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

@article{AIF_1970__20_1_37_0,
     author = {Douglas, R. G. and Shapiro, H. S. and Shields, A. L.},
     title = {Cyclic vectors and invariant subspaces for the backward shift operator},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {20},
     number = {1},
     year = {1970},
     pages = {37-76},
     doi = {10.5802/aif.338},
     zbl = {0186.45302},
     mrnumber = {42 \#5088},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1970__20_1_37_0}
}
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/item/AIF_1970__20_1_37_0/

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