Partially isometric truncated and dual truncated Toeplitz operators

Let $\theta$ be a non-constant inner function and let $\phi =\bar{u}v$, where $u$ and $v$ are inner functions such that $v$ divides $\theta$. In this paper we characterize the partially isometric truncated Toeplitz operators $A_{\phi }$ and dual truncated Toeplitz operators $D_{\phi }$ with symbols of the form $\phi =\bar{u}v$. Along with that, we obtain a few more characterization results, including the space of extremal vectors for non-zero partially isometric truncated and dual truncated Toeplitz operators.