Let $T_1 \in \mathscr B( \mathscr H_1)$ be a completely non-unitary
contraction having a non-zero characteristic function $\Theta_1$
which is a $2 \times 1$ column vector of functions in $H^\infty$.
As it is well-known, such a function $\Theta_1$ can be written as
$ \Theta_1=w_1 m_1 \left[ {a_1} \atop {b_1} \right] $ where $w_1,
m_1, a_1, b_1 \in H^\infty$ are such that $w_1$ is an outer function
with $|w_1|\leq 1$, $m_1$ is an inner function, $|a_1|^2 + |b_1|^2 =1$,
and $a_1 \wedge b_1 = 1$ (here $\wedge$ stands for the greatest
common inner divisor). Now consider a second completely non-unitary
contraction $T_2 \in \mathscr B( \mathscr H_2)$ having also a $2 \times 1$
characteristic function $ \Theta_2=w_2 m_2 \left[ {a_2} \atop {b_2} \right] $.
We prove that $T_1$ is quasi-similar to $T_2$ if, and only if, the
following conditions hold:
\begin{enumerate}
\item $m_1=m_2$,
\item $\left\{ z \in \T : \abs{w_1(z)} < 1 \right\} = \left\{ z
\in \T : \left\vert w_2(z)\right\vert < 1 \right\}$ a.e., and
\item the ideal generated by $a_1$ and $b_1$ in the Smirnov class
$\mathscr N^+$ equals the corresponding ideal generated by $a_2$ and $b_2$.
\end{enumerate}