Invariant subspaces with no generator and a problem of H. Helson

Tanaka, Jun-ichi

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1469-1491.

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.

Classification : 43A17, 46J10, 46J15, 28D10
Keywords: Compact groups with ordered duals, Invariant subspaces, Cocycles, Single generators
Mot clés : Groupes compactes avec dualité ordonnée, sous-espaces invariants, cocycles, générateurs singulier

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

[13] Petersen, Karl Ergodic theory, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1983, pp. xii+329 | DOI | MR | Zbl

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

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

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

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