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The polar decomposition for adjointable operators on Hilbert \(C^*\)-modules and \(n\)-centered operators. (English) Zbl 1431.46043

Summary: Let \(n\) be any natural number. The \(n\)-centered operator is introduced for adjointable operators on Hilbert \(C^*\)-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, \(n\)-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that, for an adjointable operator \(T\), if \(T\) is Moore-Penrose invertible and is \(n\)-centered, then its Moore-Penrose inverse is also \(n\)-centered. A Hilbert space operator \(T\) is constructed such that \(T\) is \(n\)-centered, whereas it fails to be \((n+1)\)-centered.

MSC:

46L08 \(C^*\)-modules
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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