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Normality and fixed points associated to commutative row contractions. (English) Zbl 1296.47020
Let $$H$$ be a complex separable Hilbert space. A sequence $$T=\{T_n\}_{n\geq1}$$ in $$B(H)$$ is called a row contraction if $$\sum_{n\geq1}T_nT_n^*\leq I_H.$$ A row contraction induces a completely positive map $$\Phi_T$$ on $$B(H)$$ defined by $\Phi_T(X)=\sum_{n\geq1}T_nXT_n^*, \quad X\in B(H).$ In the first part, the authors give several conditions under which a unital commuting row contraction has normal components. In the second part, they use these results in order to describe the fixed point space of $$\Phi_T$$ in terms related to the commutant of $$T$$.

##### MSC:
 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 47C15 Linear operators in $$C^*$$- or von Neumann algebras
##### Keywords:
completely positive map; normality; row contraction; fixed point
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