Quasi-similarity of contractions having a \(2 \times 1\) (Q2468418)

The paper under review concerns the quasi-similarity of contractions. Let \(T_1\in \mathcal{B}(\mathcal {H}_1)\) be a completely non-unitary contraction having a nonzero characteristic function \(\Theta_1\) which is a \(2\times 1\) column vector of functions in \(H^\infty\). It is well-known that such a function \(\Theta_1\) can be written as \(\Theta_1=w_1m_1 \left[\begin{smallmatrix} a_1\\ b_1\end{smallmatrix}\right]\), 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). Suppose that \(T_2\in \mathcal{B}(\mathcal{H}_2)\) is another completely non-unitary contraction having a \(2\times 1\) characteristic function \(\Theta_2=w_2m_2 \left[\begin{smallmatrix} a_2\\ b_2\end{smallmatrix}\right]\). In this paper, the authors prove that \(T_1\) is quasi-similar to \(T_2\) if and only if \(m_1=m_2\), \(\{z\in\mathbb{T}: | w_1(z)| \leq 1\}= \{z\in\mathbb{T}: | w_2(z)| \leq 1\}\) a.e., and the ideal generated by \(a_1\) and \(b_1\) in the Smirnov class \(\mathcal{N}^+\) equals the corresponding ideal generated by \(a_2\) and \(b_2\).