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

  • Sergio Bermudo

    Universidad Pablo de Olavide, Sevilla, Spain
  • Carmen H. Mancera

    Universidad de Sevilla, Spain
  • Pedro J. Paúl

    Universidad de Sevilla, Spain
  • Vasily Vasyunin

    Steklov Mathematical Institute, St. Petersburg, Russian Federation


Let T1B(H1)T_1 \in \mathscr B( \mathscr H_1) be a completely non-unitary contraction having a non-zero characteristic function Θ1\Theta_1 which is a 2×12 \times 1 column vector of functions in HH^\infty. As it is well-known, such a function Θ1\Theta_1 can be written as Θ1=w1m1[a1b1]\Theta_1=w_1 m_1 \left[ {a_1} \atop {b_1} \right] where w1,m1,a1,b1Hw_1, m_1, a_1, b_1 \in H^\infty are such that w1w_1 is an outer function with w11|w_1|\leq 1, m1m_1 is an inner function, a12+b12=1|a_1|^2 + |b_1|^2 =1, and a1b1=1a_1 \wedge b_1 = 1 (here \wedge stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction T2B(H2)T_2 \in \mathscr B( \mathscr H_2) having also a 2×12 \times 1 characteristic function Θ2=w2m2[a2b2]\Theta_2=w_2 m_2 \left[ {a_2} \atop {b_2} \right] . We prove that T1T_1 is quasi-similar to T2T_2 if, and only if, the following conditions hold: \begin{enumerate} \item m1=m2m_1=m_2, \item {z\T:\absw1(z)<1}={z\T:w2(z)<1}\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 a1a_1 and b1b_1 in the Smirnov class N+\mathscr N^+ equals the corresponding ideal generated by a2a_2 and b2b_2. \end{enumerate}

Cite this article

Sergio Bermudo, Carmen H. Mancera, Pedro J. Paúl, Vasily Vasyunin, Quasi-similarity of contractions having a 2 × 1 characteristic function. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 677–704

DOI 10.4171/RMI/509