Let be any natural number. The -centered operator is introduced for adjointable operators on Hilbert -modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, -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 , if is Moore–Penrose invertible and is -centered, then its Moore–Penrose inverse is also -centered. A Hilbert space operator is constructed such that is -centered, whereas it fails to be -centered.
Banach J. Math. Anal.
13(3):
627-646
(July 2019).
DOI: 10.1215/17358787-2018-0027
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