## 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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### References:

 [1] A. Aluthge, On $$p$$-hyponormal operators for \(0
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