On Toeplitz operators similar to unilateral shifts

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.