Cyclic vectors and invariant subspaces for the backward shift operator

Douglas, R. G.; Shapiro, H. S.; Shields, A. L.

Annales de l'Institut Fourier, Tome 20 (1970) no. 1, pp. 37-76.

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 $f(f\in H^2)$, c’est-à-dire, le plus petit sous-espace fermé de $H^2$ qui contient $\left\{U^{*n}f\right\}$ $(n\ge 0)$. 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 $1<\left|z\right|=1$. 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.

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 $f(f\in H^2)$, that is, the smallest closed subspace of $H^2$ that contains $\left\{U^{*n}f\right\}$ $(n\ge 0)$. 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 $1<\left|z\right|=\infty$, such that the radical limits of $f$ and $g$ coincide almost everywhere on the boundary $\left|z\right|=1.$ 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.

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