Cyclic vectors and invariant subspaces for the backward shift operator
Annales de l'Institut Fourier, Volume 20 (1970) no. 1, pp. 37-76.

The operator U of multiplication by z on the Hardy space H 2 of square summable power series has been studied by many authors. In this paper we make a similar study of the adjoint operator U * (the “backward shift”). Let K f denote the cyclic subspace generated by f(fH 2 ), that is, the smallest closed subspace of H 2 that contains {U *n f} (n0). If K f =H 2 , then f is called a cyclic vector for U * . Theorem : f is a cyclic vector if and only if there is a function g, meromorphic and of bounded Nevanlinna characteristic in the region 1<|z|=, such that the radical limits of f and g coincide almost everywhere on the boundary |z|=1. Such a g is called a “pseudo analytic continuation” of f. Other results include the following. If f has a power series with Hadamard gaps, then f is a cyclic vector. If f is not cyclic, and if f can be continued analytically across some boundary point, then every function hK f can be continued across this same point. The set of all the non-cyclic vectors is a dense F σ set of the first category that is also a vector subspace of H 2 . In addition we study the relationship of cyclic vectors to inner functions, and to approximation by rational functions.

Nous désignons par U l’opérateur de multiplication par z dans l’espace de Hardy H 2 des séries des puissances à carré sommable. Dans ce travail, nous étudions l’opérateur adjoint U * (le “backward shift”). Soit K f le sous-espace cyclique engendré par f(fH 2 ), c’est-à-dire, le plus petit sous-espace fermé de H 2 qui contient {U *n f} (n0). Si K f =H 2 , f s’appelle un vecteur cyclique pour U * . Théorème : f est un vecteur cyclique si et seulement s’il existe une fonction g, méromorphe et de caractéristique (nevanlinnienne) bornée dans la région 1<|z|=1. Une telle fonction g s’appelle une “pseudo-continuation analytique” de f. Notons aussi les résultats suivants. Si fH 2 a une série des puissances avec des lacunes de Hadamard, alors f est un vecteur cyclique. Si f n’est pas un vecteur cyclique et si f admet une continuation analytique sur un point de la frontière, alors toute fonction hK f admet une continuation sur ce point. L’ensemble de tous les vecteurs non-cycliques est un ensemble dense du type F σ de la première catégorie qui est un sous-espace vectoriel de H 2 . Enfin, nous étudions la relation entre les vecteurs cycliques et les fonctions “intérieures” de Beurling, et l’approximation par des fonctions rationnelles.

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     title = {Cyclic vectors and invariant subspaces for the backward shift operator},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
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Douglas, R. G.; Shapiro, H. S.; Shields, A. L. Cyclic vectors and invariant subspaces for the backward shift operator. Annales de l'Institut Fourier, Volume 20 (1970) no. 1, pp. 37-76. doi : 10.5802/aif.338. http://www.numdam.org/articles/10.5802/aif.338/

[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337-404. | MR | Zbl

[2] S. Banach, Théorie des opérations linéaires, Warszawa, 1932. | JFM | Zbl

[3] Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949) 239-255. | Zbl

[4] L. Bieberbach, Lehrbuch der Funktionentheorie, Bd. II, Zweite Aufl., Leipzig, (1931). | JFM | Zbl

[5] L. Bieberbach, Analytische Fortsetzung, Ergeb. der Math., Neue Folge, Heft 3, Springer-Verlag, 1955. | Zbl

[6] R.P. Boas, Entire Functions, Academic Press, New York, 1954. | MR | Zbl

[7] T. Carleman, L'Intégrale de Fourier et questions qui s'y rattachent, Uppsala, 1944. | Zbl

[8] R.G. Douglas and W. Rudin, Approximation by inner functions, Pac. J. Wath. 31 (1969) 313-320. | MR | Zbl

[9] R.G. Douglas, H.S. Shapiro and A.L. Shields, On cyclic vectors of the backward shift, Bull. Amer. Math. Soc. 73 (1967) 156-159. | MR | Zbl

[10] P.R. Halmos, A Hilbert Space Problem Book, van Nostrand, Princeton, 1967. | MR | Zbl

[11] M.T. Haplanov, On the completeness of some systems of analytic functions, Uchen. Zapiski Rostov. Gos. Ped. Inst. N° 3 (1955) 53-58 (Russian). | Zbl

[12] Kenneth Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N. J., 1962. | Zbl

[13] Yu. A. Kaz'Min, On sequences of remainders of Taylor series, Vyestnik Moskov, Univ. Ser. 1, 5 (1963) 35-46 (Russian). | Zbl

[14] Herbert Meschkowski, Hilbertsche Räume mit Kernfunktion, Springer, Berlin, 1962. | Zbl

[15] Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153-162. | MR | Zbl

[16] E. Nordgren, Composition operators, Can. J. Math. 20 (1968) 442-449. | MR | Zbl

[17] Bertil Nyman, On the one-dimensional translation group and semi-group in certain function spaces, Dissertation, Uppsala, 1950. | Zbl

[18] I.I. Privalov, Randeigenschaften Analytischer Funktionen, Zweite Aufl., Deutscher Verlag der Wiss., Berlin, 1956. | Zbl

[19] J. Ryff, Subordinate Hp functions, Duke J. Math. 33 (1966) 347-354. | MR | Zbl

[20] H.S. Shapiro, Smoothness of the boundary function of a holomorphic function of bounded type, Ark. f. Mat. 7 (1968) 443-447. | MR | Zbl

[21] H.S. Shapiro, Overconvergence of sequences of rational functions with sparse poles, Ark. f. Mat. 7 (1967) 343-349. | MR | Zbl

[22] H.S. Shapiro, Generalized analytic continuation, Symposia on Theor. Phys. and Math. vol. 8, Plenum Press, New York (1968) 151-163. | MR | Zbl

[23] H.S. Shapiro, Functions nowhere continuable in a generalized sense, Publications of the Ramanujan Institute, vol. I, Madras (in press). | Zbl

[24] H.S. Shapiro, Weighted polynomial approximation and boundary behavior of analytic functions, in “Contemporary Problems of the Theory of Analytic Functions”, Nauka, Moscow (1966) 326-335. | MR | Zbl

[25] H.S. Shapiro, Weakly invertible elements in certain function spaces and generators in l1, Michigan Math. J. 11 (1964) 161-165. | MR | Zbl

[26] Sz.-Nagy and C. Foias, Analyse Harmonique des Opérateurs de l'Espace de Hilbert, Akademiai Kiado, Budapest, 1967. | MR | Zbl

[27] G. Ts. Tumarkin, Conditions for the convergence of the boundary values of analytic functions and approximation on rectifiable curves, in “Contemporary Problems of the Theory of Analytic Functions”, Nauka, Moscow (1966) 283-295 (Russian). | Zbl

[28] G. Ts. Tumarkin, Description of a class of functions approximable by rational functions with fixed poles, Izv. Akad. Nauk Armyanskoi SSR I, 1966, No. 2 pp. 89-105 (Russian). | Zbl

[29] G. Ts. Tumarkin, Convergent sequences of Blaschke products, Sibirsk Mat. Z. V (1964), 201-233 (Russian). | Zbl

[30] G. Ts. Tumarkin, Conditions for the uniform convergence and convergence of the boundary values of analytic and meromorphic functions of uniformly bounded characteristic, Mat. Z, V (1964), 387-417 (Russian). | Zbl

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