### Cyclic vectors and invariant subspaces for the backward shift operator

R. G. Douglas, H. S. Shapiro, A. L. Shields (1970)

Annales de l'institut Fourier

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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 $f(f\in {H}^{2})$, 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...