Geodesic
On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 38, Tome 376 (2010), pp. 5-24

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Toeplitz operators with unimodular symbols on the Hardy space $H^2$ on the unit circle are considered. It is shown that the left invertibility of a Toeplitz operator with symbol $e^{it}\mapsto\theta(e^{it})e^{it/2}$, $t\in(0,2\pi)$, where $\theta$ is an inner function, depends on $\theta$. Also, Toeplitz operators that are similar to isometries are studed. Bibl. – 28 titles. 
@article{ZNSL_2010_376_a0,
     author = {M. F. Gamal'},
     title = {On {Toeplitz} operators with unimodular symbols: left invertibility and similarity to isometries},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--24},
     publisher = {mathdoc},
     volume = {376},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_376_a0/}
}
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M. F. Gamal'. On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 38, Tome 376 (2010), pp. 5-24. http://geodesic.mathdoc.fr/item/ZNSL_2010_376_a0/