Revista Matemática Iberoamericana

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

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

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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}

Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 677-704.

Dates
First available in Project Euclid: 26 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1190831225

Mathematical Reviews number (MathSciNet)
MR2371441

Zentralblatt MATH identifier
1145.47010

Subjects
Primary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A45: Canonical models for contractions and nonselfadjoint operators

Keywords
quasi-similarity contractions characteristic functions function models

Citation

Bermudo, Sergio; Mancera, Carmen H.; Paùl, Pedro J.; Vasyunin, Vasily. Quasi-similarity of contractions having a $2 \times 1$ characteristic function. Rev. Mat. Iberoamericana 23 (2007), no. 2, 677--704. https://projecteuclid.org/euclid.rmi/1190831225


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References

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