Geodesic
On Toeplitz operators similar to unilateral shifts
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 35, Tome 345 (2007), pp. 85-104

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Let $T$ be a Toeplitz operator on the Hardy space $H^2$ on the unit circle, and let the symbol of $T$ be of the form $\varphi/\psi$, where $\varphi$ is an inner function, $\psi$ is a finite Blaschke product, and $\deg\psi\le\deg\varphi$. D. N. Clark proved that such $T$ is similar to an isometry. In this paper we find necessary and sufficient conditions to such $T$ be similar to a unilateral shift. 
@article{ZNSL_2007_345_a4,
     author = {M. F. Gamal'},
     title = {On {Toeplitz} operators similar to unilateral shifts},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {85--104},
     publisher = {mathdoc},
     volume = {345},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_345_a4/}
}
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M. F. Gamal'. On Toeplitz operators similar to unilateral shifts. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 35, Tome 345 (2007), pp. 85-104. http://geodesic.mathdoc.fr/item/ZNSL_2007_345_a4/