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 be a completely non-unitary contraction having a non-zero characteristic function which is a column vector of functions in . As it is well-known, such a function can be written as where are such that is an outer function with , is an inner function, , and (here stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction having also a characteristic function . We prove that is quasi-similar to if, and only if, the following conditions hold:

  1. ,
  2. a.e., and
  3. the ideal generated by and in the Smirnov class equals the corresponding ideal generated by and .

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