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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 \(T\).

47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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