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August, 2007 Quasi-similarity of contractions having a $2 \times 1$ characteristic function
Sergio Bermudo, Carmen H. Mancera, Pedro J. Paùl, Vasily Vasyunin
Rev. Mat. Iberoamericana 23(2): 677-704 (August, 2007).


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}


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Sergio Bermudo. Carmen H. Mancera. Pedro J. Paùl. Vasily Vasyunin. "Quasi-similarity of contractions having a $2 \times 1$ characteristic function." Rev. Mat. Iberoamericana 23 (2) 677 - 704, August, 2007.


Published: August, 2007
First available in Project Euclid: 26 September 2007

zbMATH: 1145.47010
MathSciNet: MR2371441

Primary: 47A05 , 47A45

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

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 2 • August, 2007
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