Cyclic vectors and invariant subspaces for the backward shift operator

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

Annales de l'institut Fourier (1970)

  • Volume: 20, Issue: 1, page 37-76
  • ISSN: 0373-0956

Abstract

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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 ∈ H 2 ) , that is, the smallest closed subspace of H 2 that contains { U * n f } ( n ≥ 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 < | 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 h ∈ K 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.

How to cite

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Douglas, R. G., Shapiro, H. S., and Shields, A. L.. "Cyclic vectors and invariant subspaces for the backward shift operator." Annales de l'institut Fourier 20.1 (1970): 37-76. <http://eudml.org/doc/74007>.

@article{Douglas1970,
abstract = {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 $\lbrace U^\{*n\}f\rbrace $$(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&lt; \vert z\vert =\infty $, such that the radical limits of $f$ and $g$ coincide almost everywhere on the boundary $\vert z\vert =1.$ 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.},
author = {Douglas, R. G., Shapiro, H. S., Shields, A. L.},
journal = {Annales de l'institut Fourier},
keywords = {functional analysis},
language = {eng},
number = {1},
pages = {37-76},
publisher = {Association des Annales de l'Institut Fourier},
title = {Cyclic vectors and invariant subspaces for the backward shift operator},
url = {http://eudml.org/doc/74007},
volume = {20},
year = {1970},
}

TY - JOUR
AU - Douglas, R. G.
AU - Shapiro, H. S.
AU - Shields, A. L.
TI - Cyclic vectors and invariant subspaces for the backward shift operator
JO - Annales de l'institut Fourier
PY - 1970
PB - Association des Annales de l'Institut Fourier
VL - 20
IS - 1
SP - 37
EP - 76
AB - 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 $\lbrace U^{*n}f\rbrace $$(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&lt; \vert z\vert =\infty $, such that the radical limits of $f$ and $g$ coincide almost everywhere on the boundary $\vert z\vert =1.$ 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.
LA - eng
KW - functional analysis
UR - http://eudml.org/doc/74007
ER -

References

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  4. Carl C. Cowen, Eva A. Gallardo-Gutiérrez, An introduction to Rota’s universal operators: properties, old and new examples and future issues
  5. Xavier Dussau, Les shifts à poids dissymétriques sont hyper-réflexifs
  6. Guy Ruckebusch, Théorie géométrique de la Représentation Markovienne
  7. Bernard Virot, Modèles d'opérateurs linéaires et translations unilatérales simples
  8. J. Esterle, Closed ideals in certain Beurling algebras, and synthesis of hyperdistributions

NotesEmbed ?

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Bonjour, Théorème : f est un vecteur cyclique si et seulement s’il existe une fonction g, méromorphe et de caractéristique (nevanlinnienne) ... je pense qu'il ya une petite faute ,f est un vecteur non-cyclique si et seulement si ... est_ce vrai?

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