Geodesic
Cyclic sets for analytic Toeplitz operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 88-102

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Analytic Toeplitz operators $T_\varphi\colon f\mapsto\varphi f$, $\varphi\in H^\infty$, in the space $H^2$ are considered. In the case of smooth symbol and under some hypotheses of geometric nature on the curve $t\mapsto\varphi(e^{it})$, a full description of cyclic families is obtained. This description is based on the notions of an outer function and pseudocont: lnuation, which are employed to characterize cyclic vectors for the shift operator and its adjoint. 
@article{ZNSL_1987_157_a7,
     author = {B. M. Solomyak},
     title = {Cyclic sets for analytic {Toeplitz} operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {88--102},
     publisher = {mathdoc},
     volume = {157},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a7/}
}
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B. M. Solomyak. Cyclic sets for analytic Toeplitz operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 88-102. http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a7/