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}