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.
"The polar decomposition for adjointable operators on Hilbert -modules and -centered operators." Banach J. Math. Anal. 13 (3) 627 - 646, July 2019. https://doi.org/10.1215/17358787-2018-0027