Quasi-similarity of contractions having a 2 × 1 characteristic function

Abstract

Let $T_1\in \mathcal{B}({\mathcal{H}}_1)$ 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 ${\Theta }_1=w_1{m}_1\left[\genfrac{}{}{0ex}{}{a_1}{b_1}\right]$ where $w_1,m_1,a_1,b_1\in H^{\infty }$ are such that $w_1$ is an outer function with $|w_1|\le 1$, $m_1$ 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 $T_2\in \mathcal{B}({\mathcal{H}}_2)$ having also a $2\times 1$ characteristic function ${\Theta }_2=w_2{m}_2\left[\genfrac{}{}{0ex}{}{a_2}{b_2}\right]$ . We prove that $T_1$ is quasi-similar to $T_2$ if, and only if, the following conditions hold:

1. $m_1=m_2$,
2. $\left\{z\in \mathbb{T}:|w_1(z)|<1\right\}=\left\{z\in \mathbb{T}:\left|w_2(z)\right|<1\right\}$ a.e., and
3. the ideal generated by $a_1$ and $b_1$ in the Smirnov class ${\mathcal{N}}^{+}$ equals the corresponding ideal generated by $a_2$ and $b_2$.