# 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

## Abstract

Let $T_{1}∈B(H_{1})$ be a completely non-unitary contraction having a non-zero characteristic function $Θ_{1}$ which is a $2×1$ column vector of functions in $H_{∞}$. As it is well-known, such a function $Θ_{1}$ can be written as $Θ_{1}=w_{1}m_{1}[b_{1}a_{1} ]$ where $w_{1},m_{1},a_{1},b_{1}∈H_{∞}$ are such that $w_{1}$ is an outer function with $∣w_{1}∣≤1$, $m_{1}$ is an inner function, $∣a_{1}∣_{2}+∣b_{1}∣_{2}=1$, and $a_{1}∧b_{1}=1$ (here $∧$ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction $T_{2}∈B(H_{2})$ having also a $2×1$ characteristic function $Θ_{2}=w_{2}m_{2}[b_{2}a_{2} ]$ . We prove that $T_{1}$ is quasi-similar to $T_{2}$ if, and only if, the following conditions hold:

- $m_{1}=m_{2}$,
- ${z∈T:∣w_{1}(z)∣<1}={z∈T:∣w_{2}(z)∣<1}$ a.e., and
- the ideal generated by $a_{1}$ and $b_{1}$ in the Smirnov class $N_{+}$ equals the corresponding ideal generated by $a_{2}$ and $b_{2}$.

## 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