Invariant subspaces with no generator and a problem of H. Helson
[Sous-espaces invariants sans générateur et un problème de H. Helson]
Tanaka, Jun-ichi
Annales de l'Institut Fourier, Tome 65 (2015), p. 1469-1491 / Harvested from Numdam
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Dans le contexte presque périodique, aucun espace H 0 2 ne peut être engendré par un de ses éléments. En tenant compte d’un argument faisant intervenir les cocycles, on peut en déduire qu’il existe de nombreux types de sous-espaces invariants qui ne peuvent pas être engendrés par un seul de leurs éléments ; ceci permet de répondre à quelques questions de la théorie des sous-espaces invariants.

In the almost-periodic context, the H 0 2 -space cannot be generated by one of its elements. Together with a cocycle argument, this implies that there exist all kinds of invariant subspaces without a single generator, from which we answer some questions on invariant subspace theory.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2964
Classification:  43A17,  46J10,  46J15,  28D10
Mots clés: Groupes compactes avec dualité ordonnée, sous-espaces invariants, cocycles, générateurs singulier 
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     author = {Tanaka, Jun-ichi},
     title = {Invariant subspaces with no generator  and a problem of H. Helson},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {1469-1491},
     doi = {10.5802/aif.2964},
     zbl = {1332.43002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_4_1469_0}
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Tanaka, Jun-ichi. Invariant subspaces with no generator  and a problem of H. Helson. Annales de l'Institut Fourier, Tome 65 (2015) pp. 1469-1491. doi : 10.5802/aif.2964. http://gdmltest.u-ga.fr/item/AIF_2015__65_4_1469_0/

[1] Forelli, Frank Analytic and quasi-invariant measures, Acta Math., Tome 118 (1967), pp. 33-59 | Article | MR 209771 | Zbl 0171.34201

[2] Gamelin, T. W. H p spaces and extremal functions in H 1 , Trans. Amer. Math. Soc., Tome 124 (1966), pp. 158-167 | MR 213877 | Zbl 0146.37202

[3] Gamelin, Theodore W. Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J. (1969), pp. xiii+257 | MR 410387 | Zbl 0213.40401

[4] Helson, H. Analyticity on compact abelian groups, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973), Academic Press, London (1975), pp. 1-62 | MR 427959 | Zbl 0321.43009

[5] Helson, Henry Compact groups with ordered duals. VII, J. London Math. Soc. (2), Tome 20 (1979) no. 3, pp. 509-515 | Article | MR 561142 | Zbl 0475.43002

[6] Helson, Henry; Lowdenslager, David Prediction theory and Fourier series in several variables. II, Acta Math., Tome 106 (1961), pp. 175-213 | Article | MR 176287 | Zbl 0102.34802

[7] Helson, Henry; Parry, William Cocycles and spectra, Ark. Mat., Tome 16 (1978) no. 2, pp. 195-206 | Article | MR 524748 | Zbl 0401.28018

[8] Helson, Henry; Tanaka, Jun Singular cocycles and the generator problem, Operator theoretical methods (Timişoara, 1998), Theta Found., Bucharest (2000), pp. 173-186 | MR 1770322 | Zbl 0973.43004

[9] De Leeuw, K.; Glicksberg, I. Quasi-invariance and analyticity of measures on compact groups, Acta Math., Tome 109 (1963), pp. 179-205 | Article | MR 144152 | Zbl 0126.12102

[10] Moore, Calvin C.; Schmidt, Klaus Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson, Proc. London Math. Soc. (3), Tome 40 (1980) no. 3, pp. 443-475 | Article | MR 572015 | Zbl 0428.28014

[11] Muhly, Paul S. Function algebras and flows, Acta Sci. Math. (Szeged), Tome 35 (1973), pp. 111-121 | MR 331068 | Zbl 0244.54025

[12] Muhly, Paul S. Function algebras and flows. III, Math. Z., Tome 136 (1974), pp. 253-260 | Article | MR 493357 | Zbl 0267.46042

[13] Petersen, Karl Ergodic theory, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 2 (1983), pp. xii+329 | Article | MR 833286 | Zbl 0507.28010

[14] Rudin, Walter Fourier analysis on groups, Interscience Publishers (a division of John Wiley and Sons), New York-London, Interscience Tracts in Pure and Applied Mathematics, No. 12 (1962), pp. ix+285 | MR 152834 | Zbl 0107.09603

[15] Tanaka, Jun Blaschke cocycles and generators, Pacific J. Math., Tome 142 (1990) no. 2, pp. 357-378 http://projecteuclid.org/euclid.pjm/1102646351 | Article | MR 1042051 | Zbl 0731.43005

[16] Tanaka, Jun Single generator problem, Trans. Amer. Math. Soc., Tome 348 (1996) no. 10, pp. 4113-4129 | Article | MR 1340187 | Zbl 0862.46030

[17] Tanaka, Jun Dirichlet series induced by the Riemann zeta-function, Studia Math., Tome 187 (2008) no. 2, pp. 157-184 | Article | MR 2413314 | Zbl 1209.43001