On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries

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.