups and second-order differential equations. J. Funct. Anal. 4 (1969) 50-70. | MR 254668 | Zbl 0179.14605

[22] J.A. Goldstein: On a connection between first and second order differential equations in Banach spaces. J. Math. Anal. Appl. 30 (1970) 246-251. | MR 257532 | Zbl 0197.12106

[23] J.A. Goldstein: On the convergence and approximation of cosine functions. Aequationes Math. 10 (1974) 201-205. | MR 358435 | Zbl 0282.47012

[24] J.A. Goldstein, Ch. RADIN et R.E. Showalter: Convergence rates of ergodic limits for semigroups and cosine functions. Semigroup Forum 16 (1978) 89-95. | MR 487585 | Zbl 0393.47004

[25] C.C. Graham et O.C. Mcgehee: Essays in Commutative Harmonic Analysis. New York: Springer (1979). | MR 550606 | Zbl 0439.43001

[26] C.C. Graham et L.T. Ramsey: Three results on I-sets, Colloq. Math. (à paraître). | MR 891281 | Zbl 0621.43009

[27] F.P. Greenleaf: Invariant Means on Topological Groups and Their Applications. New York: Van Nostrand (1969). | MR 251549 | Zbl 0174.19001

[28] E. Hewitt et K.A. Ross: Abstract Harmonic Analysis, Volumes I et II. Berlin: Springer (1963 et 1970). | MR 262773 | Zbl 0213.40103

[29] E. Hewitt et K. Stromberg: Real and Abstract Analysis. Berlin: Springer (1969). | MR 367121 | Zbl 0225.26001

[30] Pl. Kannappan: The functional equation f(xy)+f(xy-1)=2f(x)f(y) for groups. Proc. Amer. Math. Soc. 19 (1968) 69-74. | MR 219936 | Zbl 0169.48102

[31] J. Kisyński: On second order Cauchy's problem in a Banach space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970) 371-374. | MR 269958 | Zbl 0202.10102

[32] J. Kisyński: On operator-valued solutions of d'Alembert's functional equation I. Colloq. Math. 23 (1971) 107-114. | MR 313875 | Zbl 0239.39004

[33] J. Kisyński: On operator-valued solutions of d'Alembert's functional equation II. Studia Math. 42 (1972) 43-66. | MR 313876 | Zbl 0239.39005

[34] J. Kisyński: On cosine operator functions and one-parameter groups of operators. Studia Math. 44 (1972) 93-105. | MR 312328 | Zbl 0216.42104

[35] Y. Konishi: Cosine functions of operators in locally convex spaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971/72) 443-463. | MR 324479 | Zbl 0239.47034

[36] S. Kurepa: A cosine functional equation in n-dimensional vector space. Glasnik Mat.-Fiz. Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 13 (1958) 169-189. | MR 100728 | Zbl 0082.32802

[37] S. Kurepa: A cosine functional equation in Hilbert space. Canad. J. Math. 19 (1960) 45-50. | MR 109265 | Zbl 0090.10001

[38] S. Kurepa: On some functional equations in Banach spaces. Studia Math. 19 (1960) 149-158. | MR 118986 | Zbl 0100.32702

[39] S. Kurepa: A cosine functional equation in Banach algebras. Acta Sci. Math. (Szeged) 23 (1962) 255-267. | MR 145370 | Zbl 0113.31702

[40] S. Kurepa: Uniformly bounded cosine function in a Banach space. Math. Balkanica 2 (1972) 109-115. | MR 320819 | Zbl 0261.46050

[41] S. Kurepa: Weakly measurable selfadjoint cosine function. Glas. Mat. Ser. III 8 (1973) 73-79. | MR 333825 | Zbl 0262.47011

[42] S. Kurepa: Decomposition of weakly measurable semigroups and cosine operator functions. Glas. Mat. Ser. III 11 (1976) 91-95. | MR 407659 | Zbl 0329.47013

[43] S. Kurepa: Semigroups and cosine functions. Functional Analysis; Proceedings of a conference held at Dubrovnik, Yugoslavia, 1981. Lecture Notes in Math. 948 (1982) 48-72. | MR 672792 | Zbl 0513.47028

[44] D. Lutz: Which operators generate cosine operator functions? Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 63 (1978) 314-317. | MR 548593 | Zbl 0435.47044

[45] D. Lutz: Compactness properties of operator cosine functions. C. R. Math. Rep. Acad. Sci. Canada 2 (1980) 277-280. | MR 600561 | Zbl 0448.47021

[46] D. Lutz: Über operatorwertige Lösungen der Funktionalgleichung des Cosinus. Math. Z. 171 (1980) 233-245. | MR 575242 | Zbl 0414.47021

[47] D. Lutz: Periodische operatorwertige Cosinusfunktionen. Resultate Math. 4 (1981) 75-83. | MR 625115 | Zbl 0461.47021

[48] D. Lutz: On bounded time-dependent perturbations of operator cosine functions. Aequationes Math. 23 (1981) 197-203. | MR 689033 | Zbl 0512.34047

[49] D. Lutz: Strongly continuous operator cosine functions. Functional Analysis; Proceedings of a conference held at Dubrovnik, Yugoslavia, 1981. Lecture Notes in Math. 948 (1982) 73-97. | MR 672793 | Zbl 0493.47020

[50] D. Lutz: Über die Konvergenz operatorwertiger Cosinusfunktionen mit gestortem infinitesimalen Erzeuger. Period. Math. Hungar. 14 (1983) 101-105. | MR 697362 | Zbl 0489.47025

[51] G. Maltese: Spectral representations for solutions of certain abstract functional equations. Compositio Math. 15 (1962) 1-22. | Numdam | MR 152901 | Zbl 0108.11602

[52] G. Maltese: Spectral representations for some unbounded normal operators. Trans. Amer. Math. Soc. 110 (1964) 79-87. | MR 156209 | Zbl 0134.31903

[53] S.A. Morris: Pontryagin Duality and the Structure of Locally Compact Abelian Groups. Cambridge: Cambridge University Press (1977). | MR 442141 | Zbl 0446.22006

[54] B. Nagy: On the generators of cosine operator functions. Publ. Math. Debrecen 21 (1974) 151-154. | MR 361926 | Zbl 0314.47018

[55] B. Nagy: On cosine operator functions in Banach spaces. Acta Sci. Math. (Szeged) 36 (1974) 281-289. | MR 374995 | Zbl 0273.47008

[56] B. Nagy: Cosine operator functions and the abstract Cauchy problem. Period. Math. Hungar. 7 (1976) 213-217. | MR 450730 | Zbl 0329.34051

[57] B. Nagy: Approximation theorems for cosine operator functions. Acta Math. Acad. Sci. Hungar. 29 (1977) 69-76. | MR 440419 | Zbl 0346.47041

[58] S. Nelson et R. Triggiani: Analytic properties of cosine operators. Proc. Amer. Math. Soc. 74 (1979) 101-104. | MR 521880 | Zbl 0412.47023

[59] C.R. Putnam: On normal operator in Hilbert space. Amer. J. Math. 73 (1951) 357-362. | MR 40585 | Zbl 0042.34501

[60] S.M. Rankin Iii: A remark on cosine families. Proc. Amer. Math. Soc. 79 (1980) 376-378. | MR 567976 | Zbl 0456.47033

[61] M. Rosenblum: On a theorem of Fuglede and Putnam. J. London Math. Soc. 33 (1958) 376-377. | MR 99598 | Zbl 0081.11902

[62] N. Sarapa: A note on the cosine equation for probability on compact semigroups. Glas. Mat. Ser. III 15 (1980) 383-385. | MR 608888 | Zbl 0466.60012

[63] M. Sova: Cosine operator functions. Dissertationes Math. (Rozprawy Mat.) 49 (1966) 1-47. | MR 193525 | Zbl 0156.15404

[64] M. Sova: Semigroups and cosine functions of normal operators in Hilbert spaces. Časopis Pěst. Mat. 93 (1968) 437-458. | MR 250121 | Zbl 0172.40606

[65] T. Takenaka et N. Okazawa: A Phillips-Miyadera type perturbation theorem for cosine functions of operators. Tôhuku Math. J. 30 (1978) 107-115. | MR 470751 | Zbl 0398.47030

[66] C.C. Travis et G.F. Webb: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houston J. Math. 3 (1977) 555-567. | MR 500288 | Zbl 0386.47024

[67] C.C. Travis et G.F. Webb: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. 32 (1978) 75-96. | MR 499581 | Zbl 0388.34039

[68] C.C. Travis et G.F. Webb: An abstract second order semilinear Volterra integrodifferential equation. SIAM J. Math. Anal. 10 (1978) 412-424. | MR 523855 | Zbl 0406.45014

[69] S. Tymowski: On the analogue of the formula cos t = 12eit + 12e-it for operator cosine functions. Comment. Math. Prace Mat. 23 (1983) 173-182. | MR 709186 | Zbl 0594.47035

[70] F. Vajzović: Einige Funktionalgleichungen im Fréchetschen Raum. Glas. Mat. Ser. III 3 (1968) 19-40. | MR 234159 | Zbl 0162.46302

[71] J. Wermer: Commuting spectral measures on Hilbert space. Pacific J. Math. 4 (1954) 355-361. | MR 63564 | Zbl 0056.34701