The operator of multiplication by on the Hardy space of square summable power series has been studied by many authors. In this paper we make a similar study of the adjoint operator (the “backward shift”). Let denote the cyclic subspace generated by , that is, the smallest closed subspace of that contains . If , then is called a cyclic vector for . Theorem : is a cyclic vector if and only if there is a function , meromorphic and of bounded Nevanlinna characteristic in the region , such that the radical limits of and coincide almost everywhere on the boundary Such a is called a “pseudo analytic continuation” of . Other results include the following. If has a power series with Hadamard gaps, then is a cyclic vector. If is not cyclic, and if can be continued analytically across some boundary point, then every function can be continued across this same point. The set of all the non-cyclic vectors is a dense set of the first category that is also a vector subspace of . In addition we study the relationship of cyclic vectors to inner functions, and to approximation by rational functions.
Nous désignons par l’opérateur de multiplication par dans l’espace de Hardy des séries des puissances à carré sommable. Dans ce travail, nous étudions l’opérateur adjoint (le “backward shift”). Soit le sous-espace cyclique engendré par , c’est-à-dire, le plus petit sous-espace fermé de qui contient . Si , s’appelle un vecteur cyclique pour . Théorème : est un vecteur cyclique si et seulement s’il existe une fonction , méromorphe et de caractéristique (nevanlinnienne) bornée dans la région . Une telle fonction s’appelle une “pseudo-continuation analytique” de . Notons aussi les résultats suivants. Si a une série des puissances avec des lacunes de Hadamard, alors est un vecteur cyclique. Si n’est pas un vecteur cyclique et si admet une continuation analytique sur un point de la frontière, alors toute fonction admet une continuation sur ce point. L’ensemble de tous les vecteurs non-cycliques est un ensemble dense du type de la première catégorie qui est un sous-espace vectoriel de . 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}, pages = {37--76}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {20}, number = {1}, year = {1970}, doi = {10.5802/aif.338}, mrnumber = {42 #5088}, zbl = {0186.45302}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.338/} }
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 SP - 37 EP - 76 VL - 20 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.338/ DO - 10.5802/aif.338 LA - en ID - AIF_1970__20_1_37_0 ER -
%0 Journal Article %A Douglas, R. G. %A Shapiro, H. S. %A Shields, A. L. %T Cyclic vectors and invariant subspaces for the backward shift operator %J Annales de l'Institut Fourier %D 1970 %P 37-76 %V 20 %N 1 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.338/ %R 10.5802/aif.338 %G en %F 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://archive.numdam.org/articles/10.5802/aif.338/
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