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 $f\left(f\in {H}^{2}\right)$, that is, the smallest closed subspace of ${H}^{2}$ that contains $\left\{{U}^{*n}f\right\}$ $\left(n\ge 0\right)$. 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 $1<|z|=\infty$, 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 $h\in {K}_{f}$ can be continued across this same point. The set of all the non-cyclic vectors is a dense ${F}_{\sigma }$ 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 $f\left(f\in {H}^{2}\right)$, c’est-à-dire, le plus petit sous-espace fermé de ${H}^{2}$ qui contient $\left\{{U}^{*n}f\right\}$ $\left(n\ge 0\right)$. 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 $f\in {H}^{2}$ 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 $h\in {K}_{f}$ admet une continuation sur ce point. L’ensemble de tous les vecteurs non-cycliques est un ensemble dense du type ${F}_{\sigma }$ 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},
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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/

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