Geodesic
Classification of $H^2$-functions according to the degree of their cyclicity
Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 225-242.

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The vector-valued functions $f$ in the Hardy space $H^2(E)$ are classified according to their approximation capabilities with respect to the backward shift operator $S^*$, $S^*f\overset{\operatorname{def}}=\frac{f-f(0)}z$, i.e., according to the “size” of the closed linear span $\operatorname{span}(S^{*k}f:k\geqslant0)$. Bibliography: 6 titles. 
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V. I. Vasyunin; N. K. Nikol'skii. Classification of $H^2$-functions according to the degree of their cyclicity. Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 225-242. http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a1/

[1] Gofman K., Banakhovy prostranstva analiticheskikh funktsii, IL, M., 1963

[2] Sekefalvi-Nad B., Foyash Ch., Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[3] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[4] Vasyunin V. I., Nikolskii N. K., “Upravlyayuschie podprostranstva minimalnoi razmernosti. Elementarnoe vvedenie. Discotheca”, Zap. nauchn. semin. LOMI, 113, 1981, 41–75 | MR

[5] Douglas R. G., Shapiro H. S., Shields A. L., “Cyclic vectors and invariant subspaces for the backward shift operator”, Ann. Inst. Fourier, 20:1 (1970), 37–76 | MR | Zbl

[6] Tolokonnikov V. A., “Otsenka v teoreme Karlesona o korone. Idealy algebry $H^\infty$, zadacha Sekefalvi-Nadya”, Zap. nauchn. semin. LOMI, 113, 1981, 178–198 | MR | Zbl