Quasi-similarity of contractions having a $2 \times 1$ characteristic function

Abstract

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}