Geodesic
On quasi-similarity of model contractions with non-equal defects
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 24-37 
V. I. Vasyunin; N. G. Makarov. On quasi-similarity of model contractions with non-equal defects. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 24-37. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a1/
@article{ZNSL_1986_149_a1,
     author = {V. I. Vasyunin and N. G. Makarov},
     title = {On quasi-similarity of model contractions with non-equal defects},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {24--37},
     year = {1986},
     volume = {149},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a1/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $T_\theta$ and $T_\Phi$ be $C_{10}$ contractions with characteristic functions $\theta\in H^\infty(\mathbb C^n\to\mathbb C^{n+1})$ and $\Phi\in H^\infty(\mathbb C^m\to\mathbb C^{m+1})$. The main result: $T_\theta$ and $T_\Phi$ are quasi-similar iff $$ \{\det(f,\theta)^i:f\in H_n^2\}=\{\det(g,\Phi)^i:g\in H_m^2\}. $$ The paper contains an analysis of this condition. Some examples are given.