Geodesic
She multiplicity of some contractions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 23-29

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for a unitary operator $U=U_a\oplus U_s$ for a $C_0$-contraction $T$, for the unilateral shift $S$ and for the backward shift $S^*$ the multiplicity of its direct sum is calculated: $$ \mu_{(U\oplus S^n\oplus S^{*m}\oplus T)}=\max\{\mu_{U_s}, n+\max\{\mu_{U_a}, \mu_T, 1-\delta_{m0}\}\}, $$ where $\delta_{m0}=1$ if $m=0$ and $\delta_{m0}=0$ при $m>0$. 
@article{ZNSL_1987_157_a1,
     author = {V. I. Vasyunin and M. T. Karaev},
     title = {She multiplicity of some contractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {23--29},
     publisher = {mathdoc},
     volume = {157},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a1/}
}
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V. I. Vasyunin; M. T. Karaev. She multiplicity of some contractions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 23-29. http://geodesic.mathdoc.fr/item/ZNSL_1987_157_a1/