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# Revista Matemática Iberoamericana

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Volume 23, Issue 2, 2007, pp. 677–704
DOI: 10.4171/RMI/509

Published online: 2007-08-31

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

Sergio Bermudo, Carmen H. Mancera, Pedro J. Paúl and Vasily Vasyunin

(1) Universidad Pablo de Olavide, Sevilla, Spain
(2) Universidad de Sevilla, Spain
(3) Universidad de Sevilla, Spain
(4) Steklov Mathematical Institute, St. Petersburg, Russian Federation

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}

Keywords: Quasi-similarity, contractions, characteristic functions, function models

Bermudo Sergio, Mancera Carmen, Paúl Pedro, Vasyunin Vasily: Quasi-similarity of contractions having a 2 × 1 characteristic function. Rev. Mat. Iberoam. 23 (2007), 677-704. doi: 10.4171/RMI/509