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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 characteristic...